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I NEED HELP WITH Q 10.1 ATTACHED . I AM ATTACHING BOOK AS WELL , ITS ON PAGE 409 OF THE BOOK . I AM ATTACHING BOOK SO YOU CAN CONSULT IT WHILE ANSWERING IT. please message back if you can do it.

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Springer Texts in Statistics Advisors: George Casella Stephen Fienberg Ingram Olkin E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition With 6 Illustrations E.L. Lehmann Professor of Statistics Emeritus Department of Statistics University of California, Berkeley Berkeley, CA 94720 USA Joseph P. Romano Department of Statistics Stanford University Sequoia Hall Stanford, CA 94305 USA romano@stanford.edu Editorial Board George Casella Stephen Fienberg Ingram Olkin Department of Statistics University of Florida Gainesville, FL 32611-8545 USA Department of Statistics Carnegie Mellon University Pittsburgh, PA 15213-3890 USA Department of Statistics Stanford University Stanford, CA 94305 USA Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress. ISBN 0-387-98864-5 Printed on acid-free paper. © 2005, 1986, 1959 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 10728642 Typesetting: Pages created by the author. springeronline.com (MVY) Dedicated to the Memory of Lucien Le Cam (1924-2000) and John W. Tukey (1915-2000) Preface to the Third Edition The Third Edition of Testing Statistical Hypotheses brings it into consonance with the Second Edition of its companion volume on point estimation (Lehmann and Casella, 1998) to which we shall refer as TPE2. We won’t here comment on the long history of the book which is recounted in Lehmann (1997) but shall use this Preface to indicate the principal changes from the 2nd Edition. The present volume is divided into two parts. Part I (Chapters 1–10) treats small-sample theory, while Part II (Chapters 11–15) treats large-sample theory. The preface to the 2nd Edition stated that “the most important omission is an adequate treatment of optimality paralleling that given for estimation in TPE.” We shall here remedy this failure by treating the difficult topic of asymptotic optimality (in Chapter 13) together with the large-sample tools needed for this purpose (in Chapters 11 and 12). Having developed these tools, we use them in Chapter 14 to give a much fuller treatment of tests of goodness of fit than was possible in the 2nd Edition, and in Chapter 15 to provide an introduction to the bootstrap and related techniques. Various large-sample considerations that in the Second Edition were discussed in earlier chapters now have been moved to Chapter 11. Another major addition is a more comprehensive treatment of multiple testing including some recent optimality results. This topic is now presented in Chapter 9. In order to make room for these extensive additions, we had to eliminate some material found in the Second Edition, primarily the coverage of the multivariate linear hypothesis. Except for some of the basic results from Part I, a detailed knowledge of smallsample theory is not required for Part II. In particular, the necessary background should include: Chapter 3, Sections 3.1–3.5, 3.8–3.9; Chapter 4: Sections 4.1–4.4; Chapter 5, Sections 5.1–5.3; Chapter 6, Sections 6.1–6.2; Chapter 7, Sections 7.1–7.2; Chapter 8, Sections 8.1–8.2, 8.4–8.5. viii Preface Of the two principal additions to the Third Edition, multiple comparisons and asymptotic optimality, each has a godfather. The development of multiple comparisons owes much to the 1953 volume on the subject by John Tukey, a mimeographed version which was widely distributed at the time. It was officially published only in 1994 as Volume VIII in The Collected Works of John W. Tukey. Many of the basic ideas on asymptotic optimality are due to the work of Le Cam between 1955 and 1980. It culminated in his 1986 book, Asymptotic Methods in Statistical Decision Theory. The work of these two authors, both of whom died in 2000, spans the achievements of statistics in the second half of the 20th century, from model-free data analysis to the most abstract and mathematical asymptotic theory. In acknowledgment of their great accomplishments, this volume is dedicated to their memory. Special thanks to Noureddine El Karoui, Matt Finkelman, Brit Katzen, Mee Young Park, Elizabeth Purdom, Armin Schwartzman, Azeem Shaikh and the many students at Stanford University who proofread several versions of the new chapters and worked through many of the over 300 new problems. The support and suggestions of our colleagues is greatly appreciated, especially Persi Diaconis, Brad Efron, Susan Holmes, Balasubramanian Narasimhan, Dimitris Politis, Julie Shaffer, Guenther Walther and Michael Wolf. Finally, heartfelt thanks go to friends and family who provided continual encouragement, especially Ann Marie and Mark Hodges, David Fogle, Scott Madover, David Olachea, Janis and Jon Squire, Lucy, and Ron Susek. E. L. Lehmann Joseph P. Romano January, 2005 Contents Preface I vii Small-Sample Theory 1 1 The 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 General Decision Problem Statistical Inference and Statistical Decisions Specification of a Decision Problem . . . . . Randomization; Choice of Experiment . . . Optimum Procedures . . . . . . . . . . . . . Invariance and Unbiasedness . . . . . . . . . Bayes and Minimax Procedures . . . . . . . Maximum Likelihood . . . . . . . . . . . . . Complete Classes . . . . . . . . . . . . . . . Sufficient Statistics . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 4 8 9 11 14 16 17 18 21 27 2 The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Probability Background Probability and Measure . . . . . . . . . Integration . . . . . . . . . . . . . . . . . Statistics and Subfields . . . . . . . . . . Conditional Expectation and Probability Conditional Probability Distributions . . Characterization of Sufficiency . . . . . . Exponential Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 28 31 34 36 41 44 46 . . . . . . . . . . . . . . x Contents 2.8 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Uniformly Most Powerful Tests 3.1 Stating The Problem . . . . . . . . . . . . . . 3.2 The Neyman–Pearson Fundamental Lemma . 3.3 p-values . . . . . . . . . . . . . . . . . . . . . 3.4 Distributions with Monotone Likelihood Ratio 3.5 Confidence Bounds . . . . . . . . . . . . . . . 3.6 A Generalization of the Fundamental Lemma 3.7 Two-Sided Hypotheses . . . . . . . . . . . . . 3.8 Least Favorable Distributions . . . . . . . . . 3.9 Applications to Normal Distributions . . . . . 3.9.1 Univariate Normal Models . . . . . . . 3.9.2 Multivariate Normal Models . . . . . . 3.10 Problems . . . . . . . . . . . . . . . . . . . . . 3.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Unbiasedness: Theory and First Applications 4.1 Unbiasedness For Hypothesis Testing . . . . . . . . . 4.2 One-Parameter Exponential Families . . . . . . . . . 4.3 Similarity and Completeness . . . . . . . . . . . . . . 4.4 UMP Unbiased Tests for Multiparameter Exponential 4.5 Comparing Two Poisson or Binomial Populations . . 4.6 Testing for Independence in a 2 × 2 Table . . . . . . 4.7 Alternative Models for 2 × 2 Tables . . . . . . . . . . 4.8 Some Three-Factor Contingency Tables . . . . . . . . 4.9 The Sign Test . . . . . . . . . . . . . . . . . . . . . . 4.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 55 . . . . . . . . . . . . . 56 56 59 63 65 72 77 81 83 86 86 89 92 107 . . . . . . . . . . . . . . . Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 110 111 115 119 124 127 130 132 135 139 149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Unbiasedness: Applications to Normal Distributions 150 5.1 Statistics Independent of a Sufficient Statistic . . . . . . . . . 150 5.2 Testing the Parameters of a Normal Distribution . . . . . . . 153 5.3 Comparing the Means and Variances of Two Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.4 Confidence Intervals and Families of Tests . . . . . . . . . . . 161 5.5 Unbiased Confidence Sets . . . . . . . . . . . . . . . . . . . . . 164 5.6 Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.7 Bayesian Confidence Sets . . . . . . . . . . . . . . . . . . . . . 171 5.8 Permutation Tests . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.9 Most Powerful Permutation Tests . . . . . . . . . . . . . . . . 177 5.10 Randomization As A Basis For Inference . . . . . . . . . . . . 181 5.11 Permutation Tests and Randomization . . . . . . . . . . . . . 184 5.12 Randomization Model and Confidence Intervals . . . . . . . . 187 5.13 Testing for Independence in a Bivariate Normal Distribution . 190 5.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Contents 6 Invariance 6.1 Symmetry and Invariance . . . . . . . . . . . . 6.2 Maximal Invariants . . . . . . . . . . . . . . . 6.3 Most Powerful Invariant Tests . . . . . . . . . 6.4 Sample Inspection by Variables . . . . . . . . 6.5 Almost Invariance . . . . . . . . . . . . . . . . 6.6 Unbiasedness and Invariance . . . . . . . . . . 6.7 Admissibility . . . . . . . . . . . . . . . . . . . 6.8 Rank Tests . . . . . . . . . . . . . . . . . . . . 6.9 The Two-Sample Problem . . . . . . . . . . . 6.10 The Hypothesis of Symmetry . . . . . . . . . 6.11 Equivariant Confidence Sets . . . . . . . . . . 6.12 Average Smallest Equivariant Confidence Sets 6.13 Confidence Bands for a Distribution Function 6.14 Problems . . . . . . . . . . . . . . . . . . . . . 6.15 Notes . . . . . . . . . . . . . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 212 214 218 223 225 229 232 239 242 246 248 251 255 257 276 7 Linear Hypotheses 7.1 A Canonical Form . . . . . . . . . . . . . . . . . 7.2 Linear Hypotheses and Least Squares . . . . . . 7.3 Tests of Homogeneity . . . . . . . . . . . . . . . 7.4 Two-Way Layout: One Observation per Cell . . 7.5 Two-Way Layout: m Observations Per Cell . . . 7.6 Regression . . . . . . . . . . . . . . . . . . . . . 7.7 Random-Effects Model: One-way Classification . 7.8 Nested Classifications . . . . . . . . . . . . . . . 7.9 Multivariate Extensions . . . . . . . . . . . . . . 7.10 Problems . . . . . . . . . . . . . . . . . . . . . . 7.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 277 281 285 287 290 293 297 300 304 306 317 8 The 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Minimax Principle Tests with Guaranteed Power . . . . . . . Examples . . . . . . . . . . . . . . . . . . . Comparing Two Approximate Hypotheses Maximin Tests and Invariance . . . . . . . The Hunt–Stein Theorem . . . . . . . . . . Most Stringent Tests . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 319 322 326 329 331 337 338 347 9 Multiple Testing and Simultaneous Inference 9.1 Introduction and the FWER . . . . . . . . . . 9.2 Maximin Procedures . . . . . . . . . . . . . . 9.3 The Hypothesis of Homogeneity . . . . . . . . 9.4 Scheffé’s S-Method: A Special Case . . . . . . 9.5 Scheffé’s S-Method for General Linear Models 9.6 Problems . . . . . . . . . . . . . . . . . . . . . 9.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 348 354 363 375 380 385 391 . . . . . . . . xii Contents 10 Conditional Inference 10.1 Mixtures of Experiments . 10.2 Ancillary Statistics . . . . 10.3 Optimal Conditional Tests 10.4 Relevant Subsets . . . . . 10.5 Problems . . . . . . . . . . 10.6 Notes . . . . . . . . . . . . II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large-Sample Theory 392 392 395 400 404 409 414 417 11 Basic Large Sample Theory 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic Convergence Concepts . . . . . . . . . . . . . . . . 11.2.1 Weak Convergence and Central Limit Theorems 11.2.2 Convergence in Probability and Applications . . . 11.2.3 Almost Sure Convergence . . . . . . . . . . . . . 11.3 Robustness of Some Classical Tests . . . . . . . . . . . . 11.3.1 Effect of Distribution . . . . . . . . . . . . . . . . 11.3.2 Effect of Dependence . . . . . . . . . . . . . . . . 11.3.3 Robustness in Linear Models . . . . . . . . . . . . 11.4 Nonparametric Mean . . . . . . . . . . . . . . . . . . . . 11.4.1 Edgeworth Expansions . . . . . . . . . . . . . . . 11.4.2 The t-test . . . . . . . . . . . . . . . . . . . . . . 11.4.3 A Result of Bahadur and Savage . . . . . . . . . 11.4.4 Alternative Tests . . . . . . . . . . . . . . . . . . 11.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 419 424 424 431 440 444 444 448 451 459 459 462 466 468 469 480 12 Quadratic Mean Differentiable Families 12.1 Introduction . . . . . . . . . . . . . . . . . 12.2 Quadratic Mean Differentiability (q.m.d.) . 12.3 Contiguity . . . . . . . . . . . . . . . . . . 12.4 Likelihood Methods in Parametric Models 12.4.1 Efficient Likelihood Estimation . . 12.4.2 Wald Tests and Confidence Regions 12.4.3 Rao Score Tests . . . . . . . . . . . 12.4.4 Likelihood Ratio Tests . . . . . . . 12.5 Problems . . . . . . . . . . . . . . . . . . . 12.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 482 482 492 503 504 508 511 513 517 525 13 Large Sample Optimality 13.1 Testing Sequences, Metrics, and Inequalities 13.2 Asymptotic Relative Efficiency . . . . . . . . 13.3 AUMP Tests in Univariate Models . . . . . 13.4 Asymptotically Normal Experiments . . . . 13.5 Applications to Parametric Models . . . . . 13.5.1 One-sided Hypotheses . . . . . . . . 13.5.2 Equivalence Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 527 534 540 549 553 553 559 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 567 567 570 574 582 14 Testing Goodness of Fit 14.1 Introduction . . . . . . . . . . . . . . . . . . . 14.2 The Kolmogorov-Smirnov Test . . . . . . . . . 14.2.1 Simple Null Hypothesis . . . . . . . . . 14.2.2 Extensions of the Kolmogorov-Smirnov 14.3 Pearson’s Chi-squared Statistic . . . . . . . . 14.3.1 Simple Null Hypothesis . . . . . . . . . 14.3.2 Chi-squared Test of Uniformity . . . . 14.3.3 Composite Null Hypothesis . . . . . . 14.4 Neyman’s Smooth Tests . . . . . . . . . . . . 14.4.1 Fixed k Asymptotics . . . . . . . . . . 14.4.2 Neyman’s Smooth Tests With Large k 14.5 Weighted Quadratic Test Statistics . . . . . . 14.6 Global Behavior of Power Functions . . . . . . 14.7 Problems . . . . . . . . . . . . . . . . . . . . . 14.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 583 584 584 589 590 590 594 597 599 601 603 607 616 622 629 15 General Large Sample Methods 15.1 Introduction . . . . . . . . . . . . . . . 15.2 Permutation and Randomization Tests 15.2.1 The Basic Construction . . . . . 15.2.2 Asymptotic Results . . . . . . . 15.3 Basic Large Sample Approximations . 15.3.1 Pivotal Method . . . . . . . . . 15.3.2 Asymptotic Pivotal Method . . 15.3.3 Asymptotic Approximation . . 15.4 Bootstrap Sampling Distributions . . . 15.4.1 Introduction and Consistency . 15.4.2 The Nonparametric Mean . . . 15.4.3 Further Examples . . . . . . . . 15.4.4 Stepdown Multiple Testing . . . 15.5 Higher Order Asymptotic Comparisons 15.6 Hypothesis Testing . . . . . . . . . . . 15.7 Subsampling . . . . . . . . . . . . . . . 15.7.1 The Basic Theorem in the I.I.D. 15.7.2 Comparison with the Bootstrap 15.7.3 Hypothesis Testing . . . . . . . 15.8 Problems . . . . . . . . . . . . . . . . . 15.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 631 632 632 636 643 644 646 647 648 648 653 655 658 661 668 673 674 677 680 682 690 A Auxiliary Results A.1 Equivalence Relations; Groups . . . . . . . . . . . . . . . . . . 692 692 13.6 13.7 13.8 13.5.3 Multi-sided Hypotheses . . . . . . . . . Applications to Nonparametric Models . . . . 13.6.1 Nonparametric Mean . . . . . . . . . . 13.6.2 Nonparametric Testing of Functionals . Problems . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Contents A.2 A.3 A.4 A.5 Convergence of Functions; Metric Spaces Banach and Hilbert Spaces . . . . . . . . Dominated Families of Distributions . . . The Weak Compactness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 696 698 700 References 702 Author Index 757 Subject Index 767 Part I Small-Sample Theory 1 The General Decision Problem 1.1 Statistical Inference and Statistical Decisions The raw material of a statistical investigation is a set of observations; these are the values taken on by random variables X whose distribution Pθ is at least partly unknown. Of the parameter θ, which labels the distribution, it is assumed known only that it lies in a certain set Ω, the parameter space. Statistical inference is concerned with methods of using this observational material to obtain information concerning the distribution of X or the parameter θ with which it is labeled. To arrive at a more precise formulation of the problem we shall consider the purpose of the inference. The need for statistical analysis stems from the fact that the distribution of X, and hence some aspect of the situation underlying the mathematical model, is not known. The consequence of such a lack of knowledge is uncertainty as to the best mode of behavior. To formalize this, suppose that a choice has to be made between a number of alternative actions. The observations, by providing information about the distribution from which they came, also provide guidance as to the best decision. The problem is to determine a rule which, for each set of values of the observations, specifies what decision should be taken. Mathematically such a rule is a function δ, which to each possible value x of the random variables assigns a decision d = δ(x), that is, a function whose domain is the set of values of X and whose range is the set of possible decisions. In order to see how δ should be chosen, one must compare the consequences of using different rules. To this end suppose that the consequence of taking decision d when the distribution of X is Pθ is a loss, which can be expressed as a nonnegative real number L(θ, d). Then the long-term average loss that would result from the use of δ in a number of repetitions of the experiment is the expectation 4 1. The General Decision Problem E[L(θ, δ(X))] evaluated under the assumption that Pθ is the true distribution of X. This expectation, which depends on the decision rule δ and the distribution Pθ , is called the risk function of δ and will be denoted by R(θ, δ). By basing the decision on the observations, the original problem of choosing a decision d with loss function L(θ, d) is thus replaced by that of choosing δ, where the loss is now R(θ, δ). The above discussion suggests that the aim of statistics is the selection of a decision function which minimizes the resulting risk. As will be seen later, this statement of aims is not sufficiently precise to be meaningful; its proper interpretation is in fact one of the basic problems of the theory. 1.2 Specification of a Decision Problem The methods required for the solution of a specific statistical problem depend quite strongly on the three elements that define it: the class P = {Pθ , θ ∈ Ω} to which the distribution of X is assumed to belong; the structure of the space D of possible decisions d; and the form of the loss function L. In order to obtain concrete results it is therefore necessary to make specific assumptions about these elements. On the other hand, if the theory is to be more than a collection of isolated results, the assumptions must be broad enough either to be of wide applicability or to define classes of problems for which a unified treatment is possible. Consider first the specification of the class P. Precise numerical assumptions concerning probabilities or probability distributions are usually not warranted. However, it is frequently possible to assume that certain events have equal probabilities and that certain other are statistically independent. Another type of assumption concerns the relative order of certain infinitesimal probabilities, for example the probability of occurrences in an interval of time or space as the length of the internal tends to zero. The following classes of distributions are derived on the basis of only such assumptions, and are therefore applicable in a great variety of situations. The binomial distribution b(p, n) with   n x P (X = x) = p (1 − p)n−x , x = 0, . . . , n. 0 ≤ p ≤ 1. (1.1) x This is the distribution of the total number of successes in n independent trials when the probability of success for each trial is p. The Poisson distribution P (τ ) with P (X = x) = τ x −τ e , x! x = 0, 1, . . . , 0 < τ. (1.2) This is the distribution of the number of events occurring in a fixed interval of time or space if the probability of more than one occurrence in a very short interval is of smaller order of magnitude than that of a single occurrence, and if the numbers of events in nonoverlapping intervals are statistically independent. Under these assumptions, the process generating the events is called a Poisson 1.2. Specification of a Decision Problem 5 process. Such processes are discussed, for example, in the books by Feller (1968), Ross (1996), and Taylor and Karlin (1998). The normal distribution N (ξ, σ 2 ) with probability density   1 1 p(x) = √ −∞ < x, ξ < ∞, 0 < σ. (1.3) exp − 2 (x − ξ)2 , 2σ 2πσ Under very general conditions, which are made precise by the central limit theorem, this is the approximate distribution of the sum of a large number of independent random variables when the relative contribution of each term to the sum is small. We consider next the structure of the decision space D. The great variety of possibilities is indicated by the following examples. Example 1.2.1 Let X1 , . . . , Xn be a sample from one of the distributions (1.1)– (1.3), that is let the X’s be distributed independently and identically according to one of these distributions. Let θ be p, τ , or the pair (ξ, σ) respectively, and let γ = γ(θ) be a real-valued function of θ. (i) If one wishes to decide whether or not γ exceeds some specified value γ0 , the choice lies between the two decisions d0 : γ > γ0 and d1 : γ ≤ γ0 . In specific applications these decisions might correspond to the acceptance or rejection of a lot of manufactured goods, of an experimental airplane as ready for flight testing, of a new treatment as an improvement over a standard one, and so on. The loss function of course depends on the application to be made. Typically, the loss is 0 if the correct decision is chosen, while for an incorrect decision the losses L(γ, d0 ) and L(γ, d1 ) are increasing functions of |γ − γ0 |. (ii) At the other end of the scale is the much more detailed problem of obtaining a numerical estimate of γ. Here a decision d of the statistician is a real number, the estimate of γ, and the losses might be L(γ, d) = v(γ)w(|d − γ|), where w is a strictly increasing function of the error |d − γ|. (iii) An intermediate case is the choice between the three alternatives d0 : γ < γ0 , d1 : γ > γ1 , d2 : γ0 ≤ γ ≤ γ1 , for example accepting a new treatment, rejecting it, or recommending it for further study. The distinction illustrated by this example is the basis for one of the principal classifications of statistical methods. Two-decision problems such as (i) are usually formulated in terms of testing a hypothesis which is to be accepted or rejected (see Chapter 3). It is the theory of this class of problems with which we shall be mainly concerned here. The other principal branch of statistics is the theory of point estimation dealing with problems such as (ii). This is the subject of TPE2. The intermediate problem (iii) is a special case of a multiple decision procedure. Some problems of this kind are treated in Ferguson (1967, Chapter 6); a discussion of some others is given in Chapter 9. Example 1.2.2 Suppose that the data consist of samples Xij , j = 1, . . . , ni , from normal populations N (ξi , σ 2 ), i = 1, . . . , s. (i) Consider first the case s = 2 and the question of whether or not there is a material difference between the two populations. This has the same structure as problem (iii) of the previous example. Here the choice lies between the three 6 1. The General Decision Problem decisions d0 : |ξ2 − ξ1 | ≤ ∆, d1 : ξ2 > ξ1 + ∆, d2 : ξ2 < ξ1 − ∆, where ∆ is preassigned. An analogous problem, involving k + 1 possible decisions, occurs in the general case of k populations. In this case one must choose between the decision that the k distributions do not differ materially, d0 : max |ξj − ξi | ≤ ∆, and the decisions dk : max |ξj − ξi | > ∆ and ξk is the largest of the means. (ii) A related problem is that of ranking the distributions in increasing order of their mean ξ. (iii) Alternatively, a standard ξ0 may be given and the problem is to decide which, if any, of the population means exceed the standard. Example 1.2.3 Consider two distributions—to be specific, two Poisson distributions P (τ1 ), P (τ2 )—and suppose that τ1 is known to be less than τ2 but that otherwise the τ ’s are unknown. Let Z1 , . . . , Zn be independently distributed, each according to either P (τ1 ) or P (τ2 ). Then each Z is to be classified as to which of the two distributions it comes from. Here the loss might be the number of Z’s that are incorrectly classified, multiplied by a suitable function of τ1 and τ2 . An example of the complexity that such problems can attain and the conceptual as well as mathematical difficulties that they may involve is provided by the efforts of anthropologists to classify the human population into a number of homogeneous races by studying the frequencies of the various blood groups and of other genetic characters. All the problems considered so far could be termed action problems. It was assumed in all of them that if θ were known a unique correct decision would be available, that is, given any θ, there exists a unique d for which L(θ, d) = 0. However, not all statistical problems are so clear-cut. Frequently it is a question of providing a convenient summary of the data or indicating what information is available concerning the unknown parameter or distribution. This information will be used for guidance in various considerations but will not provide the sole basis for any specific decisions. In such cases the emphasis is on the inference rather than on the decision aspect of the problem. Although formally it can still be considered a decision problem if the inferential statement itself is interpreted as the decision to be taken, the distinction is of conceptual and practical significance despite the fact that frequently it is ignored.1 An important class of such problems, estimation by interval, is illustrated by the following example. (For the more usual formulation in terms of confidence intervals, see Sections 3.5, 5.4 and 5.5.) Example 1.2.4 Let X = (X1 , . . . , Xn ) be a sample from N (ξ, σ 2 ) and let a decision consist in selecting an interval [L, L] and stating that it contains ξ. Suppose that decision procedures are restricted to intervals [L(X), L̄(X)] whose expected length for all ξ and σ does not exceed kσ where k is some preassigned constant. An appropriate loss function would be 0 if the decision is correct and would otherwise depend on the relative position of the interval to the true value of ξ. In this case there are many correct decisions corresponding to a given distribution N (ξ, σ 2 ). 1 For a more detailed discussion of this distinction see, for example, Cox (1958), Blyth (1970), and Barnett (1999). 1.2. Specification of a Decision Problem 7 It remains to discuss the choice of loss function, and of the three elements defining the problem this is perhaps the most difficult to specify. Even in the simplest case, where all losses eventually reduce to financial ones, it can hardly be expected that one will be able to evaluate all the short- and long-term consequences of an action. Frequently it is possible to simplify the formulation by taking into account only certain aspects of the loss function. As an illustration consider Example 1.2.1(i) and let L(θ, d0 ) = a for γ(θ) ≤ γ0 and L(θ, d1 ) = b for γ(θ) > γ0 . The risk function becomes  R(θ, δ) = aPθ {δ(X) = d0 } bPθ {δ(X) = d1 } if if γ ≤ γ0 , γ > γ0 , (1.4) and is seen to involve only the two probabilities of error, with weights which can be adjusted according to the relative importance of these errors. Similarly, in Example 1.2.3 one may wish to restrict attention to the number of misclassifications. Unfortunately, such a natural simplification is not always available, and in the absence of specific knowledge it becomes necessary to select the loss function in some conventional way, with mathematical simplicity usually an important consideration. In point estimation problems such as that considered in Example 1.2.1(ii), if one is interested in estimating a real-valued function γ = γ(θ), it is customary to take the square of the error, or somewhat more generally to put L(θ, d) = v(θ)(d − γ)2 . (1.5) Besides being particularly simple mathematically, this can be considered as an approximation to the true loss function L provided that for each fixed θ, L(θ, d) is twice differentiable in d, that L(θ, γ(θ)) = 0 for all θ, and that the error is not large. It is frequently found that, within one problem, quite different types of losses may occur, which are difficult to measure on a common scale. Consider once more Example 1.2.1(i) and suppose that γ0 is the value of γ when a standard treatment is applied to a situation in medicine, agriculture, or industry. The problem is that of comparing some new process with unknown γ to the standard one. Turning down the new method when it is actually superior, or adopting it when it is not, clearly entails quite different consequences. In such cases it is sometimes convenient to treat the various loss components, say L1 , L2 , . . . , Lr , separately. Suppose in particular that r = 2 and the L1 represents the more serious possibility. One can then assign a bound to this risk component, that is, impose the condition EL1 (θ, δ(X)) ≤ α, (1.6) and subject to this condition minimize the other component of the risk. Example 1.2.4 provides an illustration of this procedure. The length of the interval [L, L̄] (measured in σ-units) is one component of the loss function, the other being the loss that results if the interval does not cover the true ξ. 8 1. The General Decision Problem 1.3 Randomization; Choice of Experiment The description of the general decision problem given so far is still too narrow in certain respects. It has been assumed that for each possible value of the random variables a definite decision must be chosen. Instead, it is convenient to permit the selection of one out of a number of decisions according to stated probabilities, or more generally the selection of a decision according to a probability distribution defined over the decision space; which distribution depends of course on what x is observed. One way to describe such a randomized procedure is in terms of a nonrandomized procedure depending on X and a random variable Y whose values lie in the decision space and whose conditional distribution given x is independent of θ. Although it may run counter to one’s intuition that such extra randomization should have any value, there is no harm in permitting this greater freedom of choice. If the intuitive misgivings are correct, it will turn out that the optimum procedures always are of the simple nonrandomized kind. Actually, the introduction of randomized procedures leads to an important mathematical simplification by enlarging the class of risk functions so that it becomes convex. In addition, there are problems in which some features of the risk function such as its maximum can be improved by using a randomized procedure. Another assumption that tacitly has been made so far is that a definite experiment has already been decided upon so that it is known what observations will be taken. However, the statistical considerations involved in designing an experiment are no less important than those concerning its analysis. One question in particular that must be decided before an investigation is undertaken is how many observations should be taken so that the risk resulting from wrong decisions will not be excessive. Frequently it turns out that the required sample size depends on the unknown distribution and therefore cannot be determined in advance as a fixed number. Instead it is then specified as a function of the observations and the decision whether or not to continue experimentation is made sequentially at each stage of the experiment on the basis of the observations taken up to that point. Example 1.3.1 On the basis of a sample X1 , . . . , Xn from a normal distribution N (ξ, σ 2 ) one wishes to estimate ξ. Here the risk function of an estimate, for example its expected squared error, depends on σ. For large σ the sample contains only little information in the sense that two distributions N (ξ1 , σ 2 ) and N (ξ2 , σ 2 ) with fixed difference ξ2 − ξ1 become indistinguishable as σ → ∞, with the result that the risk tends to infinity. Conversely, the risk approaches zero as σ → 0, since then effectively the mean becomes known. Thus the number of observations needed to control the risk at a given level is unknown. However, as soon as some observations have been taken, it is possible to estimate σ 2 and hence to determine the additional number of observations required. Example 1.3.2 In a sequence of trials with constant probability p of success, one wishes to decide whether p ≤ 12 or p > 12 . It will usually be possible to reach a decision at an early stage if p is close to 0 or 1 so that practically all observations are of one kind, while a larger sample will be needed for intermediate values of p. This difference may be partially balanced by the fact that for intermediate 1.4. Optimum Procedures 9 values a loss resulting from a wrong decision is presumably less serious than for the more extreme values. Example 1.3.3 The possibility of determining the sample size sequentially is important not only because the distributions Pθ can be more or less informative but also because the same is true of the observations themselves. Consider, for example, observations from the uniform distribution over the interval (θ − 12 , θ + 1 ) and the problem of estimating θ. Here there is no difference in the amount 2 of information provided by the different distributions Pθ . However, a sample X1 , X2 , . . . , Xn can practically pinpoint θ if max |Xj − Xi | is sufficiently close to 1, or it can give essentially no more information then a single observation if max |Xj − Xi | is close to 0. Again the required sample size should be determined sequentially. Except in the simplest situations, the determination of the appropriate sample size is only one aspect of the design problem. In general, one must decide not only how many but also what kind of observations to take. In clinical trials, for example, when a new treatment is being compared with a standard procedure, a protocol is required which specifies to which of the two treatments each of the successive incoming patients is to be assigned. Formally, such questions can be subsumed under the general decision problem described at the beginning of the chapter, by interpreting X as the set of all available variables, by introducing the decisions whether or not to stop experimentation at the various stages, by specifying in case of continuance which type of variable to observe next, and by including the cost of observation in the loss function. The determination of optimum sequential stopping rules and experimental designs is outside the scope of this book. An introduction to this subject is provided, for example, by Siegmund (1985). 1.4 Optimum Procedures At the end of Section 1.1 the aim of statistical theory was stated to be the determination of a decision function δ which minimizes the risk function R(θ, δ) = Eθ [L(θ, δ(X))]. (1.7) Unfortunately, in general the minimizing δ depends on θ, which is unknown. Consider, for example, some particular decision d0 , and the decision procedure δ(x) ≡ d0 according to which decision d0 is taken regardless of the outcome of the experiment. Suppose that d0 is the correct decision for some θ0 , so that L(θ0 , d0 ) = 0. Then δ minimizes the risk at θ0 since R(θ0 , δ) = 0, but presumably at the cost of a high risk for other values of θ. In the absence of a decision function that minimizes the risk for all θ, the mathematical problem is still not defined, since it is not clear what is meant by a best procedure. Although it does not seem possible to give a definition of optimality that will be appropriate in all situations, the following two methods of approach frequently are satisfactory. The nonexistence of an optimum decision rule is a consequence of the possibility that a procedure devotes too much of its attention to a single parameter value 10 1. The General Decision Problem at the cost of neglecting the various other values that might arise. This suggests the restriction to decision procedures which possess a certain degree of impartiality, and the possibility that within such a restricted class there may exist a procedure with uniformly smallest risk. Two conditions of this kind, invariance and unbiasedness, will be discussed in the next section. Instead of restricting the class of procedures, one can approach the problem somewhat differently. Consider the risk functions corresponding to two different decision rules δ1 and δ2 . If R(θ, δ1 ) < R(θ, δ2 ) for all θ, then δ1 is clearly preferable to δ2 , since its use will lead to a smaller risk no matter what the true value of θ is. However, the situation is not clear when the two risk functions intersect as in Figure 1.1. What is needed is a principle which in such cases establishes a preference of one of the two risk functions over the other, that is, which introduces an ordering into the set of all risk functions. A procedure will then be optimum if its risk function is best according to this ordering. Some criteria that have been suggested for ordering risk functions will be discussed in Section 1.6. R(␪,␦) ␪ Figure 1.1. A weakness of the theory of optimum procedures sketched above is its dependence on an extraneous restricting or ordering principle, and on knowledge concerning the loss function and the distributions of the observable random variables which in applications is frequently unavailable or unreliable. These difficulties, which may raise doubt concerning the value of an optimum theory resting on such shaky foundations, are in principle no different from those arising in any application of mathematics to reality. Mathematical formulations always involve simplification and approximation, so that solutions obtained through their use cannot be relied upon without additional checking. In the present case a check consists in an overall evaluation of the performance of the procedure that the theory produces, and an investigation of its sensitivity to departure from the assumptions under which it was derived. The optimum theory discussed in this book should therefore not be understood to be prescriptive. The fact that a procedure δ is optimal according to some optimality criterion does not necessarily mean that it is the right procedure to use, or even a satisfactory procedure. It does show how well one can do in this particular direction and how much is lost when other aspects have to be taken into account. 1.5. Invariance and Unbiasedness 11 The aspect of the formulation that typically has the greatest influence on the solution of the optimality problem is the family P to which the distribution of the observations is assumed to belong. The investigation of the robustness of a proposed procedure to departures from the specified model is an indispensable feature of a suitable statistical procedure, and although optimality (exact or asymptotic) may provide a good starting point, modifications are often necessary before an acceptable solution is found. It is possible to extend the decision-theoretic framework to include robustness as well as optimality. Suppose robustness is desired against some class P  of distributions which is larger (possibly much larger) than the give P. Then one may assign a bound M to the risk to be tolerated over P  . Within the class of procedures satisfying this restriction, one can then optimize the risk over P as before. Such an approach has been proposed and applied to a number of specific problems by Bickel (1984) and Kempthorne (1988). Another possible extension concerns the actual choice of the family P, the model used to represent the actual physical situation. The problem of choosing a model which provides an adequate description of the situation without being unnecessarily complex can be treated within the decision-theoretic formulation of Section 1.1 by adding to the loss function a component representing the complexity of the proposed model. Such approaches to model selection are discussed in Stone (1981), de Leeuw (1992) and Rao and Wu (2001). 1.5 Invariance and Unbiasedness2 A natural definition of impartiality suggests itself in situations which are symmetric with respect to the various parameter values of interest: The procedure is then required to act symmetrically with respect to these values. Example 1.5.1 Suppose two treatments are to be compared and that each is applied n times. The resulting observations X11 , . . . , X1n and X21 , . . . , X2n are samples from N (ξ1 , σ 2 ) and N (ξ2 , σ 2 ) respectively. The three available decisions are d0 : |ξ2 − ξ1 | ≤ ∆, d1 : ξ2 > ξ1 + ∆, d2 : ξ2 < ξ1 − ∆, and the loss is wij if decision dj is taken when di would have been correct. If the treatments are to be compared solely in terms of the ξ’s and no outside considerations are involved, the losses are symmetric with respect to the two treatments so that w01 = w02 , w10 = w20 , w12 = w21 . Suppose now that the labeling of the two treatments as 1 and 2 is reversed, and correspondingly also the labeling of the X’s, the ξ’s, and the decisions d1 and d2 . This changes the meaning of the symbols, but the formal decision problem, because of its symmetry, remains unaltered. It is then natural to require the corresponding symmetry from the procedure δ and ask that δ(x11 , . . . , x1n , x21 , . . . , x2n ) = d0 , d1 , or d2 as δ(x21 , . . . , x2n , x11 , . . . , x1n ) = d0 , d2 , or d1 respectively. If this condition were not satisfied, the decision as to which population has the greater mean would depend on the presumably quite 2 The concepts discussed here for general decision theory will be developed in more specialized form in later chapters. The present section may therefore be omitted at first reading. 12 1. The General Decision Problem accidental and irrelevant labeling of the samples. Similar remarks apply to a number of further symmetries that are present in this problem. Example 1.5.2 Consider a sample X1 , . . . , Xn from a distribution with density σ −1 f [(x − ξ)/σ] and the problem of estimating the location parameter ξ, say the mean of the X’s, when the loss is (d − ξ)2 /σ 2 , the square of the error expressed in σ-units. Suppose that the observations are originally expressed in feet, and let Xi = aX with a = 12 be the corresponding observations in inches. In the transformed problem the density is σ −1 f [(x − ξ  )/σ  ] with ξ  = aξ, σ  = aσ. Since (d − ξ  )2 /σ 2 = (d − ξ)2 /σ 2 , the problem is formally unchanged. The same estimation procedure that is used for the original observations is therefore appropriate after the transformation and leads to δ(aX1 , . . . , aXn ) as an estimate of ξ  = aξ, the parameter ξ expressed in inches. On reconverting the estimate into feet one finds that if the result is to be independent of the scale of measurements, δ must satisfy the condition of scale invariance δ(aX1 , . . . , aXn ) = δ(X1 , . . . , Xn ) . a The general mathematical expression of symmetry is invariance under a suitable group of transformations. A group G of transformations g of the sample space is said to leave a statistical decision problem invariant if it satisfies the following conditions: (i) It leaves invariant the family of distributions P = {Pθ , θ ∈ Ω}, that is, for any possible distribution Pθ of X the distribution of gX, say Pθ , is also in P. The resulting mapping θ = ḡθ of Ω is assumed to be onto3 Ω and 1:1. (ii) To each g ∈ G, there corresponds a transformation g ∗ = h(g) of the decision space D onto itself such that h is a homomorphism, that is, satisfies the relation h(g1 g2 ) = h(g1 )h(g2 ), and the loss function L is unchanged under the transformation, so that L(ḡθ, g ∗ d) = L(θ, d). Under these assumptions the transformed problem, in terms of X  = gX, θ = ḡθ, and d = g ∗ d, is formally identical with the original problem in terms of X, θ, and d. Given a decision procedure δ for the latter, this is therefore still appropriate after the transformation. Interpreting the transformation as a change of coordinate system and hence of the names of the elements, one would, on observing x , select the decision which in the new system has the name δ(x ), so that its old name is g ∗−1 δ(x ). If the decision taken is to be independent of the particular coordinate system adopted, this should coincide with the original decision δ(x), that is, the procedure must satisfy the invariance condition δ(gx) = g ∗ δ(x) for all x ∈ X, g ∈ G. (1.8) Example 1.5.3 The model described in Example 1.5.1 is invariant also under  the transformations Xij = Xij + c, ξi = ξi + c. Since the decisions d0 , d1 , and d2 3 The term onto is used in indicate that ḡΩ is not only contained in but actually equals Ω; that is, given any θ  in Ω, there exists θ in Ω such that ḡθ = θ  . 1.5. Invariance and Unbiasedness 13 concern only the differences ξ2 − ξ1 , they should remain unchanged under these transformations, so that one would expect to have g ∗ di = di for i = 0, 1, 2. It is in fact easily seen that the loss function does satisfy L(ḡθ, d) = L(θ, d), and hence that g ∗ d = d. A decision procedure therefore remains invariant in the present case if it satisfies δ(gx) = δ(x) for all g ∈ G, x ∈ X. It is helpful to make a terminological distinction between situations like that of Example 1.5.3 in which g ∗ d = d for all d, and those like Examples 1.5.1 and 1.5.2 where invariance considerations require δ(gx) to vary with g. In the former case the decision procedure remains unchanged under the transformations X  = gX and is thus truly invariant; in the latter, the procedure varies with g and may then more appropriately be called equivariant rather than invariant. Typically, hypothesis testing leads to procedures that are invariant in this sense; estimation problems (whether by point or interval estimation), to equivariant ones. Invariant tests and equivariant confidence sets will be discussed in Chapter 6. For a brief discussion of equivariant point estimation, see Bondessen (1983); a fuller treatment is given in TPE2, Chapter 3. Invariance considerations are applicable only when a problem exhibits certain symmetries. An alternative impartiality restriction which is applicable to other types of problems is the following condition of unbiasedness. Suppose the problem is such that for each θ there exists a unique correct decision and that each decision is correct for some θ. Assume further that L(θ1 , d) = L(θ2 , d) for all d whenever the same decision is correct for both θ1 and θ2 . Then the loss L(θ, d ) depends only on the actual decision taken, say d , and the correct decision d. The loss can thus be denoted by L(d, d ) and this function measures how far apart d and d are. Under these assumptions a decision function δ is said to be unbiased with respect to the loss function L, or L-unbiased, if for all θ and d Eθ L(d , δ(X)) ≥ Eθ L(d, δ(X)) where the subscript θ indicates the distribution with respect to which the expectation is taken and where d is the decision that is correct for θ. Thus δ is unbiased if on the average δ(X) comes closer to the correct decision than to any wrong one. Extending this definition, δ is said to be L-unbiased for an arbitrary decision problem if for all θ and θ Eθ L(θ , δ(X)) ≥ Eθ L(θ, δ(X)). (1.9) Example 1.5.4 Suppose that in the problem of estimating a real-valued parameter θ by confidence intervals, as in Example 1.2.4, the loss is 0 or 1 as the interval [L, L̄] does or does not cover the true θ. Then the set of intervals [L(X), L̄(X)] is unbiased if the probability of covering the true value is greater than or equal to the probability of covering any false value. Example 1.5.5 In a two-decision problem such as that of Example 1.2.1(i), let ω0 and ω1 be the sets of θ-values for which d0 and d1 are the correct decisions. Assume that the loss is 0 when the correct decision is taken, and otherwise is given by L(θ, d0 ) = a for θ ∈ ω1 , and L(θ, d1 ) = b for θ ∈ ω0 . Then  aPθ {δ(X) = d0 } if θ ∈ ω1 , Eθ L(θ , δ(X)) = bPθ {δ(X) = d1 } if θ ∈ ω0 , 14 1. The General Decision Problem so that (1.9) reduces to aPθ {δ(X) = d0 } ≥ bPθ {δ(X) = d1 } for θ  ∈ ω0 , with the reverse inequality holding for θ ∈ ω1 . Since Pθ {δ(X) = d0 } + Pθ {δ(X) = d1 } = 1, the unbiasedness condition (1.9) becomes Pθ {δ(X) = d1 } ≤ Pθ {δ(X) = d1 } ≥ a a+b a a+b for for θ ∈ ω0 , θ ∈ ω1 . (1.10) Example 1.5.6 In the problem of estimating a real-valued function γ(θ) with the square of the error as loss, the condition of unbiasedness becomes Eθ [δ(X) − γ(θ )]2 ≥ Eθ [δ(X) − γ(θ)]2 for all θ, θ  . On adding and subtracting h(θ) = Eθ δ(X) inside the brackets on both sides, this reduces to [h(θ) − γ(θ )]2 ≥ [h(θ) − γ(θ)]2 for all θ, θ  . If h(θ) is one of the possible values of the function γ, this condition holds if and only if Eθ δ(X) = γ(θ) . (1.11) In the theory of point estimation, (1.11) is customarily taken as the definition of unbiasedness. Except under rather pathological conditions, it is both a necessary and sufficient condition for δ to satisfy (1.9). (See Problem 1.2.) 1.6 Bayes and Minimax Procedures We now turn to a discussion of some preference orderings of decision procedures and their risk functions. One such ordering is obtained by assuming that in repeated experiments the parameter itself is a random variable Θ, the distribution of which is known. If for the sake of simplicity one supposes that this distribution has a probability density ρ(θ), the overall average loss resulting from the use of a decision procedure δ is   r(ρ, δ) = Eθ L(θ, δ(X))ρ(θ) dθ = R(θ, δ)ρ(θ) dθ (1.12) and the smaller r(ρ, δ), the better is δ. An optimum procedure is one that minimizes r(ρ, δ), and is called a Bayes solution of the given decision problem corresponding to a priori density ρ. The resulting minimum of r(ρ, δ) is called the Bayes risk of δ. Unfortunately, in order to apply this principle it is necessary to assume not only that θ is a random variable but also that its distribution is known. This assumption is usually not warranted in applications. Alternatively, the right-hand side of (1.12) can be considered as a weighted average of the risks; for ρ(θ) ≡ 1 in particular, it is then the area under the risk curve. With this interpretation the choice of a weight function ρ expresses the importance the experimenter attaches to the various values of θ. A systematic Bayes theory has been developed which 1.6. Bayes and Minimax Procedures 15 interprets ρ as describing the state of mind of the investigator towards θ. For an account of this approach see, for example, Berger (1985a) and Robert (1994). If no prior information regarding θ is available, one might consider the maximum of the risk function its most important feature. Of two risk functions the one with the smaller maximum is then preferable, and the optimum procedures are those with the minimax property of minimizing the maximum risk. Since this maximum represents the worst (average) loss that can result from the use of a given procedure, a minimax solution is one that gives the greatest possible protection against large losses. That such a principle may sometimes be quite unreasonable is indicated in Figure 1.2, where under most circumstances one would prefer δ1 to δ2 although its risk function has the larger maximum. R(␪,␦) ␦2 ␦1 ␪ Figure 1.2. Perhaps the most common situation is one intermediate to the two just described. On the one hand, past experience with the same or similar kind of experiment is available and provides an indication of what values of θ to expect; on the other, this information is neither sufficiently precise nor sufficiently reliable to warrant the assumptions that the Bayes approach requires. In such circumstances it seems desirable to make use of the available information without trusting it to such an extent that catastrophically high risks might result if it is inaccurate or misleading. To achieve this one can place a bound on the risk and restrict consideration to decision procedures δ for which R(θ, δ) ≤ C for all θ. (1.13) [Here the constant C will have to be larger than the maximum risk C0 of the minimax procedure, since otherwise there will exist no procedures satisfying (1.13).] Having thus assured that the risk can under no circumstances get out of hand, the experimenter can now safely exploit his knowledge of the situation, which may be based on theoretical considerations as well as on past experience; he can follow his hunches and guess at a distribution ρ for θ. This leads to the selection of a procedure δ (a restricted Bayes solution), which minimizes the average risk (1.12) for this a priori distribution subject to (1.13). The more certain one is of ρ, the larger one will select C, thereby running a greater risk in case of a poor guess but improving the risk if the guess is good. Instead of specifying an ordering directly, one can postulate conditions that the ordering should satisfy. Various systems of such conditions have been investigated 16 1. The General Decision Problem and have generally led to the conclusion that the only orderings satisfying these systems are those which order the procedures according to their Bayes risk with respect to some prior distribution of θ. For details, see for example Blackwell and Girshick (1954), Ferguson (1967), Savage (1972), Berger (1985a), and Bernardo and Smith (1994). 1.7 Maximum Likelihood Another approach, which is based on considerations somewhat different from those of the preceding sections, is the method of maximum likelihood. It has led to reasonable procedures in a great variety of problems, and is still playing a dominant role in the development of new tests and estimates. Suppose for a moment that X can take on only a countable set of values x1 , x2 , . . . , with Pθ (x) = Pθ {X = x}, and that one wishes to determine the correct value of θ, that is, the value that produced the observed x. This suggests considering for each possible θ how probable the observed x would be if θ were the true value. The higher this probability, the more one is attracted to the explanation that the θ in question produced x, and the more likely the value of θ appears. Therefore, the expression Pθ (x) considered for fixed x as a function of θ has been called the likelihood of θ. To indicate the change in point of view, let it be denoted by Lx (θ). Suppose now that one is concerned with an action problem involving a countable number of decisions, and that it is formulated in terms of a gain function (instead of the usual loss function), which is 0 if the decision taken is incorrect and is a(θ) > 0 if the decision taken is correct and θ is the true value. Then it seems natural to weight the likelihood Lx (θ) by the amount that can be gained if θ is true, to determine the value of θ that maximizes a(θ)Lx (θ) and to select the decision that would be correct if this were the true value of θ. Essentially the same remarks apply in the case in which Pθ (x) is a probability density rather than a discrete probability. In problems of point estimation, one usually assumes that a(θ) is independent of θ. This leads to estimating θ by the value that maximizes the likelihood Lx (θ), the maximum-likelihood estimate of θ. Another case of interest is the class of two-decision problems illustrated by Example 1.2.1(i). Let ω0 and ω1 denote the sets of θ-values for which d0 and d1 are the correct decisions, and assume that a(θ) = a0 or a1 as θ belongs to ω0 or ω1 respectively. Then decision d0 or d1 is taken as a1 supθ∈ω1 Lx (θ) < or > a0 supθ∈ω0 Lx (θ), that is as sup Lx (θ) θ∈ω0 sup Lx (θ) > or < a1 . a0 (1.14) θ∈ω1 This is known as a likelihood ratio procedure.4 4 This definition differs slightly from the usual one where in the denominator on the left-hand side of (1.14) the supremum is taken over the set ω 0 ∪ ω 1 . The two definitions agree whenever the left-hand side of (1.14) is ≤ 1, and the procedures therefore agree is a1 < a0 . 1.8. Complete Classes 17 Although the maximum likelihood principle is not based on any clearly defined optimum considerations, it has been very successful in leading to satisfactory procedures in many specific problems. For wide classes of problems, maximum likelihood procedures will be shown in Chapter 13 to possess various asymptotic optimum properties as the sample size tends to infinity; also see TPE2, Chapter 6. On the other hand, there exist examples for which the maximum-likelihood procedure is worse than useless; where it is, in fact, so bad that one can do better without making any use of the observations (see Problem 6.28). 1.8 Complete Classes None of the approaches described so far is reliable in the sense that the resulting procedure is necessarily satisfactory. There are problems in which a decision procedure δ0 exists with uniformly minimum risk among all unbiased or invariant procedures, but where there exists a procedure δ1 not possessing this particular impartiality property and preferable to δ0 . (Cf. Problems 1.14 and 1.16.) As was seen earlier, minimax procedures can also be quite undesirable, while the success of Bayes and restricted Bayes solutions depends on a priori information which is usually not very reliable if it is available at all. In fact, it seems that in the absence of reliable a priori information no principle leading to a unique solution can be entirely satisfactory. This suggests the possibility, at least as a first step, of not insisting on a unique solution but asking only how far a decision problem can be reduced without loss of relevant information. It has already been seen that a decision procedure δ can sometimes be eliminated from consideration because there exists a procedure δ  dominating it in the sense that R(θ, δ  ) ≤ R(θ, δ) R(θ, δ  ) < R(θ, δ) for all θ for some θ. (1.15) In this case δ is said to be inadmissible; δ is called admissible if no such dominating δ  exists. A class C of decision procedures is said to be complete if for any δ not in C there exists δ  in C dominating it. A complete class is minimal if it does not contain a complete subclass. If a minimal complete class exists, as is typically the case, it consists exactly of the totality of admissible procedures. It is convenient to define also the following variant of the complete class notion. A class C is said to be essentially complete if for any procedure δ there exists δ  in C such that R(θ, δ  ) ≤ R(θ, δ) for all θ. Clearly, any complete class is also essentially complete. In fact, the two definitions differ only in their treatment of equivalent decision rules, that is, decision rules with identical risk function. If δ belongs to the minimal complete class C, any equivalent decision rule must also belong to C. On the other hand, a minimal essentially complete class need contain only one member from such a set of equivalent procedures. In a certain sense a minimal essentially complete class provides the maximum possible reduction of a decision problem. On the one hand, there is no reason to consider any of the procedures that have been weeded out. For each of them, there is included one in C that is as good or better. On the other hand, it is not possible to reduce the class further. Given any two procedures in C, each of them 18 1. The General Decision Problem is better in places than the other, so that without additional information it is not known which of the two is preferable. The primary concern in statistics has been with the explicit determination of procedures, or classes of procedures, for various specific decision problems. Those studied most extensively have been estimation problems, and problems involving a choice between only two decisions (hypothesis testing), the theory of which constitutes the subject of the present volume. However, certain conclusions are possible without such specialization. In particular, two results concerning the structure of complete classes and minimax procedures have been proved to hold under very general assumptions.5 (i) The totality of Bayes solutions and limits of Bayes solutions constitute a complete class. (ii) Minimax procedures are Bayes solutions with respect to a least favorable a priori distribution, that is, an a priori distribution that maximizes the associated Bayes risk, and the minimax risk equals this maximum Bayes risk. Somewhat more generally, if there exists no least favorable a priori distribution but only a sequence for which the Bayes risk tends to the maximum, the minimax procedures are limits of the associated sequence of Bayes solutions. 1.9 Sufficient Statistics A minimal complete class was seen in the preceding section to provide the maximum possible reduction of a decision problem without loss of information. Frequently it is possible to obtain a less extensive reduction of the data, which applies simultaneously to all problems relating to a given class P = {Pθ , θ ∈ Ω} of distributions of the given random variable X. It consists essentially in discarding that part of the data which contains no information regarding the unknown distribution Pθ , and which is therefore of no value for any decision problem concerning θ. Example 1.9.1 Trials are performed with constant unknown probability p of success. If Xi is 1 or 0 as the ith trial is a success or failure, the sample (X1 , . . . , Xn ) shows how many successes there were and in which trials they occurred. The second of these pieces of information  contains no evidence as to the value of p.Once the total number of successes Xi is known to be equal to t, each of the nt possible positions of these successes is equally likely regardless  of p. It follows that knowing Xi but neither the individual Xi nor p, one can, from a table of random numbers, construct a set of random variables X1 , . . . , Xn whose joint distribution is the same as that of X1 , . . . , Xn . Therefore, the infor mation contained in the Xi is the same as that contained in Xi and a table of random numbers. 5 Precise statements and proofs of these results are given in the book by Wald (1950). See also Ferguson (1967) and Berger (1985a). Additional results and references are given in Brown and Marden (1989) and Kowalski (1995). 1.9. Sufficient Statistics 19 Example 1.9.2 If X1 , . . . , Xn are independently normally distributed with zero 2 mean and variance point over each  σ2 , the conditional distribution of the sample of the spheres, Xi = constant, is uniform irrespective of σ 2 . One can  therefore construct an equivalent sample X1 , . . . , Xn from a knowledge of Xi2 and a mechanism that can produce a point randomly distributed over a sphere. More generally, a statistic T is said to be sufficient for the family P = {Pθ , θ ∈ Ω} (or sufficient for θ, if it is clear from the context what set Ω is being considered) if the conditional distribution of X given T = t is independent of θ. As in the two examples it then follows under mild assumptions6 that it is not necessary to utilize the original observations X. If one is permitted to observe only T instead of X, this does not restrict the class of available decision procedures. For any value t of T let Xt be a random variable possessing the conditional distribution of X given t. Such a variable can, at least theoretically, be constructed by means of a suitable random mechanism. If one then observes T to be t and Xt to be x , the random variable X  defined through this two-stage process has the same distribution as X. Thus, given any procedure based on X, it is possible to construct an equivalent one based on X  which can be viewed as a randomized procedure based solely on T . Hence if randomization is permitted (and we shall assume throughout that this is the case), there is no loss of generality in restricting consideration to a sufficient statistic. It is inconvenient to have to compute the conditional distribution of X given t in order to determine whether or not T is sufficient. A simple check is provided by the following factorization criterion. Consider first the case that X is discrete, and let Pθ (x) = Pθ {X = x}. Then a necessary and sufficient condition for T to be sufficient for θ is that there exists a factorization Pθ (x) = gθ [T (x)]h(x), (1.16) where the first factor may depend on θ but depends on x only through T (x), while the second factor is independent of θ.  Suppose that (1.16) holds, and let T (x) = t. Then Pθ {T = t} = P0 (x )   summed over all points x with T (x ) = t, and the conditional probability Pθ {X = x | T = t} = Pθ (x) h(x) =  Pθ {T = t} h(x ) is independent of θ. Conversely, if this conditional distribution does not depend on θ and is equal to, say k(x, t), then Pθ (x) = Pθ {T = t}k(x, t), so that (1.16) holds. Example 1.9.3 Let X1 , . . . , Xn be independently and identically distributed according to the Poisson distribution (1.2). Then Pτ (x1 , . . . , xn ) = τ  xi −nτ e n , xj ! j=1 6 These are connected with difficulties concerning the behavior of conditional probabilities. For a discussion of these difficulties see Sections 2.3–2.5. 20 1. The General Decision Problem and it follows that  Xi is a sufficient statistic for τ . In the case that the distribution of X is continuous and has probability density pX θ (x), let X and T be vector-valued, X = (X1 , . . . , Xn ) and T = (T1 , . . . Tt ) say. Suppose that there exist functions Y = (Y1 , . . . , Yn−r ) on the sample space such that the transformation (x1 , . . . , xn ) ↔ (T1 (x), . . . , Tr (x), Y1 (x), . . . , Yn−r (x)) (1.17) is 1:1 on a suitable domain, and that the joint density of T and Y exists and is related to that of X by the usual formula T,Y (T (x), Y (x)) · |J|, pX θ (x) = pθ (1.18) . . , Tr , Y1 , . . . , Yn−r ) with respect to (x1 , . . . , xn ). where J is the Jacobian of (T1 , . Thus in Example 1.9.2, T = Xi2 , Y1 , . . . , Yn−1 can be taken to be the polar (t, y) of T and Y , coordinates of the sample point. From the joint density pT,Y θ the conditional density of Y given T = t is obtained as Y |t pθ (y) = pT,Y (t, y) θ (t, y  ) dy  pT,Y θ (1.19) provided the denominator is different from zero. Regularity conditions for the validity of (1.18) are given by Tukey (1958b). Since in the conditional distribution given t only the Y ’s vary, T is sufficient for θ if the conditional distribution of Y given t is independent of θ. Suppose that T satisfies (1.19). Then analogously to the discrete case, a necessary and sufficient condition for T to be sufficient is a factorization of the density of the form pX θ (x) = gθ [T (x)]h(x). (1.20) (See Problem 1.19.) The following two examples illustrate the application of the criterion in this case. In both examples the existence of functions Y satisfying (1.17)–(1.19) will be assumed but not proved. As will be shown later (Section 2.6), this assumption is actually not needed for the validity of the factorization criterion. Example 1.9.4 Let X1 , . . . , Xn be independently distributed with normal probability density  1  2 ξ  n pξ,σ (x) = (2πσ 2 )−n/2 exp − 2 xi + 2 xi − 2 ξ 2 . 2σ σ 2σ  2  Then the factorization criterion shows ( Xi , Xi ) to be sufficient for (ξ, σ). Example 1.9.5 Let X1 , . . . , Xn be independently distributed according to the uniform distribution U (0, θ) over the interval (0, θ). Then pθ (x) = θ−n (max xi , θ), where u(a, b) is 1 or 0 as a ≤ b or a > b, and hence max Xi is sufficient for θ. An alternative criterion of Bayes sufficiency, due to Kolmogorov (1942), provides a direct connection between this concept and some of the basic notions of decision theory. As in the theory of Bayes solutions, consider the unknown parameter θ as a random variable Θ with an a priori distribution, and assume 1.10. Problems 21 for simplicity that it has a density ρ(θ). Then if T is sufficient, the conditional distribution of Θ given X = x depends only on T (x). Conversely, if ρ(θ) = 0 for all θ and if the conditional distribution of Θ given x depends only on T (x), then T is sufficient for θ. In fact, under the assumptions made, the joint density of X and Θ is pθ (x)ρ(θ). If T is sufficient, it follows from (1.20) that the conditional density of Θ given x depends only on T (x). Suppose, on the other hand, that for some a priori distribution for which ρ(θ) = 0 for all θ the conditional distribution of Θ given x depends only on T (x). Then pθ (x)ρ(θ) = fθ [T (x)] pθ (x)ρ(θ ) dθ and by solving for pθ (x) it is seen that T is sufficient. Any Bayes solution depends only on the conditional distribution of Θ given x (see Problem 1.8) and hence on T (x). Since typically Bayes solutions together with their limits form an essentially complete class, it follows that this is also true of the decision procedures based on T . The same conclusion had already been reached more directly at the beginning of the section. For a discussion of the relation of these different aspects of sufficiency in more general circumstances and references to the literature see Le Cam (1964), Roy and Ramamoorthi (1979) and Yamada and Morimoto (1992). An example of a statistic which is Bayes sufficient in the Kolmogorov sense but not according to the definition given at the beginning of this section is provided by Blackwell and Ramamoorthi (1982). By restricting attention to a sufficient statistic, one obtains a reduction of the data, and it is then desirable to carry this reduction as far as possible. To illustrate the different possibilities, consider once  more the binomial nExample m 1.9.1. If m is any integer less than n and T1 = i=1 Xi , T2 = i=m+1 Xi , then (T1 , T2 ) constitutes a sufficient statistic, since the conditional distribution of X1 , . . . , Xn given T1 = t1 , T2 = t2 is independent of p. For the same reason,  the full sample (X1 , . . . , Xn ) itself is also a sufficient statistic. However, T = n i=1 Xi provides a more thorough reduction than either of these and than various others that can be constructed. A sufficient statistic T is said to be minimal sufficient if the data cannot be reduced  beyond T without losing sufficiency. For the binomial example in particular, n i=1 Xi can be shown to be minimal (Problem 1.17). This illustrates the fact that in specific examples the sufficient statistic determined by inspection through the factorization criterion usually turns out to be minimal. Explicit procedures for constructing minimal sufficient statistics are discussed in Section 1.5 of TPE2. 1.10 Problems Section 1.2 Problem 1.1 The following distributions arise on the basis of assumptions similar to those leading to (1.1)–(1.3). 22 1. The General Decision Problem (i) Independent trials with constant probability p of success are carried out until a preassigned number m of successes has been obtained. If the number of trials required is X + m, then X has the negative binomial distribution N b(p, m):   m+x−1 m P {X = x} = x = 0, 1, 2 . . . . p (1 − p)x , x (ii) In a sequence of random events, the number of events occurring in any time interval of length τ has the Poisson distribution P (λτ ), and the numbers of events in nonoverlapping time intervals are independent. Then the “waiting time” T , which elapses from the starting point, say t = 0, until the first event occurs, has the exponential probability density p(t) = λe−λτ , t ≥ 0. Let Ti , i ≥ 2, be the time elapsing from the occurrence of the (i − 1)st event to that of the ith event. Then it is also true, although more difficult to prove, that T1 , T2 , . . . are identically and independently distributed. A proof is given, for example, in Karlin and Taylor (1975). (iii) A point X is selected “at random” in the interval (a, b), that is, the probability of X falling in any subinterval of (a, b) depends only on the length of the subinterval, not on its position. Then X has the uniform distribution U (a, b) with probability density p(x) = 1/(b − a), a < x < b. Section 1.5 Problem 1.2 Unbiasedness in point estimation. Suppose that γ is a continuous real-valued function defined over Ω which is not constant in any open subset of Ω, and that the expectation h(θ) = Eθ δ(X) is a continuous function of θ for every estimate δ(X) of γ(θ). Then (1.11) is a necessary and sufficient condition for δ(X) to be unbiased when the loss function is the square of the error. [Unbiasedness implies that γ 2 (θ ) − γ 2 (θ) ≥ 2h(θ)[γ(θ ) − γ(θ)] for all θ, θ . If θ is neither a relative minimum nor maximum of γ, it follows that there exist points θ arbitrarily close to θ both such that γ(θ) + γ(θ ) ≥ and ≤ 2h(θ), and hence that γ(θ) = h(θ). That this equality also holds for an extremum of γ follows by continuity, since γ is not constant in any open set.] Problem 1.3 Median unbiasedness. (i) A real number m is a median for the random variable Y if P {Y ≥ m} ≥ 12 , P {Y ≤ m} ≥ 12 . Then all real a1 , a2 such that m ≤ a1 ≤ a2 or m ≥ a1 ≥ a2 satisfy E|Y − a1 | ≤ E|Y − a2 |. (ii) For any estimate δ(X) of γ(θ), let m− (θ) and m+ (θ) denote the infimum and supremum of the medians of δ(X), and suppose that they are continuous functions of θ. Let γ(θ) be continuous and not constant in any open subset of Ω. Then the estimate δ(X) of γ(θ) is unbiased with respect to the loss function L(θ, d) = |γ(θ) − d| if and only if γ(θ) is a median of δ(X) for each θ. An estimate with this property is said to be median-unbiased. 1.10. Problems 23 Problem 1.4 Nonexistence of unbiased procedures. Let X1 , . . . , Xn be independently distributed with density (1/a)f ((x − ξ)/a), and let θ = (ξ, a). Then no estimator of ξ exists which is unbiased with respect to the loss function (d − ξ)k /ak . Note. For more general results concerning the nonexistence of unbiased procedures see Rojo (1983). Problem 1.5 Let C be any class of procedures that is closed under the transformations of a group G in the sense that δ ∈ C implies g ∗ δg −1 ∈ C for all g ∈ G. If there exists a unique procedure δ0 that uniformly minimizes the risk within the class C, then δ0 is invariant.7 If δ0 is unique only up to sets of measure zero, then it is almost invariant, that is, for each g it satisfies the equation δ(gx) = g ∗ δ(x) except on a set Ng of measure 0. Problem 1.6 Relation of unbiasedness and invariance. (i) If δ0 is the unique (up to sets of measure 0) unbiased procedure with uniformly minimum risk, it is almost invariant. (ii) If Ḡ is transitive and G∗ commutative, and if among all invariant (almost invariant) procedures there exists a procedure δ0 with uniformly minimum risk, then it is unbiased. (iii) That conclusion (ii) need not hold without the assumptions concerning G∗ and Ḡ is shown by the problem of estimating the mean ξ of a normal distribution N (ξ, σ 2 ) with loss function (ξ − d)2 /σ 2 . This remains invariant under the groups G1 : gx = x + b, −∞ < b < ∞ and G2 : gx = ax + b, 0 < a < ∞, −∞ < b < ∞. The best invariant estimate relative to both groups is X, but there does not exist an estimate which is unbiased with respect to the given loss function. [(i): This follows from the preceding problem and the fact that when δ is unbiased so is g ∗ δg −1 . (ii): It is the defining property of transitivity that given θ, θ  there exists ḡ such that θ = ḡθ. Hence for any θ, θ  Eθ L(θ , δ0 (X)) = Eθ L(ḡθ, δ0 (X)) = Eθ L(θ, g ∗−1 δ0 (X)). Since G∗ is commutative, g ∗−1 δ0 is invariant, so that R(θ, g ∗−1 δ0 ) ≥ R(θ, δ0 ) = Eθ L(θ, δ0 (X)).] Section 1.6 Problem 1.7 Unbiasedness in interval estimation. Confidence intervals I = (L, L̄) are unbiased for estimating θ with loss function L(θ, I) = (θ−L)2 +(L̄−θ)2 provided E[ 12 (L + L̄)] = θ for all θ, that is, provided the midpoint of I is an unbiased estimate of θ in the sense of (1.11). Problem 1.8 Structure of Bayes solutions. (i) Let Θ be an unobservable random quantity with probability density ρ(θ), and let the probability density of X be pθ (x) when Θ = θ. Then δ is a Bayes solution 7 Here and in Problems 1.6, 1.7, 1.11, 1.15, and 1.16 the term “invariant” is used in the general sense (1.8) of “invariant or equivalent.” 24 1. The General Decision Problem of a given decision problem if for each x the decision δ(x) is chosen so as to minimize L(θ, δ(x))π(θ | x) dθ, where π(θ | x) = ρ(θ)pθ (x)/ ρ(θ )pθ (x) dθ is the conditional (a posteriori) probability density of Θ given x. (i) Let the problem be a two-decision problem with the losses as given in Example 1.5.5. Then the Bayes solution consists in choosing decision d0 if aP {Θ ∈ ω1 | x} < bP {Θ ∈ ω0 | x} and decision d1 if the reverse inequality holds. The choice of decision is immaterial in case of equality. (iii) In the case of point estimation of a real-valued function g(θ) with loss function L(θ, d) = (g(θ) − d)2 , the Bayes solution becomes δ(x) = E[g(Θ) | x]. When instead the loss function is L(θ, d) = |g(θ) − d|, the Bayes estimate δ(x) is any median of the conditional distribution of g(Θ) given x. [(i): The Bayes risk r(ρ, δ) can be written as [ L(θ, δ(x))π(θ | x) dθ] × p(x) dx, where p(x) = ρ(θ )pθ (x) dθ . (ii): The conditional expectation L(θ, d0 )π(θ | x) dθ reduces to aP {Θ ∈ ω1 | x}, and similarly for d1 .] Problem 1.9 (i) As an example in which randomization reduces the maximum risk, suppose that a coin is known to be either standard (HT) or to have heads on both sides (HH). The nature of the coin is to be decided on the basis of a single toss, the loss being 1 for an incorrect decision and 0 for a correct one. Let the decision be HT when T is observed, whereas in the contrary case the decision is made at random, with probability ρ for HT and 1−ρ for HH. Then the maximum risk is minimized for ρ = 13 . (ii) A genetic setting in which such a problem might arise is that of a couple, of which the husband is either dominant homozygous (AA) or heterozygous (Aa) with respect to a certain characteristic, and the wife is homozygous recessive (aa). Their child is heterozygous, and it is of importance to determine to which genetic type the husband belongs. However, in such cases an a priori probability is usually available for the two possibilities. One is then dealing with a Bayes problem, and randomization is no longer required. In fact, if the a priori probability is p that the husband is dominant, then the Bayes procedure classifies him as such if p > 13 and takes the contrary decision if p < 13 . Problem 1.10 Unbiasedness and minimax. Let Ω = Ω0 ∪ Ω1 where Ω0 , Ω1 are mutually exclusive, and consider a two-decision problem with loss function L(θ, di ) = ai for θ ∈ Ωj (j = i) and L(θ, di ) = 0 for θ ∈ Ωi (i = 0, 1). (i) Any minimax procedure is unbiased. (ii) The converse of (i) holds provided Pθ (A) is a continuous function of θ for all A, and if the sets Ω0 and Ω1 have at least one common boundary point. [(i): The condition of unbiasedness in this case is equivalent to sup Rδ (θ) ≤ a0 a1 /(a0 + a1 ). That this is satisfied by any minimax procedure is seen by comparison with the procedure δ(x) = d0 or = d1 with probabilities a1 /(a0 + a1 ) and a0 /(a0 + a1 ) respectively. (ii): If θ0 , is a common boundary point, continuity of the risk function implies that any unbiased procedure satisfies Rδ (θ0 ) = a0 a1 /(a0 + a1 ) and hence sup Rδ (θ0 ) = a0 a1 /(a0 + a1 ).] 1.10. Problems 25 Problem 1.11 Invariance and minimax. Let a problem remain invariant relative to the groups G, Ḡ, and G∗ over the spaces X , Ω, and D respectively. Then a randomized procedure Yx is defined to be invariant if for all x and g the conditional distribution of Yx given x is the same as that of g ∗−1 Ygx . (i) Consider a decision procedure which remains invariant under a finite group G = {g1 , . . . , gN }. If a minimax procedure exists, then there exists one that is invariant. (ii) This conclusion does not necessarily hold for infinite groups, as is shown by the following example. Let the parameter space Ω consist of all elements θ of the free group with two generators, that is, the totality of formal products π1 . . . πn (n = 0, 1, 2, . . .) where each πi is one of the elements a, a−1 , b, b−1 and in which all products aa−1 , a−1 a, bb−1 , and b−1 b have been canceled. The empty product (n = 0) is denoted by e. The sample point X is obtained by multiplying θ on the right by one of the four elements a, a−1 , b, b−1 with probability 14 each, and canceling if necessary, that is, if the random factor equals πn−1 . The problem of estimating θ with L(θ, d) equal to 0 if d = θ and equal to 1 otherwise remains invariant under multiplication of X, θ, and d on the left by an arbitrary sequence π−m . . . π−2 π−1 (m = 0, 1, . . .). The invariant procedure that minimizes the maximum risk has risk function R(θ, δ) ≡ 34 . However, there exists a noninvariant procedure with maximum risk 14 . [(i): If Yx is a (possibly randomized) minimax procedure, an invariant minimax  procedure Yx is defined by P (Yx = d) = N P (Ygi x = gi∗ d)/N . i=1 (ii): The better procedure consists in estimating θ to be π1 . . . πk−1 when π1 . . . πk is observed (k ≥ 1), and estimating θ to be a, a−1 , b, b−1 with probability 14 each in case the identity is observed. The estimate will be correct unless the last element of X was canceled, and hence will be correct with probability ≥ 34 .] Section 1.7 Problem 1.12 (i) Let X have probability density pθ (x) with θ one of the values θ1 , . . . , θn , and consider the problem of determining the correct value of θ, so that the choice lies between the n decisions d1 = θ1 , . . . , dn = θn with gain a(θi ) if di = θi and 0 otherwise. Then the Bayes solution (which maximizes the average gain) when θ is a random variable taking on each of the n values with probability 1/n coincides with the maximum-likelihood procedure. (ii) Let X have probability density pθ (x) with 0 ≤ θ ≤ 1. Then the maximum-likelihood estimate is the mode (maximum value) of the a posteriori density of Θ given x when Θ is uniformly distributed over (0, 1). consider the Problem 1.13 (i) Let X1 , . . . , Xn be a sample from N (ξ, σ 2 ), and  problem of deciding between ω0 : ξ < 0 and ω1 : ξ ≥ 0. If x̄ = xi /n and C = (a1 /a0 )2/n , the likelihood-ratio procedure takes decision d0 or d, as √ nx̄ 1 and k = or > k, (1 − C)/C if C < 1. 26 1. The General Decision Problem (ii) For the problem of deciding between ω0 : σ < σ0 and ω1 : σ ≥ σ0 the likelihood ratio procedure takes decision d0 or d, as  (xi − x̄)2 < or > k, nσ02 where k is the smaller root of the equation Cx = ex−1 if C > 1, and the larger root of x = Cex−1 if C < 1, where C is defined as in (i). Section 1.8 Problem 1.14 Admissibility of unbiased procedures. (i) Under the assumptions of Problem 1.10, if among the unbiased procedures there exists one with uniformly minimum risk, it is admissible. (ii) That in general an unbiased procedure with uniformly minimum risk need not be admissible is seen by the following example. Let X have a Poisson distribution truncated at 0, so that Pθ {X = x} = θx e−θ /[x!(1 − e−θ )] for x = 1, 2, . . . . For estimating γ(θ) = e−θ with loss function L(θ, d) = (d − e−θ )2 , there exists a unique unbiased estimate, and it is not admissible. [(ii): The unique unbiased estimate δ0 (x) = (−1)x+1 is dominated by δ1 (x) = 0 or 1 as x is even or odd.] Problem 1.15 Admissibility of invariant procedures. If a decision problem remains invariant under a finite group, and if there exists a procedure δ0 that uniformly minimizes the risk among all invariant procedures, then δ0 is admissible. [This follows from the identity R(θ, δ) = R(ḡθ, g ∗ δg −1 ) and the hint given in Problem 1.11(i).] Problem 1.16 (i) Let X take on the values θ − 1 and θ + 1 with probability 1 each. The problem of estimating θ with loss function L(θ, d) = min(|θ − d|, 1) 2 remains invariant under the transformation gX = X + c, ḡθ = θ + c, g ∗ d = d + c. Among invariant estimates, those taking on the values X − 1 and X + 1 with probabilities p and q (independent of X) uniformly minimize the risk. (ii) That the conclusion of Problem 1.15 need not hold when G is infinite follows by comparing the best invariant estimates of (i) with the estimate δ1 (x) which is X + 1 when X < 0 and X − 1 when X ≥ 0. Section 1.9 Problem 1.17 In n independent trials with constant probability p of success, n let Xi = 1 or 0 as the ith trial is a success or not. Then X i is minimal i=1 sufficient.  [Let T = Xi and suppose that U = f (T ) is sufficient and that f (k1 ) = · · · = f (kr ) = u. Then P {T = t | U = u} depends on p.] Problem 1.18 (i) Let X1 , . . . , Xn be a sample from the uniform distribution U (0, θ), 0 < θ < ∞, and let T = max(X1 , . . . , Xn ). Show that T is sufficient, 1.11. Notes 27 once by using the definition of sufficiency and once by using the factorization criterion and assuming the existence of statistics Yi satisfying (1.17)–(1.19). (ii) Let X1 , . . . , Xn be a sample from the exponential distribution E(a, b) with density (1/b)e−(x−a)/b when x ≥ a (−∞ <  a < ∞, 0 < b). Use the factorization criterion to prove that (min(X1 , . . . , Xn ), n i=1 Xi ) is sufficient for a, b, assuming the existence of statistics Yi satisfying (1.17)–(1.19). Problem 1.19 A statistic T satisfying (1.17)–(1.19) is sufficient if and only if it satisfies (1.20). 1.11 Notes Some of the basic concepts of statistical theory were initiated during the first quarter of the 19th century by Laplace in his fundamental Théorie Analytique des Probabilités (1812), and by Gauss in his papers on the method of least squares. Loss and risk functions are mentioned in their discussions of the problem of point estimation, for which Gauss also introduced the condition of unbiasedness. A period of intensive development of statistical methods began toward the end of the century with the work of Karl Pearson. In particular, two areas were explored in the researches of R. A. Fisher, J. Neyman, and many others: estimation and the testing of hypotheses. The work of Fisher can be found in his books (1925, 1935, 1956) and in the five volumes of his collected papers (1971–1973). An interesting review of Fisher’s contributions is provided by Savage (1976), and his life and work are recounted in the biography by his daughter Joan Fisher Box (1978). Many of Neyman’s principal ideas are summarized in his Lectures and Conferences (1938b). Collections of his early papers and of his joint papers with E. S. Pearson have been published [Neyman (1967) and Neyman and Pearson (1967)], and Constance Reid (1982) has written his biography. An influential synthesis of the work of this period by Cramér appeared in 1946. Further concepts were introduced in Lehmann (1950, 1951ab). More recent surveys of the modern theories of estimation and testing are contained, for example, in the books by Strasser (1985), Stuart and Ord (1991, 1999), Schervish (1995), Shao (1999) and Bickel and Doksum (2001). A formal unification of the theories of estimation and hypothesis testing, which also contains the possibility of many other specializations, was achieved by Wald in his general theory of decision procedures. An account of this theory, which is closely related to von Neumann’s theory of games, is found in Wald’s book (1950) and in those of Blackwell and Girshick (1954), Ferguson (1967), and Berger (1985b). 2 The Probability Background 2.1 Probability and Measure The mathematical framework for statistical decision theory is provided by the theory of probability, which in turn has its foundations in the theory of measure and integration. The present chapter serves to define some of the basic concepts of these theories, to establish some notation, and to state without proof some of the principal results which will be used throughout Chapters 3–9. In the remainder of this chapter, certain special topics are treated in more detail. Basic notions of convergence in probability theory which will be needed for large sample statistical theory are deferred to Section 11.2. Probability theory is concerned with situations which may result in different outcomes. The totality of these possible outcomes is represented abstractly by the totality of points in a space Z. Since the events to be studied are aggregates of such outcomes, they are represented by subsets of Z. The union of two sets C1 , C2 will be denoted by C1 ∪ C2 , their intersection by C1 ∩ C2 , the complement of C by C c = Z − C, and the empty set by 0. The probability P (C) of an event C is a real number between 0 and 1; in particular P (0) = 0 and P (Z) = 1 Probabilities have the property of countable additivity,    if Ci ∩ Cj = 0 for all P Ci = P (Ci ) (2.1) i = j. (2.2) Unfortunately it turns out that the set functions with which we shall be concerned usually cannot be defined in a reasonable manner for all subsets of Z if they are to satisfy (2.2). It is, for example, not possible to give a reasonable definition of “area” for all subsets of a unit square in the plane. 2.1. Probability and Measure 29 The sets for which the probability function P will be defined are said to be “measurable.” The domain of definition of P should include with any set C its complement C c , and with any countable number of events their union. By (2.1), it should also include Z. A class of sets that contains Z and is closed under complementation and countable unions is a σ-field. Such a class is automatically also closed under countable intersections. The starting point of any probabilistic considerations is therefore a space Z, representing the possible outcomes, and a σ-field C of subsets of Z, representing the events whose probability is to be defined. Such a couple (Z, C) is called a measurable space, and the elements of C constitute the measurable sets. A countably additive nonnegative (not necessarily finite) set function µ defined over C and such that µ(0) = 0 is called a measure. If it assigns the value 1 to Z, it is a probability measure. More generally, µ is finite if µ(Z) < ∞ and σ-finite if there exist C1 , C2 , . . . in C (which may always be taken to be mutually exclusive) such that ∪Ci = Z and µ(Ci ) < ∞ for i = 1, 2, . . . . Important special cases are provided by the following examples. Example 2.1.1 (Lebesgue measure) Let Z be the n-dimensional Euclidean space En , and C the smallest σ-field containing all rectangles1 R = {(z1 , . . . , zn ) : ai < zi ≤ bi , i = 1, . . . , n}. The elements of C are called the Borel sets of En . Over C a unique measure µ can be defined, which to any rectangle R assigns as its measure the volume of R, µ(R) = n  (bi − ai ). i=1 The measure µ can be completed by adjoining to C all subsets of sets of measure zero. The domain of µ is thereby enlarged to a σ-field C  , the class of Lebesguemeasurable sets. The term Lebesgue-measure is used for µ both when it is defined over the Borel sets and when it is defined over the Lebesgue-measurable sets. This example can be generalized to any nonnegative set function ν, which is defined and countably additive over the class of rectangles R. There exists then, as before, a unique measure µ over (Z, C) that agrees with ν for all R. This measure can again be completed; however, the resulting σ-field depends on µ and need not agree with the σ-field C  obtained above. Example 2.1.2 (Counting measure) Suppose the Z is countable, and let C be the class of all subsets of Z. For any set C, define µ(C) as the number of elements of C if that number is finite, and otherwise as +∞. This measure is sometimes called counting measure. In applications, the probabilities over (Z, C) refer to random experiments or observations, the possible outcomes of which are the points z ∈ Z. When recording the results of an experiment, one is usually interested only in certain of its 1 If π(z) is a statement concerning certain objects z, then {z : π(z)} denotes the set of all those z for which π(z) is true. 30 2. The Probability Background aspects, typically some counts or measurements. These may be represented by a function T taking values in some space T . Such a function generates in T the σ-field B of sets B whose inverse image C = T −1 (B) = {z : z ∈ Z, T (z) ∈ B} is in C, and for any given probability measure P over (Z, C) a probability measure Q over (T , B ) defined by Q(B) = P (T −1 (B)). (2.3) Frequently, there is given a σ-field B of sets in T such that the probability of B should be defined if and only if B ∈ B. This requires that T −1 (B) ∈ C for all B ∈ B, and the function (or transformation) T from (Z, C) into2 (T , B) is then said to be C-measurable. Another implication is the sometimes convenient restriction of probability statements to the sets B ∈ B even though there may exist sets B ∈ / B for which T −1 (B) ∈ C and whose probability therefore could be defined. Of particular interest is the case of a single measurement in which the function of T is real-valued. Let us denote it by X, and let A be the class of Borel sets on the real line X . Such a measurable real-valued X is called a random variable, and the probability measure it generates over (X , A) will be denoted by P X and called the probability distribution of X. The value this measure assigns to a set A ∈ A will be denoted interchangeably by P X (A) and P (X ∈ A). Since the intervals {x : x ≤ a} are in A, the probabilities F (a) = P (X ≤ a) are defined for all a. The function F , the cumulative distribution function (cdf) of X, is nondecreasing and continuous on the right, and F (−∞) = 0, F (+∞) = 1. Conversely, if F is any function with these properties, a measure can be defined over the intervals by P {a < X ≤ b} = F (b) − F (a). It follows from Example 2.1.1 that this measure uniquely determines a probability distribution over the Borel sets. Thus the probability distribution P X and the cumulative distribution function F uniquely determine each other. These remarks extend to probability distributions over n-dimensional Euclidean space, where the cumulative distribution function is defined by F (a1 , . . . , an ) = P {X1 ≤ a1 , . . . , Xn ≤ an }. In concrete problems, the space (Z, C), corresponding to the totality of possible outcomes, is usually not specified and remains in the background. The real starting point is the set X of observations (typically vector-valued) that are being recorded and which constitute the data, and the associated measurable space (X , A), the sample space. Random variables or vectors that are measurable transformations T from (X , A) into some (T , B) are called statistics. The distribution of T is then given by (2.3) applied to all B ∈ B. With this definition, a statistic is specified by the function T and the σ-field B. We shall, however, adopt the convention that when a function T takes on its values in a Euclidean space, unless otherwise stated the σ-field B of measurable sets will be taken to be the class of 2 The term into indicates that the range of T is in T ; if T (Z) = T , the transformation is said to be from Z onto T . 2.2. Integration 31 Borel sets. It then becomes unnecessary to mention it explicitly or to indicate it in the notation. The distinction between statistics and random variables as defined here is slight. The term statistic is used to indicate that the quantity is a function of more basic observations; all statistics in a given problem are functions defined over the same sample space (X , A). On the other hand, any real-valued statistic T is a random variable, since it has a distribution over (T , B), and it will be referred to as a random variable when its origin is irrelevant. Which term is used therefore depends on the point of view and to some extent is arbitrary. 2.2 Integration According to the convention of the preceding section, a real-valued function f defined over (X , A) is measurable if f −1 (B) ∈ A for every Borel set B on the real line. Such a function f is said to be simple if it takes on only a finite number of values. Let µ be a measure defined over (X , A), and let f be a simple function taking on the distinct values a1 , . . . , am on the sets A1 , . . . , Am , which are in A, since f is measurable. If µ(Ai ) < ∞ when ai = 0, the integral of f with respect to µ is defined by   f dµ = ai µ(Ai ). (2.4) Given any nonnegative measurable function f , there exists a nondecreasing sequence of simple functions fn converging to f . Then the integral of f is defined as   f dµ = lim fn dµ, (2.5) n→∞ which can be shown to be independent of the particular sequence of fn ’s chosen. For any measurable function f its positive and negative parts f + (x) = max[f (x), 0] and f − (x) = max[−f (x), 0] (2.6) are also measurable, and f (x) = f + (x) − f − (x). If the integrals of f + and f − are both finite, then f is said to be integrable, and its integral is defined as    f dµ = f + dµ − f − dµ. If of the two integrals one is finite and one infinite, then the integral of f is defined to be the appropriate infinite value; if both are infinite, the integral is not defined. Example 2.2.1 Let X be the closed interval [a, b], A be the class of Borel sets or of Lebesgue measurable sets in X , and µ be Lebesgue measure. Then the integral b of f with respect to µ is written as a f (x) dx, and is called the Lebesgue integral of f . This integral generalizes the Riemann integral in that it exists and agrees with the Riemann integral of f whenever the latter exists. 32 2. The Probability Background Example 2.2.2 Let X be countable and consist of the points x1 , x2 , . . . ; let A be the class of all subsets of X , and let µ assign measure bi to the point xi . Then f is integrable provided f (xi )bi converges absolutely, and f dµ is given by this sum. Let P X be the probability distribution of a random variable X, and let T be a real-valued statistic. If the function T (x) is integrable, its expectation is defined by  E(T ) = T (x) dP X (x). (2.7) It will be seen from Lemma 2.3.2 in Section 2.3 below that the integration can be carried out alternatively in t-space with respect to the distribution of T defined by (2.3), so that also  E(T ) = t dP T (t). (2.8) The definition (2.5) of the integral permits the basic convergence theorems. Theorem 2.2.1 Fatou’s Lemma Let fn be a sequence of measurable functions such that fn (x) ≥ 0 and fn (x) → f (x), except possibly on a set of x values having µ measure 0. Then,   f dµ ≤ lim inf fn dµ . Theorem 2.2.2 Let fn be a sequence of measurable functions, and let fn (x) → f (x), except possibly on a set of x values having µ measure 0. Then   fn dµ → f dµ if any one of the following conditions holds: (i) Lebesgue Monotone Convergence Theorem: the fn ’s are nonnegative and the sequence is nondecreasing; or (ii) Lebesgue Dominated Convergence Theorem: there exists an integrable function g such that |fn (x)| ≤ g(x) for n and x. or (iii) General Form: there exist gn and g with |fn | ≤ gn , gn (x) → g(x) except possibly on a µ null set, and gn dµ → gdµ. Corollary 2.2.1 Vitali’s Theorem Suppose fn and f are real-valued measurable functions with fn (x) → f (x), except possibly on a set having µ measure 0. Assume   lim sup fn2 (x)dµ(x) ≤ f 2 (x)dµ(x) < ∞ . n 2.2. Integration Then,  33 |fn (x) − f (x)|2 dµ(x) → 0 . For a proof of this result, see Theorem 6.1.3 of Hájek, Sidák, and Sen (1999). For any set A ∈ A, let IA be its indicator function defined by IA (x) = 1 or 0 and let x ∈ A or x ∈ Ac , as (2.9)   f dµ = f IA dµ. (2.10) A If µ is a measure and f a nonnegative measurable function over (X , A), then  ν(A) = f dµ (2.11) A defines a new measure over (X , A). The fact that (2.11) holds for all A ∈ A is expressed by writing dν = f dµ or f= dν . dµ (2.12) Let µ and ν be two given σ-finite measures over (X , A). If there exists a function f satisfying (2.12), it is determined through this relation up to sets of measure zero, since   f dµ = g dµ for all A ∈ A A A 3 implies that f = g a.e. µ. Such an f is called the Radon–Nikodym derivative of ν with respect to µ, and in the particular case that ν is a probability measure, the probability density of ν with respect to µ. The question of existence of a function f satisfying (2.12) for given measures µ and ν is answered in terms of the following definition. A measure ν is absolutely continuous with respect to µ if µ(A) = 0 implies ν(A) = 0. Theorem 2.2.3 (Radon–Nikodym) If µ and ν are σ-finite measures over (X , A), then there exists a measurable function f satisfying (2.12) if and only if ν is absolutely continuous with respect to µ. The direct (or Cartesian) product A × B of two sets A and B is the set of all pairs (x, y) with x ∈ A, y ∈ B. Let (X , A) and (Y, B) be two measurable spaces, and let A × B be the smallest σ-field containing all sets A × B with A ∈ A and B ∈ B. If µ and ν are two σ-finite measures over (X , A) and (Y, B) respectively, 3 A statement that holds for all points x except possibly on a set of µ-measure zero is said to hold almost everywhere µ, abbreviated a.e. µ; or to hold a.e. (A, µ) if it is desirable to indicate the σ-field over which µ is defined. 34 2. The Probability Background then there exists a unique measure λ = µ × ν over (X × Y, A × B), the product of µ and ν, such that for any A ∈ A, B ∈ B, λ(A × B) = µ(A)ν(B). (2.13) Example 2.2.3 Let X , Y be Euclidean spaces of m and n dimensions, and let A, B be the σ-fields of Borel sets in these spaces. Then X × Y is an (m + n)dimensional Euclidean space, and A × B the class of its Borel sets. Example 2.2.4 Let Z = (X, Y ) be a random variable defined over (X × Y, A × B), and suppose that the random variables X and Y have distributions P X , P Y over (X , A) and (Y, B). Then X and Y are said to be independent if the probability distribution P Z of Z is the product P X × P Y . In terms of these concepts the reduction of a double integral to a repeated one is given by the following theorem. Theorem 2.2.4 (Fubini) Let µ and ν be σ-finite measures over (X , A) and (Y, B) respectively, and let λ = µ × ν. If f (x, y) is integrable with respect to λ, then (i) for almost all (ν) fixed y, the function f (x, y) is integrable with respect to µ, (ii) the function f (x, y) dµ(x) is integrable with respect to ν, and     f (x, y) dλ(x, y) = f (x, y) dµ(x) dν(y). (2.14) 2.3 Statistics and Subfields According to the definition of Section 2.1, a statistic is a measurable transformation T from the sample space (X , A) into a measurable space (T , B). Such a transformation induces in the original sample space the subfield4   A0 = T −1 (B) = T −1 (B) : B ∈ B . (2.15) Since the set T −1 [T (A)] contains A but is not necessarily equal to A, the σ-field A0 need not coincide with A and hence can be a proper subfield of A. On the other hand, suppose for a moment that T = T (X ), that is, that the transformation T is onto rather than into T . Then   T T −1 (B) = B for all B ∈ B, (2.16) so that the relationship A0 = T −1 (B) establishes a 1:1 correspondence between the sets of A0 and B, which is an isomorphism—that is, which preserves the set operations of intersection, union, and complementation. For most purposes it is therefore immaterial whether one works in the space (X , A0 ) or in (T , B). These generate two equivalent classes of events, and therefore of measurable functions, possible decision procedures, etc. If the transformation T is only into T , the above 4 We shall use this term in place of the more cumbersome “sub-σ-field.” 2.3. Statistics and Subfields 35 1:1 correspondence applies to the class B of subsets of T  = T (X ) which belong to B, rather than to B itself. However, any set B ∈ B is equivalent to B  = B ∩ T  in the sense that any measure over (X , A) assigns the same measure to B  as to B. Considered as classes of events, A0 and B therefore continue to be equivalent, with the only difference that B contains several (equivalent) representations of the same event. As an example, let X be the real line and A the class of Borel sets, and let T (x) = x2 . Let T be either the positive real axis or the whole real axis, and let B be the class of Borel subsets of T . Then A0 is the class of Borel sets that are symmetric with respect to the origin. When considering, for example, real-valued measurable functions, one would, when working in T -space, restrict attention to measurable function of x2 . Instead, one could remain in the original space, where the restriction would be to the class of even measurable functions of x. The equivalence is clear. Which representation is more convenient depends on the situation. That the correspondence between the sets A0 = T −1 (B) ∈ A0 and B ∈ B establishes an analogous correspondence between measurable functions defined over (X , A0 ) and (T , B) is shown by the following lemma. Lemma 2.3.1 Let the statistic T from (X , A) into (T , B) induce the subfield A0 . Then a real-valued A-measurable function f is A0 -measurable if and only if there exists a B-measurable function g such that f (x) = g[T (x)] for all x. Proof. Suppose first that such a function g exists. Then the set {x : f (x) < r} = T −1 ({t : g(t) < r}) is in A0 , and f is A0 -measurable. Conversely, if f is A0 -measurable, then the sets   i i+1 Ain = x : n < f (x) ≤ n , i = 0, ±1, ±2, . . . , 2 2 are (for fixed n) disjoint sets in A0 whose union is X , and there exist Bin ∈ B such that Ain = T −1 (Bin ). Let  ∗ Bin = Bin ∩ { Bjn }c . j=i Since Ain and Ajn are mutually exclusive for i = j, the set T −1 (Bin ∩ Bjn ) is ∗ c ∗ empty and so is the set T −1 (Bin ∩ {Bin } ). Hence, for fixed n, the sets Bin are ∗ disjoint, and still satisfy Ain = T −1 (Bin ). Defining fn (x) = i 2n if x ∈ Ain , i = 0 ± 1, ±2, . . . , one can write fn (x) = gn [T (x)], 36 2. The Probability Background where ⎧ ⎨ gn (t) = i 2n ⎩ 0 ∗ for t ∈ Bin , i = 0 ± 1, ±2, . . . , otherwise. Since the functions gn are B-measurable, the set B on which gn (t) converges to a finite limit is in B. Let R = T (X ) be the range of T . Then for t ∈ R, lim gn [T (x)] = lim fn (x) = f (x) for all x ∈ X so that R is contained in B. Therefore, the function g defined by g(t) = lim gn (t) for t ∈ B and g(t) = 0 otherwise possesses the required properties. The relationship between integrals of the functions f and g above is given by the following lemma. Lemma 2.3.2 Let T be a measurable transformation from (X , A) into (T , B), µ a σ-finite measure over (X , A), and g a real-valued measurable function of t. If µ∗ is the measure defined over (T , B) by   for all B ∈ B, (2.17) µ∗ (B) = µ T −1 (B) then for any B ∈ B,   g[T (x)] dµ(x) = T −1 (B) g(t) dµ∗ (t) (2.18) B in the sense that if either integral exists, so does the other and the two are equal. Proof. Without loss of generality let B be the whole space T . If g is the indicator of a set B0 ∈ B, the lemma holds, since the left- and right-hand sides of (2.18) reduce respectively to µ[T −1 (B0 )] and µ∗ (B0 ), which are equal by the definition of µ∗ . If follows that (2.18) holds successively for all simple functions, for all nonnegative measurable functions, and hence finally for all integrable functions. 2.4 Conditional Expectation and Probability If two statistics induce the same subfield A0 , they are equivalent in the sense of leading to equivalent classes of measurable events. This equivalence is particularly relevant to considerations of conditional probability. Thus if X is normally distributed with zero mean, the information carried by the statistics |X|, X 2 , 2 2 2 e−X , and so on, is the same. Given that |X| = t, X 2 = t2 , e−X = e−t , it follows that X is ±t, and any reasonable definition of conditional probability will assign probability 12 to each of these values. The general definition of conditional probability to be given below will in fact involve essentially only A0 and not the range space T of T . However, when referred to A0 alone the concept loses much of its intuitive meaning, and the gap between the elementary definition and that of the general case becomes unnecessarily wide. For these reasons it is frequently more convenient to work with a particular representation of a statistic, involving a definite range space (T , B). 2.4. Conditional Expectation and Probability 37 Let P be a probability measure over (X , A), T a statistic with range space (T , B), and A0 the subfield it induces. Consider a nonnegative function f which is integrable (A, P ), that is A-measurable and P -integrable. Then A f dP is defined for all A ∈ A and therefore for all A0 ∈ A0 . If follows from the Radon–Nikodym theorem (Theorem 2.2.3) that there exists a function f0 which is integrable (A0 , P ) and such that   f dP = f0 dP for all A0 ∈ A0 , (2.19) A0 A0 and that f0 is unique (A0 , P ). By Lemma 2.3.1, f0 depends on x only through T (x). In the example of a normally distributed variable X with zero mean, and T = X 2 , the function f0 is determined by (2.19) holding for all sets A0 that are symmetric with respect to the origin, so that f0 (x) = 12 [f (x) + f (−x)]. The function f0 defined through (2.19) is determined by two properties: (i) Its average value over any set A0 with respect to P is the same as that of f ; (ii) It depends on x only through T (x) and hence is constant on the sets Dx over which T is constant. Intuitively, what one attempts to do in order to construct such a function is to define f0 (x) as the conditional P -average of f over the set Dx . One would thereby replace the single averaging process of integrating f represented by the left-hand side with a two-stage averaging process such as an iterated integral. Such a construction can actually be carried out when X is a discrete variable and in the regular case considered in Section 1.9; f0 (x) is then just the conditional expectation of f (X) given T (x). In general, it is not clear how to define this conditional expectation directly. Since it should, however, possess properties (i) and (ii), and since these through (2.19) determine f0 uniquely (A0 , P ), we shall take f0 (x) of (2.19) as the general definition of the conditional expectation E[f (X) | T (x)]. Equivalently, if f0 (x) = g[T (x)], one can write E[f (X) | t] = E[f (X) | T = t] = g(t), so that E[f (X) | t] is a B-measurable function defined up to equivalence (B, P T ). In the relationship of integrals given in Lemma 2.3.2, if µ = P X , then µ∗ = P T , and it is seen that the function g can be defined directly in terms of f through   X f (x) dP (x) = g(t) dP T (t) for all B ∈ B, (2.20) T −1 (B) B which is equivalent to (2.19). So far, f has been assumed to be nonnegative. In the general case, the conditional expectation of f is defined as E[f (X) | t] = E[f + (X) | t] − E[f − (X) | t]. Example 2.4.1 (Order statistics) Let X1 , . . . , Xn be identically and independently distributed random variables with continuous distribution function, and let T (x1 , . . . , xn ) = (x(1) , . . . , x(n) ) 38 2. The Probability Background where x(1) ≤ · · · ≤ x(n) denote the ordered x’s. Without loss of generality one can restrict attention to the points with x(1) < · · · < x(n) , since the probability of two coordinates being equal is 0. Then X is the set of all n-tuples with distinct coordinates, T the set of all ordered n-tuples, and A and B are the classes of Borel subsets of X and T . Under T −1 the set consisting of the single point a = (a1 , . . . , an ) is transformed into the set consisting of the n! points (ai1 , . . . , ain ) that are obtained from a by permuting the coordinates in all possible ways. It follows that A0 is the class of all sets that are symmetric in the sense that if A0 contains a point x = (x1 , . . . , xn ), then it also contains all points (xi1 , . . . , xin ). For any integrable function f , let 1  f0 (x) = f (xi1 , . . . , xin ), n! where the summation extends over the n! permutations of (x1 , . . . , xn ). Then f0 is A0 -measurable, since it is symmetric in its n arguments. Also   f (x1 , . . . , xn ) dP (x1 ) . . . dP (xn ) = f (xi1 , . . . , xin ) dP (x1 ) . . . dP (xn ), A0 A0 so that f0 satisfies (2.19). It follows that f0 (x) is the conditional expectation of f (X) given T (x). The conditional expectation of f (X) given the above statistic T (x) can also be found without assuming the X’s to be identically and independently distributed. Suppose that X has a density h(x) with respect to a measure µ (such as Lebesgue measure), which is symmetric in the variables x1 , . . . , xn in the sense that for any A ∈ A it assigns to the set {x : (xi1 , . . . , xin ) ∈ A} the same measure for all permutations (i1 , . . . , in ). Let  f (xi1 , . . . , xin )h(xi1 , . . . , xin )  f0 (x1 , . . . , xn ) = ; h(xi1 , . . . , xin ) here and in the sums below the summation extends over the n! permutations of (x1 , . . . , xn ). The function f0 is symmetric in its n arguments and hence A0 measurable. For any symmetric set A0 , the integral  f0 (x1 , . . . , xn )h(xj1 , . . . , xjn ) dµ(x1 , . . . , xn ) A0 has the same value for each permutation (xj1 , . . . , xjn ), and therefore  f0 (x1 , . . . , xn )h(x1 , . . . , xn ) dµ(x1 , . . . , xn ) A0  1  = f0 (x1 , . . . , xn ) h(xi1 , . . . , xin ) dµ(x1 , . . . , xn ) n! A  0 = f (x1 , . . . , xn )h(x1 , . . . , xn ) dµ(x1 , . . . , xn ). A0 It follows that f0 (x) = E[f (X) | T (x)]. Equivalent the statistic T (x) = (x(1) , . . . , x(n) ), the set of order statistics, is  to   U (x) = xi , x2i , . . . , xn . This is an immediate consequence of the fact, i to be shown below, that if T (x0 ) = t0 and U (x0 ) = u0 , then     T −1 t0 = U −1 u0 = S 2.4. Conditional Expectation and Probability 39     where t0 and u0 denote the sets consisting of the single point t0 and u0 respectively, and where S consists of the totality of points x = (x1 , . . . , xn ) obtained by permutingthecoordinates of x0 = (x01 , . . . , x0n ) in all possible ways. That T −1 t0 = S is obvious. To see the corresponding fact for U −1 , let ⎛ ⎞    xi , xi xj , xi xj xk , . . . , x1 x2 · · · xn ⎠ , V (x) = ⎝ i i 0 for all y. Then under P1 the marginal distribution of T and a version of the conditional distribution of Y given t are given by   Y |t dP1T (t) = b(t) a(y) dP0 (y) dP0T (t) and Y |t dP1 Y |t (y) = a(y) dP0 (y) Y |t a(y  ) dP0 Y (y  ) . Proof. The first statement of the lemma follows from the equation P1 {T ∈ B} = E1 [IB (T )] = = E0 [IB (T )a(Y )b(T )]    Y |t b(T ) a(y) dP0 (y) dP0T (t). B Y To check the second statement, one need only show that for any integrable f the expectation E1 f (Y, T ) satisfies (2.28), which is immediate. The denominator of Y |t dP1 is positive, since a(y) > 0 for all y. 2.6 Characterization of Sufficiency We can now generalize the definition of sufficiency given in Section 1.9. If P = {Pθ , θ ∈ Ω} is any family of distributions defined over a common sample space (X , A), a statistic T is sufficient for P (or for θ) if for each A in A there exists a determination of the conditional probability function Pθ (A | t) that is independent of θ. As an example suppose that X1 , . . . , Xn are identically and independently distributed with continuous distribution function Fθ , θ ∈ Ω. Then it follows from Example 2.4.1 that the set of order statistics T (X) = (X(1) , . . . , X(n) ) is sufficient for θ. Theorem 2.6.1 If X is Euclidean, and if the statistic T is sufficient for P, then there exist determinations of the conditional probability distributions Pθ (A | t) which are independent of θ and such that for each fixed t, Pθ (A | t) is a probability measure over A. Proof. This is seen from the proof of Theorem 2.5.1. By the definition of sufficiency one can, for each rational number r, take the functions F (r, t) to be independent of θ, and the resulting conditional distributions will then also not depend on θ. 2.6. Characterization of Sufficiency 45 In Chapter 1 the definition of sufficiency was justified by showing that in a certain sense a sufficient statistic contains all the available information. In view of Theorem 2.6.1 the same justification applies quite generally when the sample space is Euclidean. With the help of a random mechanism one can then construct from a sufficient statistic T a random vector X  having the same distribution as the original sample vector X. Another generalization of the earlier result, not involving the restriction to a Euclidean sample space, is given in Problem 2.13. The factorization criterion of sufficiency, derived in Chapter 1, can be extended to any dominated family of distributions, that is, any family P = {Pθ , θ ∈ Ω} possessing probability densities pθ with respect to some σ-finite measure µ over (X , A). The proofof this statement is based on the existence of a probability distribution λ = ci Pθi (Theorem 2.2.3 of the Appendix), which is equivalent to P in the sense that for any A ∈ A λ(A) = 0 if and only if Pθ = 0 for all θ ∈ Ω. (2.29) Theorem 2.6.2 Let P = {Pθ , θ ∈ Ω} be a dominated family of probability distributions over (X , A), and let λ = ci Pθi satisfy (2.29). Then a statistic T with range space (T , B) is sufficient for P if and only if there exist nonnegative B-measurable functions gθ (t) such that dPθ (x) = gθ [T (x)] dλ(x) (2.30) for all θ ∈ Ω. Proof. Let A0 be the subfield induced by T , and suppose that T is sufficient for θ. Then for all θ ∈ Ω, A0 ∈ A0 , and A ∈ A  P (A | T (x)) dPθ (x) = Pθ (A ∩ A0 ); and since λ =  A0 ci Pθi ,  P (A | T (x)) dλ(x) = λ(A ∩ A0 ), A0 so that P (A | T (x)) serves as conditional probability function also for λ. Let gθ (T (x)) be the Radon–Nikodym derivative dPθ (x)/dλ(x) for (A0 , λ). To prove (2.30) it is necessary to show that gθ (T (x)) is also the derivative of Pθ for (A, λ). If A0 is put equal to X in the first displayed equation, this follows from the relation   Pθ (A) = P (A | T (x)) dPθ (x) = Eλ [IA (x) | T (x)] dPθ (x)  = Eλ [IA (x) | T (x)] gθ (T (x)) dλ(x)  = Eλ [gθ (T (x))IA (x) | T (x)] dλ(x)   gθ (T (x)) dλ(x). = gθ (T (x))IA (x) dλ(x) = A Here the second equality uses the fact, established at the beginning of the proof, that P (A | T (x)) is also the conditional probability for λ; the third equality holds 46 2. The Probability Background because the function being integrated is A0 -measurable and because dPθ = gθ dλ for (A0 , λ); the fourth is an application of Lemma 2.4.1(ii); and the fifth employs the defining property of conditional expectation. Suppose conversely that (2.30) holds. We shall then prove that the conditional probability function Pλ (A | t) serves as a conditional probability function for all P ∈ P. Let gθ (T (x)) = dPθ (x)/ dλ(x) on A and for fixed A and θ define a measure ν over A by the equation dν = IA dPθ . Then over A0 , dν(x)/ dPθ (x) = Eθ [IA (X) | T (x)], and therefore dν(x) = Pθ [A | T (x)]gθ (T (x)) dλ(x) over A0 . On the other hand, dν(x)/dλ(x) = IA (x)gθ (T (x)) over A, and hence dν(x) dλ(x) = Eλ [IA (X)gθ (T (X)) | T (x)] = Pλ [A | T (x)]gθ (T (x)) over A0 . It follows that Pλ (A | T (x))gθ (T (x)) = Pθ (A | T (x))gθ (T (x)) (A0 , λ) and hence (A0 , Pθ ). Since gθ (T (x)) = 0 (A0 , Pθ ), this shows that Pθ (A | T (x)) = Pλ (A | T (x)) (A0 , Pθ ), and hence that Pλ (A | T (x)) is a determination of Pθ (A | T (x)). Instead of the above formulation, which explicitly involves the distribution λ, it is sometimes more convenient to state the result with respect to a given dominating measure µ. Corollary 2.6.1 (Factorization theorem) If the distributions Pθ of P have probability densities pθ = dPθ /dµ with respect to a σ-finite measure µ, then T is sufficient for P if and only if there exist nonnegative B-measurable functions gθ on T and a nonnegative A-measurable function h on X such that pθ (x) = gθ [T (x)]h(x) (A, µ). (2.31)  Proof. Let λ = ci Pθi satisfy (2.29). Then if T is sufficient, (2.31) follows from (2.30) with h = dλ/dµ. Conversely, if (2.31) holds,  dλ(x) = ci gθi [T (x)]h(x) dµ(x) = k[T (x)]h(x) dµ(x) and therefore dPθ (x) = gθ∗ (T (x)) dλ(x) where gθ∗ (t) = gθ (t)/k(t) when k(t) > 0 and may be defined arbitrarily when k(t) = 0. For extensions of the factorizations theorem to undominated families, see Ghosh, Morimoto, and Yamada (1981) and the literature cited there. 2.7 Exponential Families An important family of distributions which admits a reduction by means of sufficient statistics is the exponential family, defined by probability densities of the form ! k "  pθ (x) = C(θ) exp Qj (θ)Tj (x) h(x) (2.32) j=1 2.7. Exponential Families 47 with respect to a σ-finite measure µ over a Euclidean sample space (X , A). Particular cases are the distributions of a sample X = (X1 , . . . , Xn ) from a binomial, Poisson, or normal distribution. In the binomial case, for example, the density (with respect to counting measure) is       p n x n n−x n = (1 − p) exp x log p (1 − p) . 1−p x x Example 2.7.1 If Y1 , . . . , Yn are independently distributed, each with density (with respect to Lebesgue measure)    y [(f /2)−1] exp −y/ 2σ 2 pσ (y) = , y > 0, (2.33) (2σ 2 )f /2 Γ(f /2) then the joint distribution of the Y ’s constitutes an exponential family. For σ = 1, (2.33) is the density of the χ2 -distribution  with f degrees of freedom; in particular for f an integer this is the density of fj=1 Xj2 , where the X’s are a sample from the normal distribution N (0, 1). Example 2.7.2 Consider n independent trials, each of them resulting in one of the s outcomes E1 , . . . , Es with probabilities p1 , . . . , ps respectively. If Xij is 1 when the outcome of the ith trial is Ej and 0 otherwise, the joint distribution of the X’s is  x  x  P {X11 = x11 , . . . , Xns } = p1 i1 p2 i2 · · · ps xis ,  where all  xij = 0 or 1 and j xij = 1. this forms an exponential family with n Tj (x) = x (j = 1, . . . , s − 1). The joint distribution of the T ’s is the ij i=1 multinomial distribution M (n; p1 , . . . , ps ) given by P {T1 = t1 , . . . , Ts−1 = ts−1 } = (2.34) n! t1 ! . . . ts−1 !(n − t1 − · · · − ts−1 )! s−1 ×pt11 . . . ps−1 (1 − p1 − · · · − ps−1 )n−t1 −···−ts−1 . t If X1 , . . . , Xn is a sample from a distribution with density (2.32), the joint distribution  of the X’s constitutes an exponential family with the sufficient statistics n i=1 Tj (Xi ), j = 1, . . . , k. Thus there exists a k-dimensional sufficient statistic for (X1 , . . . , Xn ) regardless of the sample size. Suppose conversely that X1 , . . . , Xn is a sample from a distribution with some density pθ (x) and that the set over which this density is positive is independent of θ. Then under regularity assumptions which make the concept of dimensionality meaningful, if there exists a k-dimensional sufficient statistic with k < n, the densities pθ (x) constitute an exponential family. For proof of this result, see Darmois (1935), Koopman (1936) and Pitman (1937). Regularity conditions of the result are discussed in Barankin and Maitra (1963), Brown (1964), Barndorff–Nielsen and Pedersen (1968), and Hipp (1974). 48 2. The Probability Background Employing a more natural parametrization and absorbing the factor h(x) into µ, we shall write an exponential family in the form dPθ (x) = pθ (x) dµ(x) with ! k "  θj Tj (x) . (2.35) pθ (x) = C(θ) exp j=1 For suitable choice of the constant C(θ), the right-hand side of (2.35) is a probability density provided its integral is finite. The set Ω of parameter points θ = (θ1 , . . . , θk ) for which this is the case is the natural parameter space of the exponential family (2.35). Optimum tests of certain hypotheses concerning any θj are obtained in Chapter 4. We shall now consider some properties of exponential families required for this purpose. Lemma 2.7.1 The natural parameter space of an exponential family is convex. Proof. Let (θ1 , . . . , θk ) and (θ1 , . . . , θk ) be two parameter points for which the integral of (2.35) is finite. Then by Hölder’s inequality,  #  $  exp αθj + (1 − α)θj Tj (x) dµ(x)  ≤ exp # α  1−α $ $ #  θj Tj (x) dµ(x) <∞ exp θj Tj (x) dµ(x) for any 0 < α < 1. If the convex set Ω lies in a linear space of dimension < k, then (2.35) can be rewritten in a form involving fewer than k components of T . We shall therefore, without loss of generality, assume Ω to be k-dimensional. It follows from the factorization theorem that T (x) = (T1 (x), . . . , Tk (x)) is sufficient for P = {Pθ , θ ∈ Ω}. Lemma 2.7.2 Let X be distributed according to the exponential family ! r " s   T θi Ui (x) + ϑj Tj (x) dµ(x). dPθ,ϑ (x) = C(θ, ϑ) exp i=1 j=1 Then there exist measures λθ and νt over s- and r-dimensional Euclidean space respectively such that (i) the distribution of T = (T1 , . . . , Ts ) is an exponential family of the form  s   T dPθ,ϑ (t) = C(θ, ϑ) exp ϑj tj dλθ (t), (2.36) j=1 (ii) the conditional distribution of U = (U1 , . . . , Ur ) given T = t is an exponential family of the form  r   U |t dPθ· (u) = C(θ) exp θi ui dνt (u), (2.37) i=1 and hence in particular is independent of ϑ. 2.7. Exponential Families 49 Proof. Let (θ0 , ϑ0 ) be a point of the natural parameter space, and let µ∗ = PθX0 ,ϑ0 . Then dPθX0 ,ϑ0 (x) = C(θ, ϑ) C(θ0 , ϑ0 ) " ! r s   0 0 (θi − θi )Ui (x) + (ϑj − ϑj )Tj (x) dµ∗ (x), × exp i=1 j=1 and the result follows from Lemma 2.5.1, with " ! ! r "      U |t 0 0 dλθ (t) = exp − ϑ i ti (θi − θi )ui dPθ0 ,ϑ0 (u) dPθT0 ,ϑ0 (t) exp i=1 and    U |t θi0 ui dPθ0 ,ϑ0 (u). dνt (u) = exp − Theorem 2.7.1 Let φ be any function on (X , A) for which the integral ! k "   φ(x) exp θj Tj (x) dµ(x) (2.38) j=1 considered as a function of the complex variables θj = ξj + iηj (j = 1, . . . , k) exists for all (ξ1 , . . . , ξk ) ∈ Ω and is finite. Then (i) the integral is an analytic function of each of the θ’s in the region R of parameter points for which (ξ1 , . . . , ξk ) is an interior point of the natural parameter space Ω; (ii) the derivatives of all orders with respect to the θ’s of the integral (2.38) can be computed under the integral sign. Proof. Let (ξ1 , . . . , ξk ) be any fixed point in the interior of Ω, and consider one of the variables in question, say θ1 . Breaking up the factor    φ(x) exp ξ20 + iη20 T2 (x) + · · · + ξk0 + iηk0 Tk (x) into its real and complex part and each of these into its positive and negative part, and absorbing this factor in each of the four terms thus obtained into the measure µ, one sees that as a function of θ1 the integral (2.38) can be written as   exp [θ1 T1 (x)] dµ1 (x) − exp [θ1 T1 (x)] dµ2 (x)   + i exp [θ1 T1 (x)] dµ3 (x) − i exp [θ1 T1 (x)] dµ4 (x). It is therefore sufficient to prove the result for integrals of the form  ψ(θ1 ) = exp [θ1 T1 (x)] dµ(x). Since (ξ10 , . . . , ξk0 ) is in the interior of Ω, there exists δ > 0 such that ψ(θ1 ) exists and is finite for all θ1 with |ξ1 − ξ10 | ≤ δ. Consider the difference    exp [θ1 T1 (x)] − exp θ10 T1 (x) ψ(θ1 ) − ψ(θ10 ) = dµ(x). θ1 − θ10 θ1 − θ10 50 2. The Probability Background The integrand can be written as exp   θ10 T1 (x) ! "   exp (θ1 − θ10 )T1 (x) − 1 . θ1 − θ10 Applying to the second factor the inequality % % % % % exp(az) − 1 % % ≤ exp(δ|a|) % % % z δ % % for |z| ≤ δ, the integrand is seen to be bounded above in absolute value by % % % % % % %   0  %%  0  0 1% 1% % % exp θ1 T1 + δ|T1 | % ≤ % exp θ1 + δ T1 + exp θ1 − δ T1 % % % δ% δ% for |θ1 −θ10 | ≤ δ. Since the right-hand side integrable, it follows from the Lebesgue dominated-convergence theorem [Theorem 2.2.2(ii)] that for any sequence of (n) points θ1 tending to θ10 , the difference quotient of ψ tends to    T1 (x) exp θ10 T1 (x) dµ(x). This completes the proof of (i), and proves (ii) for the first derivative. The proof for the higher derivatives is by induction and is completely analogous. 2.8 Problems Section 2.1 Problem 2.1 Monotone class. A class F of subsets of a space is a field if it contains the whole space and is closed under complementation and under finite unions; a class M is monotone if the union and intersection of every increasing and decreasing sequence of sets of M is again in M. The smallest monotone class M0 containing a given field F coincides with the smallest σ-field A containing F . [One proves first that M0 is a field. To show, for example, that A ∩ B ∈ M0 when A and B are in M0 , consider, for a fixed set A ∈ F , the class MA of all B in M0 for which A ∩ B ∈ M0 . Then MA is a monotone class containing F, and hence MA = M0 . Thus A ∩ B ∈ MA for all B. The argument can now be repeated with a fixed set B ∈ M0 and the class MB of sets A in M0 for which A ∩ B ∈ M0 . Since M0 is a field and monotone, it is a σ-field containing F and hence contains A. But any σ-field is a monotone class so that also M0 is contained in A.] Section 2.2 Problem 2.2 Prove Corollary 2.2.1 using Theorems 2.2.1 and 2.2.2. Problem 2.3 Radon–Nikodym derivatives. 2.8. Problems 51 (i) If λ and µ are σ-finite measures over (X , A) and µ is absolutely continuous with respect to λ, then   dµ f dµ = f dλ dλ for any µ-integrable function f . (ii) If λ, µ, and ν are σ-finite measures over (X , A) such that ν is absolutely continuous with respect to µ and µ with respect to λ, then dν dν dµ = dλ dµ dλ a.e. λ. (iii) If µ and ν are σ-finite measures,, which are equivalent in the sense that each is absolutely continuous with respect to the other, then −1 dν dµ a.e. µ, ν. = dµ dν (iv) ∞ If µk , k = 1, 2, . . . , and µ are finite measures over (X , A) such that k=1 µk (A) = µ(A) for all A ∈ A, and if the µk are absolutely continuous with respect to a σ-finite measure λ, then µ is absolutely continuous with respect to λ, and d n  k=1 dλ µk n  dµk = , dλ k=1 d lim n→∞ n  k=1 dλ µk = dµ dλ a.e. λ. [(i): The equation in question holds when f is the indicator of a set, hence when f is simple, and therefore for all integrable f . (ii): Apply (i) with f = dν/dµ.] Problem 2.4 If f (x) > 0 for all x ∈ S and µ is σ-finite, then S f dµ = 0 implies µ(S) = 0.  [Let Sn be the subset of S on which f (x) ≥ 1/n Then µ(S) ≤ µ(Sn ) and µ(Sn ) ≤ n Sn f dµ ≤ n S f dµ = 0.] Section 2.3 Problem 2.5 Let (X , A) be a measurable space, and A0 a σ-field contained in A. Suppose that for any function T , the σ-field B is taken as the totality of sets B such that T −1 (B) ∈ A. Then it is not necessarily true that there exists a function T such that T −1 (B) ∈ A0 . [An example is furnished by any A0 such that for all x the set consisting of the single point x is in A0 .] Section 2.4 Problem 2.6 that (i) Let P be any family of distributions X = (X1 , . . . , Xn ) such P {(Xi , Xi+1 , . . . , Xn , X1 , . . . , Xi−1 ) ∈ A} = P {(X1 , . . . , Xn ) ∈ A} 52 2. The Probability Background for all Borel sets A and all i = 1, . . . , n. For any sample point (x1 , . . . , xn ) define (y1 , . . . , yn ) = (xi , xi+1 , . . . , xn , x1 , . . . , xi−1 ), where xi = x(1) = min(x1 , . . . , xn ). Then the conditional expectation of f (X) given Y = y is f0 (y1 , . . . , yn ) = 1 [f (y1 , . . . , yn ) + f (y2 , . . . , yn , y1 ) n + · · · + f (yn , y1 , . . . , yn−1 )]. (ii) Let G = {g1 , . . . , gr } be any group of permutations of the coordinates x1 , . . . , xn of a point x in n-space, and denote by gx the point obtained by applying g to the coordinates of x. Let P be any family of distributions P of X = (X1 , . . . , Xn ) such that P {gX ∈ A} = P {X ∈ A} g ∈ G. for all (2.39) For any point x let t = T (x) be any rule that selects a unique point from the r points gk x, k = 1, . . . , r (for example the smallest first coordinate if this defines it uniquely, otherwise also the smallest second coordinate, etc.). Then E[f (X) | t] = r 1 f (gk t). r k=1 (iii) Suppose that in (ii) the distributions P do not satisfy the invariance condition (2.39) but are given by dP (x) = h(x) dµ(x), where µ is invariant in the sense that µ{x : gx ∈ A} = µ(A). Then r  E[f (X) | t] = f (gk t)h(gk t) k=1 r  . h(gk t) k=1 Section 2.5 Problem 2.7 Prove Theorem 2.5.1 for the case of an n-dimensional sample space. [The condition that the cumulative distribution function is nondecreasing is replaced by P {x1 < X1 ≤ x1 , . . . , xn < Xn ≤ xn } ≥ 0; the condition that it is continuous on the right can be stated as limm→∞ F (x1 + 1/m, . . . , xn + 1/m) = F (x1 , . . . , xn ).] Problem 2.8 Let X = Y × T , and suppose that P0 , P1 are two probability distributions given by dP0 (y, t) = f (y)g(t) dµ(y) dν(t), dP1 (y, t) = h(y, t) dµ(y) dν(t), where h(y, t)/f (y)g(t) < ∞. Then under P1 the probability density of Y with respect to µ is %   h(y, T ) %% pY1 (y) = f (y)E0 Y = y . f (y)g(T ) % 2.8. Problems [We have  pY1 (y) =  h(y, t) dν(t) = f (y) T T 53 h(y, t) g(t) dν(t).] f (y)g(t) Section 2.6 Problem 2.9 Symmetric distributions. (i) Let P be any family of distributions of X = (X1 , . . . , Xn ) which are symmetric in the sense that P {(Xi1 , . . . , Xin ) ∈ A} = P {(X1 , . . . , Xn ) ∈ A} for all Borel sets A and all permutations (i1 , . . . , in ) of (1, . . . , n). Then the statistic T of Example 2.4.1 is sufficient for P, and the formula given in the first part of the example for the conditional expectation E[f (X) | T (x)] is valid. (ii) The statistic Y of Problem 2.6 is sufficient. (iii) Let X1 , . . . , Xn be identically and independently distributed according to a continuous distribution P ∈ P, and suppose that the distributions of P are symmetric with respect to the origin. Let Vi = |Xi | and Wi = V(i) . Then (W1 , . . . , Wn ) is sufficient for P. Problem 2.10 Sufficiency of likelihood ratios. Let P0 , P1 be two distributions with densities p0 , p1 . Then T (x) = p1 (x)/p0 (x) is sufficient for P = {P0 , P1 }. [This follows from the factorization criterion by writing p1 = T · p0 , p0 = 1 · p0 .] Problem 2.11 Pairwise sufficiency. A statistic T is pairwise sufficient for P if it is sufficient for every pair of distributions in P. (i) If P is countable and T is pairwise sufficient for P, then T is sufficient for P. (ii) If P is a dominated family and T is pairwise sufficient for P, then T is sufficient for P. [(i): Let  P = {P0 , P1 , . . .}, and let A0 be the sufficient subfield induced by T . Let λ = ci Pi (ci > 0) be equivalent to P. For each j = 1, 2, . . . the probability measure λj that is proportional to (c0 /n)P0 + cj Pj is equivalent to {P0 , Pj }. Thus by pairwise sufficiency, the derivative fj = dP0 /[(c0 /n) dP0 + cj dPj ] is A0 -measurable. Let Sj = {x : fj (x) = 0} and S = n j=1 Sj . Then S ∈ A0 ,   −1 P0 (S) = 0, and on X − S the derivative dP0 /d n c P equals ( n j j j=1 j=1 1/fj ) which is A0 -measurable. It then follows from Problem 2.3 that n  d cj Pj dP0 dP0 j=0 =  n dλ dλ d cj Pj j=0  is also A0 -measurable. (ii): Let λ = ∞ j=1 cj Pθj be equivalent to P. Then pairwise sufficiency of T implies for any θ0 that dPθ0 /(dPθ0 + dλ) and hence dPθ0 /dλ is a measurable function of T .] 54 2. The Probability Background Problem 2.12 If a statistic T is sufficient for P, then for every function f which is (A, Pθ )-integrable for all θ ∈ Ω there exists a determination of the conditional expectation function Eθ [f (X) | t] that is independent of θ. [If X is Euclidean, this follows from Theorems 2.5.2 and 2.6.1. In general, if f is nonnegative there exists a nondecreasing sequence of simple nonnegative functions fn tending to f . Since the conditional expectation of a simple function can be taken to be independent of θ by Lemma 2.4.1(i), the desired result follows from Lemma 2.4.1(iv).] Problem 2.13 For a decision problem with a finite number of decisions, the class of procedures depending on a sufficient statistic T only is essentially complete. [For Euclidean sample spaces this follows from Theorem 2.5.1 without any restriction on the decision space. For the present case, let a decision procedure be given by δ(x) = (δ (1) (x), . . . , δ (m) (x)) where δ (i) (x) is the probability with which decision di is taken when x is observed. If T is sufficient and η (i) (t) = E[δ (i) (X) | t], the procedures δ and η have identical risk functions.] [More general versions of this result are discussed, for example, by Elfving (1952), Bahadur (1955), Burkholder (1961), LeCam (1964), and Roy and Ramamoorthi (1979).] Section 2.7 Problem 2.14 Let Xi (i = 1, . . . , s) be independently distributed with Poisson  distribution P (λi ), and let T0 = Xj , Ti = Xi , λ = λj . Then T0 has the Poisson distribution P (λ), and the conditional distribution of T1 , . . . , Ts−1 given T0 = t0 is the multinomial distribution (2.34) with n = t0 and pi = λi /λ. Problem 2.15 Life testing. Let X1 , . . . , Xn be independently distributed with exponential density (2θ)−1 e−x/2θ for x ≥ 0, and let the ordered X’s be denoted by Y1 ≤ Y2 ≤ · · · ≤ Yn . It is assumed that Y1 becomes available first, then Y2 , and so on, and that observation is continued until Yr has been observed. This might arise, for example, in life testing where each X measures the length of life of, say, an electron tube, and n tubes are being tested simultaneously. Another application is to the disintegration of radioactive material, where n is the number of atoms, and observation is continued until r α-particles have been emitted. (i) The joint distribution of Y1 , . . . , Yr is an exponential family with density ⎡  ⎤ r yi + (n − r)yr ⎢ i=1 ⎥ n! 1 ⎥, exp ⎢ 0 ≤ y1 ≤ · · · ≤ yr . ⎣− ⎦ (2θ)r (n − r)! 2θ (ii) The distribution of [ r i=1 Yi +(n−r)Yr ]/θ is χ2 with 2r degrees of freedom. (iii) Let Y1 , Y2 , . . . denote the time required until the first, second, . . . event occurs in a Poisson process with parameter 1/2θ (see Problem 1.1). Then Z1 = Y1 /θ , Z2 = (Y2 − Y1 )/θ , Z3 = (Y3 − Y2 )/θ , . . . are independently distributed as χ2 with 2 degrees of freedom, and the joint density Y1 , . . . , Yr is an exponential family with density  y  1 r exp −  , 0 ≤ y1 ≤ · · · ≤ yr .  r 2θ (2θ ) 2.9. Notes 55 The distribution of Yr /θ is again χ2 with 2r degrees of freedom. (iv) The same model arises in the application to life testing if the number n of tubes is held constant by replacing each burned-out tube with a new one, and if Y1 denotes the time at which the first tube burns out, Y2 the time at which the second tube burns out, and so on, measured from some fixed time. [(ii): The random variables Zi = (n − i + 1)(Yi − Yi−1 )/θ (i = 1, 2, . . . , r) are r 2 independently r distributed as χ with 2 degrees of freedom, and [ i=1 Yi + (n − r)Yr /θ = i=1 Zi .] Problem 2.16 For any θ which is an interior point of the natural parameter space, the expectations and covariances of the statistics Tj in the exponential family (2.35) are given by E [Tj (X)] = − ∂ log C(θ) ∂θj E [Ti (X)Tj (X)] − [ETi (X)ETj (X)] = − ∂ 2 log C(θ) ∂θi ∂θj (j = 1, . . . , k), (i, j = 1, . . . , k). Problem 2.17 Let Ω be the natural parameter space of the exponential family (2.35), and for any fixed tr+1 , . . . , tk (r < k) let Ωθ1 ...θr be the natural parameter space of the family of conditional distributions given Tr+1 = tr+1 , . . . , Tk = tk . (i) Then Ωθ1 ,...,θr contains the projection Ωθ1 ,...,θr of Ω onto θ1 , . . . , θr . (ii) An example in which Ωθ1 ,...,θr is a proper subset of Ωθ1 ,...,θr is the family of densities pθ1 θ2 (x, y) = C(θ1 , θ2 ) exp(θ1 x + θ2 y − xy), x, y > 0. 2.9 Notes The theory of measure and integration in abstract spaces and its application to probability theory, including in particular conditional probability and expectation, is treated in a number of books, among them Dudley (1989), Williams (1991) and Billingsley (1995). The material on sufficient statistics and exponential families is complemented by the corresponding sections in TPE2. Much fuller treatments of exponential families (as well as sufficiency) are provided by Barndorff–Nielsen (1978) and Brown (1986). 3 Uniformly Most Powerful Tests 3.1 Stating The Problem We now begin the study of the statistical problem that forms the principal subject of this book, the problem of hypothesis testing. As the term suggests, one wishes to decide whether or not some hypothesis that has been formulated is correct. The choice here lies between only two decisions: accepting or rejecting the hypothesis. A decision procedure for such a problem is called a test of the hypothesis in question. The decision is to be based on the value of a certain random variable X, the distribution Pθ of which is known to belong to a class P = {Pθ , θ ∈ Ω}. We shall assume that if θ were known, one would also know whether or not the hypothesis is true. The distributions of P can then be classified into those for which the hypothesis is true and those for which it is false. The resulting two mutually exclusive classes are denoted by H and K, and the corresponding subsets of Ω by ΩH and ΩK respectively, so that H ∪ K = P and ΩH ∪ ΩK = Ω. Mathematically, the hypothesis is equivalent to the statement that Pθ is an element of H. It is therefore convenient to identify the hypothesis with this statement and to use the letter H also to denote the hypothesis. Analogously we call the distributions in K the alternatives to H, so that K is the class of alternatives. Let the decisions of accepting or rejecting H be denoted by d0 and d1 respectively. A nonrandomized test procedure assigns to each possible value x of X one of these two decisions and thereby divides the sample space into two complementary regions S0 and S1 . If X falls into S0 , the hypothesis is accepted; otherwise it is rejected. The set S0 is called the region of acceptance, and the set S1 the region of rejection or critical region. 3.1. Stating The Problem 57 When performing a test one may arrive at the correct decision, or one may commit one of two errors: rejecting the hypothesis when it is true (error of the first kind) or accepting it when it is false (error of the second kind). The consequences of these are often quite different. For example, if one tests for the presence of some disease, incorrectly deciding on the necessity of treatment may cause the patient discomfort and financial loss. On the other hand, failure to diagnose the presence of the ailment may lead to the patient’s death. It is desirable to carry out the test in a manner which keeps the probabilities of the two types of error to a minimum. Unfortunately, when the number of observations is given, both probabilities cannot be controlled simultaneously. It is customary therefore to assign a bound to the probability of incorrectly rejecting H when it is true and to attempt to minimize the other probability subject to this condition. Thus one selects a number α between 0 and 1, called the level of significance, and imposes the condition that Pθ {δ(X) = d1 } = Pθ {X ∈ S1 } ≤ α for all θ ∈ ΩH . (3.1) Subject to this condition, it is desired to minimize Pθ {δ(X) = d0 } for θ in ΩK or, equivalently, to maximize Pθ {δ(X) = d1 } = Pθ {X ∈ S1 } for all θ ∈ ΩK . (3.2) Although usually (3.2) implies that sup Pθ {X ∈ S1 } = α, (3.3) ΩH it is convenient to introduce a term for the left-hand side of (3.3): it is called the size of the test or critical region S1 . The condition (3.1) therefore restricts consideration to test whose size does not exceed the given level of significance. The probability of rejection (3.2) evaluated for a given θ in ΩK is called the power of the test against the alternative θ. Considered as a function of θ for all θ ∈ Ω, the probability (3.2) is called the power function of the test and is denoted by β(θ). The choice of a level of significance α is usually somewhat arbitrary, since in most situations there is no precise limit to the probability of an error of the first kind that can be tolerated.1 Standard values, such as .01 or .05, were originally chosen to effect a reduction in the tables needed for carrying out various test. By habit, and because of the convenience of standardization in providing a common frame of reference, these values gradually became entrenched as the conventional levels to use. This is unfortunate, since the choice of significance level should also take into consideration the power that the test will achieve against the alternatives of interest. There is little point in carrying out an experiment which has only a small chance of detecting the effect being sought when it exists. Surveys by Cohen (1962) and Freiman et al. (1978) suggest that this is in fact the case for many studies. Ideally, the sample size should then be increased to permit adequate values for both significance level and power. If that is not feasible one may wish to use higher values of α than the customary ones. The opposite possibility, 1 The standard way to remove the arbitrary choice of α is to report the p-value of the test, defined as the smallest level of significance leading to rejection of the null hypothesis. This approach will discussed toward the end of Section 3.3. 58 3. Uniformly Most Powerful Tests that one would like to decrease α, arises when the latter is so close to 1 that α can be lowered appreciably without a significant loss of power (cf. Problem 3.11). Rules for choosing α in relation to the attainable power are discussed by Lehmann (1958), Arrow (1960), and Sanathanan (1974), and from a Bayesian point of view by Savage (1962, pp. 64–66). See also Rosenthal and Rubin (1985). Another consideration that may enter into the specification of a significance level is the attitude toward the hypothesis before the experiment is performed. If one firmly believes the hypothesis to be true, extremely convincing evidence will be required before one is willing to give up this belief, and the significance level will accordingly be set very low. (A low significance level results in the hypothesis being rejected only for a set of values of the observations whose total probability under hypothesis is small, so that such values would be most unlikely to occur if H were true.) Let us next consider the structure of a randomized test. For any values x, such a test chooses between the two decisions, rejection or acceptance, with certain probabilities that depend on x and will be denoted by φ(x) and 1 − φ(x) respectively. If the value of X is x, a random experiment is performed with two possible outcomes R and R̄, the probabilities of which are φ(x) and 1 − φ(x). If in this experiment R occurs, the hypothesis is rejected, otherwise it is accepted. A randomized test is therefore completely characterized by a function φ, the critical function, with 0 ≤ φ(x) ≤ 1 for all x. If φ takes on only the values 1 and 0, one is back in the case of a nonrandomized test. The set of points x for which φ(x) = 1 is then just the region of rejection, so that in a nonrandomized test φ is simply the indicator function of the critical region. If the distribution of X is Pθ , and the critical function φ is used, the probability of rejection is  Eθ φ(X) = φ(x) dPθ (x), the conditional probability φ(x) of rejection given x, integrated with respect to the probability distribution of X. The problem is to select φ so as to maximize the power βφ (θ) = Eθ φ(X) for all θ ∈ ΩK (3.4) subject to the condition Eθ φ(X) ≤ α for all θ ∈ ΩH . (3.5) The same difficulty now arises that presented itself in the general discussion of Chapter 1. Typically, the test that maximized the power against a particular alternative in K depends on this alternative, so that some additional principal has to be introduced to define what is meant by an optimum test. There is one important exception: if K contains only one distribution, that is, if one is concerned with a single alternative, the problem is completely specified by (3.4) and (3.5). It then reduces to the mathematical problem of maximizing an integral subject to certain side conditions. The theory of this problem, and its statistical applications, constitutes the principle subject of the present chapter. In special cases it may of course turn out that the same test maximizes the power of all alternatives in K even when there is more than one. Examples of such uniformly most powerful (UMP) tests will be given in Section 3.4 and 3.7. 3.2. The Neyman–Pearson Fundamental Lemma 59 In the above formulation the problem can be considered as special case of the general decision problem with two types of losses. Corresponding to the two kinds of error, one can introduce the two component loss functions, L1 (θ, d1 ) = 1 or 0 L1 (θ, d0 ) = 0 as θ ∈ ΩH or θ ∈ ΩK , for all θ L2 (θ, d0 ) = 0 or 1 L2 (θ, d1 ) = 0 as θ ∈ ΩH or θ ∈ ΩK , for all θ . and With this definition the minimization of EL2 (θ, δ(X)) subject to the restriction EL1 (θ, δ(X)) ≤ α is exactly equivalent to the problem of hypothesis testing as given above. The formal loss functions L1 and L2 clearly do not represent in general the true losses. The loss resulting from an incorrect acceptance of the hypothesis, for example, will not be the same for all alternatives. The more the alternative differs from the hypothesis, the more serious are the consequences of such an error. As was discussed earlier, we have purposely foregone the more detailed approach implied by this criticism. Rather than working with a loss function which in practice one does not know, it seems preferable to base the theory on the simpler and intuitively appealing notion of error. It will be seen later that at least some of the results can be justified also in the more elaborate formulation. 3.2 The Neyman–Pearson Fundamental Lemma A class of distributions is called simple if it contains a single distribution, and otherwise it is said to be composite. The problem of hypothesis testing is completely specified by (3.4) and (3.5) if K is simple. Its solution is easiest and can be given explicitly when the same is true of H. Let the distributions under a simple hypothesis H and alternative K be P0 and P1 , and suppose for a moment that these distributions are discrete with Pi {X = x} = Pi (x) for i = 0, 1. If at first one restricts attention to nonrandomized tests, the optimum test is defined as the critical region S satisfying  P0 (x) ≤ α (3.6) x∈S and  P1 (x) = maximum . x∈S It is easy to see which points should be included in S. To each point are attached two values, its probability under P0 and under P1 . The selected points are to have a total value not exceeding α on the one scale, and as large as possible on the other. This is a situation that occurs in many contexts. A buyer with a limited budget who wants to get “the most for his money” will rate the items according to their value per dollar. In order to travel a given distance in the shortest possible time, one must choose the quickest mode of transportation, that is, the one that 60 3. Uniformly Most Powerful Tests yields the largest number of miles per hour. Analogously in the present problem the most valuable points x are those with the highest value of r(x) = P1 (x) . P0 (x) The points are therefore rated according to the value of this ratio and selected for S in this order, as many as one can afford under restriction (3.6). Formally this means that S is the set of all points x for which r(x) > c, where c is determined by the condition  P0 {X ∈ S} = P0 (x) = α . x:r(x)>c Here a difficulty is seen to arise. It may happen that when a certain point is included, the value α has not yet been reached but that it would be exceeded if the point were also included. The exact value α can then either not be achieved at all, or it can be attained only by breaking the preference order established by r(x). The resulting optimization problem has no explicit solution. (Algorithms for obtaining the maximizing set S are given by the theory of linear programming.) The difficulty can be avoided, however, by a modification which does not require violation of the r-order and which does lead to a simple explicit solution, namely by permitting randomization.2 This makes it possible to split the next point, including only a portion of it, and thereby to obtain the exact value α without breaking the order of preference that has been established for inclusion of the various sample points. These considerations are formalized in the following theorem, the fundamental lemma of Neyman and Pearson. Theorem 3.2.1 Let P0 and P1 be probability distributions possessing densities p0 and p1 respectively with respect to a measure µ.3 (i) Existence. For testing H : p0 against the alternative K : p1 there exists a test φ and a constant k such that E0 φ(X) = α and  φ(x) = 1 0 when when p1 (x) > kp0 (x), p1 (x) < kp0 (x). (3.7) (3.8) (ii) Sufficient condition for a most powerful test. If a test satisfies (3.7) and (3.8) for some k, then it is most powerful for testing p0 against p1 at level α. (iii) Necessary condition for a most powerful test. If φ is most powerful at level α for testing p0 against p1 , then for some k it satisfies (3.8) a.e. µ. It also satisfies (3.7) unless there exists a test of size < α and with power 1. Proof. For α = 0 and α = 1 the theorem is easily seen to be true provided the value k = + ∞ is admitted in (3.8) and 0 · ∞ is interpreted as 0. Throughout the proof we shall therefore assume 0 < α < 1. 2 In practice, typically neither the breaking of the r-order nor randomization is considered acceptable. The common solution, instead, is to adopt a value of α that can be attained exactly and therefore does not present this problem. 3 There is no loss of generality in this assumption, since one can take µ = P + P . 0 1 3.2. The Neyman–Pearson Fundamental Lemma 61 (i): Let α(c) = P0 {p1 (X) > cp0 (X)}. Since the probability is computed under P0 , the inequality need be considered only for the set where p0 (x) > 0, so that α(c) is the probability that the random variable p1 (X)/p0 (X) exceeds c. Thus 1 − α(c) is a cumulative distribution function, and α(c) is nonincreasing and continuous on the right, α(c − 0) − α(c) = P0 {p1 (X)/p0 (X) = c}, α(−∞) = 1, and α(∞) = 0. Given any 0 < α < 1, let c0 be such that α(c0 ) ≤ α ≤ α(c0 − 0), and consider the test φ defined by ⎧ when p1 (x) > c0 p0 (x), ⎨ 1 α−α(c0 ) when p1 (x) = c0 p0 (x), φ(x) = ⎩ α(c0 −0)−α(c0 ) 0 when p1 (x) < c0 p0 (x). Here the middle expression is meaningful unless α(c0 ) = α(c0 − 0); since then P0 {p1 (X) = c0 p0 (X)} = 0, φ is defined a.e. The size of φ is     p1 (X) p1 (X) α − α(c0 ) E0 φ(X) = P0 > c0 + P0 = c0 = α, p0 (X) α(c0 − 0) − α(c0 ) p0 (X) so that c0 can be taken as the k of the theorem. (ii): Suppose that φ is a test satisfying (3.7) and (3.8) and that φ∗ is any other test with E0 φ∗ (X) ≤ α. Denote by S + and S − the sets in the sample space where φ(x) − φ∗ (x) > 0 and < 0 respectively. If x is in S + , φ(x) must be > 0 and p1 (x) ≥ kp0 (x). In the same way p1 (x) ≤ kp0 (x) for all x in S − , and hence   (φ − φ∗ )(p1 − kp0 ) dµ = (φ − φ∗ )(p1 − kp0 ) dµ ≥ 0. S + ∪S − The difference in power between φ and φ∗ therefore satisfies   (φ − φ∗ )p1 dµ ≥ k (φ − φ∗ )p0 dµ ≥ 0, as was to be proved. (iii): Let φ∗ be most powerful at level α for testing p0 against p1 , and let φ satisfy (3.7) and (3.8). Let S be the intersection of the set S + ∪ S − , on which φ and φ∗ differ, with the set {x : p1 (x) = kp0 (x)}, and suppose that µ(S) > 0. Since (φ − φ∗ )(p1 − kp0 ) is positive on S, it follows from Problem 2.4 that   (φ − φ∗ )(p1 − kp0 ) dµ = (φ − φ∗ )(p1 − kp0 ) dµ > 0 S + ∪S − S and hence that φ is more powerful against p1 than φ∗ . This is a contradiction, and therefore µ(S) = 0, as was to be proved. If φ∗ were of size < α and power < 1, it would be possible to include in the rejection region additional points or portions of points and thereby to increase the power until either the power is 1 or the size is α. Thus either E0 φ∗ (X) = α or E1 φ∗ (X) = 1. The proof of part (iii) shows that the most powerful test is uniquely determined by (3.7) and (3.8) except on the set on which p1 (x) = kp0 (x). On this set, φ can be defined arbitrarily provided the resulting test has size α. Actually, we have shown that it is always to define φ to be constant over this boundary set. In the trivial case that there exists a test of power 1, the constant k of (3.8) is 0, and one will accept H for all points for which p1 (x) = kp0 (x) even though the test may then have size < α. 62 3. Uniformly Most Powerful Tests It follows from these remarks that the most powerful test is determined uniquely (up to sets of measure zero) by (3.7) and (3.8) whenever the set on which p1 (x) = kp0 (x) has µ-measure zero. This unique test is then clearly nonrandomized. More generally, it is seen that randomization is not required except possibly on the boundary set, where it may be necessary to randomize in order to get the size equal to α. When there exists a test of power 1, (3.7) and (3.8) will determine a most powerful test, but it may not be unique in that there may exist a test also most powerful and satisfying (3.7) and (3.8) for some α < α. Corollary 3.2.1 Let β denote the power of the most powerful level-α test (0 < α < 1) for testing P0 against P1 . Then α < β unless P0 = P1 . Proof. Since the level-α test given by φ(x) ≡ α has power α, it is seen that α ≤ β. If α = β < 1, the test φ(x) ≡ α is most powerful and by Theorem 3.2.1(iii) must satisfy (3.8). Then p0 (x) = p1 (x) a.e. µ and hence P0 = P1 . An alternative method for proving some of the results of this section is based on the following geometric representation of the problem of testing a simple hypothesis against a simple alternative. Let N be the set of all points (α, β) for which there exists a test φ such that α = E0 φ(X), β = E1 φ(X). This set is convex, contains the points (0,0) and (1,1), and is symmetric with respect to the point ( 12 , 12 ) in the sense that with any point (α, β) it also contains the point (1 − α, 1 − β). In addition, the set N is closed. [This follows from the weak compactness theorem for critical functions, Theorem A.5.1 of the Appendix; the argument is the same as that in the proof of Theorem 3.6.1(i).] For each value 0 < α0 < 1, the level-α0 tests are represented by the points whose abscissa is ≤ αo . The most powerful of these tests (whose existence follows from the fact that N is closed) corresponds to the point on the upper boundary of N with abscissa α0 . This is the only point corresponding to a most powerful level-α0 test unless there exists a point (α, 1) in N with α < α0 (Figure 3.1b). ␤ 1 (1,1) ␤ 1 (1–2, 1–2) (1–2, 1–2) 0 (1,1) 1 ␣ 0 1 (b) (a) Figure 3.1. ␣ 3.3. p-values 63 As a example of this geometric approach, consider the following alternative proof of Corollary 3.2.1. Suppose that for some 0 < α0 < 1 the power of the most powerful level-α0 test is α0 . Then it follows from the convexity of N that (α, β) ∈ N implies β ≤ α, and hence from the symmetry of N that N consists exactly of the line segment connecting the points (0,0) and (1,1). This means that φpo dµ = φp1 dµ for all φ and hence that p0 = p1 (a.e.µ), as was to be proved. A proof of Theorem 3.2.1 along these lines is given in a more general setting in the proof of Theorem 3.6.1. Example 3.2.1 Suppose X is an observation from N (ξ, σ 2 ), with σ 2 known. The null hypothesis specifies ξ = 0 and the alternative specifies ξ = ξ1 for some ξ1 > 0. Then, the likelihood ratio is given by exp[− 2σ1 2 (x − ξ1 )2 ] p1 (x) ξ2 ξ1 x = exp[ 2 − 12 ] . = 1 2 p0 (x) σ 2σ exp[− 2σ2 x ] (3.9) Since the exponential function is strictly increasing and ξ1 > 0, the set of x where p1 (x)/p0 (x) > k is equivalent to the set of x where x > k . In order to determine k , the level constraint P0 {X > k  } = α must be satisfied, and so k = σz1−α , where z1−α is the 1 − α quantile of the standard normal distribution. Therefore, the most powerful level α test rejects if X > σz1−α . 3.3 p-values Testing at a fixed level α as described in Sections 3.1 and 3.2 is one of two standard (non-Bayesian) approaches to the evaluation of hypotheses. To explain the other, suppose that, under P0 , the distribution of p1 (X)/p0 (X) is continuous. Then, the most powerful level α test is nonrandomized and rejects if p1 (X)/p0 (X) > k, where k = k(α) is determined by (3.7). For varying α, the resulting tests provide an example of the typical situation in which the rejection regions Sα are nested in the sense that Sα ⊂ Sα if α < α . (3.10) When this is the case,4 it is good practice to determine not only whether the hypothesis is accepted or rejected at the given significance level, but also to determine the smallest significance level, or more formally p̂ = p̂(X) = inf{α : X ∈ Sα } , (3.11) at which the hypothesis would be rejected for the given observation. This number, the so-called p-value gives an idea of how strongly the data contradict the 4 See Problems 3.17 and 3.58 for examples where optimal nonrandomized tests need not be nested. 64 3. Uniformly Most Powerful Tests hypothesis.5 It also enables others to reach a verdict based on the significance level of their choice. Example 3.3.1 (Continuation of Example 3.2.1) Let Φ denote the standard normal c.d.f. Then, the rejection region can be written as X X ) > 1 − α} = {X : 1 − Φ( ) < α} . σ σ For a given observed value of X, the inf over all α where the last inequality holds is X p̂ = 1 − Φ( ) . σ Alternatively, the p-value is P0 {X ≥ x}, where x is the observed value of X. Note that, under ξ = 0, the distribution of p̂ is given by Sα = {X : X > σz1−α } = {X : Φ( X X ) ≤ u} = P0 {Φ( ) ≥ 1 − u} = u , σ σ because Φ(X/σ) is uniformly distributed on (0,1) (see Problem 3.22); therefore, p̂ is uniformly distributed on (0,1). P0 {p̂ ≤ u} = P0 {1 − Φ( A general property of p-values is given in the following lemma, which applies to both simple and composite null hypotheses. Lemma 3.3.1 Suppose X has distribution Pθ for some θ ⊂ Ω, and the null hypothesis H specifies θ ∈ ΩH . Assume the rejection regions satisfy (3.10). (i) If sup Pθ {X ∈ Sα } ≤ α for all 0 < α < 1, (3.12) θ∈ΩH then the distribution of p̂ under θ ∈ ΩH satisfies Pθ {p̂ ≤ u} ≤ u for all 0 ≤ u ≤ 1 . (3.13) (ii) If, for θ ∈ ΩH , Pθ {X ∈ Sα } = α for all 0 < α < 1 , (3.14) then Pθ {p̂ ≤ u} = u for all 0 ≤ u ≤ 1 ; i.e. p̂ is uniformly distributed over (0, 1). Proof. (i) If θ ∈ ΩH , then the event {p̂ ≤ u} implies {X ∈ Sv } for all u < v. The result follows by letting v → u. (ii) Since the event {X ∈ Su } implies {p̂ ≤ u}, it follows that Pθ {p̂ ≤ u} ≥ Pθ {X ∈ Su } . Therefore, if (3.14) holds, then Pθ {p̂ ≤ u} ≥ u, and the result follows from (i). 5 One could generalize the definition of p-value to include randomized level α tests φ α assuming that they are nested in the sense that φ α (x) ≤ φ α (x) for all x and α < α  . Simply define p̂ = inf{α : φ α (X ) = 1}; in words, p̂ is the smallest level of significance where the hypothesis is rejected with probability one. 3.4. Distributions with Monotone Likelihood Ratio 65 Example 3.3.2 Suppose X takes values 1, 2, . . . , 10. Under H, the distribution 1 for j = 1, . . . , 10. Under K, suppose p1 (j) = j/55. is uniform, i.e., p0 (j) = 10 The MP level α = i/10 test rejects if X ≥ 11 − i. However, unless α is a multiple of 1/10, the MP level α test is randomized. If we want to restrict attention to nonrandomized procedures, consider the conservative approach by defining i i+1 ≤α< . 10 10 If the observed value of X is x, then the p-value is given by (11 − x)/10. Then, the distribution of p̂ under H is given by Sα = {X ≥ 11 − i} if 11 − X ≤ u} = P {X ≥ 11 − 10u} ≤ u , (3.15) 10 and the last inequality is an equality if and only if u is of the form i/10 for some integer i = 0, 1, . . . , 10, i.e. the levels for which the MP test is nonrandomized (Problem 3.21). P {p̂ ≤ u} = P { P -values, with the additional information they provide, are typically more appropriate than fixed levels in scientific problems, whereas a fixed predetermined α is unavoidable when acceptance or rejection of H implies an imminent concrete decision. A review of some of the issues arising in this context, with references to the literature, is given in Kruskal (1978). 3.4 Distributions with Monotone Likelihood Ratio The case that both the hypothesis and the class of alternatives are simple is mainly of theoretical interest, since problems arising in applications typically involve a parametric family of distributions depending on one or more parameters. In the simplest situation of this kind the distributions depend on a single realvalued parameter θ, and the hypothesis is one-sided, say H : θ ≤ θ0 . In general, the most powerful test of H against an alternative θ1 > θ0 depends on θ1 and is then not UMP. However, a UMP test does exist if an additional assumption is satisfied. The real-parameter family of densities pθ (x) is said to have monotone likelihood ratio6 if there exists a real-valued function T (x) such that for any θ < θ the distributions Pθ and Pθ are distinct, and the ratio pθ (x)/pθ (x) is a nondecreasing function of T (x). Theorem 3.4.1 Let θ be a real parameter, and let the random variable X have probability density pθ (x) with monotone likelihood ratio in T (x). (i) For testing H : θ ≤ θ0 against K : θ > θ0 , there exists a UMP test, which is given by ⎧ ⎨ 1 when T (x) > C, γ when T (x) = C, (3.16) φ(x) = ⎩ 0 when T (x) < C, 6 This definition is in terms of specific versions of the densities p . If instead the θ definition is to be given in terms of the distribution P θ , various null-set considerations enter which are discussed in Pfanzagl (1967). 66 3. Uniformly Most Powerful Tests where C and γ are determined by Eθ0 φ(X) = α. (3.17) (ii) The power function β(θ) = Eθ φ(X) of this test is strictly increasing for all points θ for which 0 < β(θ) < 1. (iii) For all θ , the test determined by (3.16) and (3.17) is UMP for testing  H : θ ≤ θ against K  : θ > θ at level α = β(θ ). (iv) For any θ < θ0 the test minimizes β(θ) (the probability of an error of the first kind) among all tests satisfying (3.17). Proof. (i) and (ii): Consider first the hypothesis H0 : θ = θ0 and some simple alternative θ1 > θ0 . The most desirable points for rejection are those for which r(x) = pθ1 (x)/pθ0 (x) = g[T (x)] is sufficiently large. If T (x) < T (x ), then r(x) ≤ r(x ) and x is at least as desirable as x. Thus the test which rejects for large values of T (x) is most powerful. As in the proof of Theorem 3.2.1(i), it is seen that there exist C and γ such that (3.16) and (3.17) hold. By Theorem 3.2.1(ii), the resulting test is also most powerful for testing Pθ against Pθ at level α = β(θ ) provided θ < θ . Part (ii) of the present theorem now follows from Corollary 3.2.1. Since β(θ) is therefore nondecreasing the test satisfies Eθ φ(X) ≤ α for θ ≤ θ0 . (3.18) The class of tests satisfying (3.18) is contained in the class satisfying Eθ0 φ(X) ≤ α. Since the given test maximizes β(θ1 ) within this wider class, it also maximizes β(θ1 ) subject to (3.18); since it is independent of the particular alternative θ1 > θ0 chosen, it is UMP against K. (iii) is proved by an analogous argument. (iv) follows from the fact that the test which minimizes the power for testing a simple hypothesis against a simple alternative is obtained by applying the fundamental lemma (Theorem 3.2.1) with all inequalities reversed. By interchanging inequalities throughout, one obtains in an obvious manner the solution of the dual problem, H : θ ≥ θ0 , K : θ < θ0 . The proof of (i) and (ii) exhibits the basic property of families with monotone likelihood ratio: every pair of parameter values θ0 < θ1 establishes essentially the same preference order of the sample points (in the sense of the preceding section). A few examples of such families, and hence of UMP one-sided tests, will be given below. However, the main applications of Theorem 3.4.1 will come later, when such families appear as the set of conditional distributions given a sufficient statistic (Chapters 4 and 5) and as distributions of a maximal invariant (Chapters 6 and 7). Example 3.4.1 (Hypergeometric) From a lot containing N items of a manufactured product, a sample of size n is selected at random, and each item in the sample is inspected. If the total number of defective items in the lot is D, the number X of defectives found in the sample has the hypergeometric distribution D N −D P {X = x} = PD (x) = x Nn−x , max(0, n + D − N ) ≤ x ≤ min(n, D). n 3.4. Distributions with Monotone Likelihood Ratio 67 Interpreting PD (x) as a density with respect to the measure µ that assigns to any set on the real line as measure the number of integers 0, 1, 2, . . . that it contains, and nothing that for values of x within its range  D+1 N −D−n+x PD+1 (x) if n + D + 1 − N ≤ x ≤ D, N −D D+1−x = 0 or ∞ if x = n + D − N or D + 1, PD (x) it is seen that the distributions satisfy the assumption of monotone likelihood ratios with T (x) = x. Therefore there exists a UMP test for testing the hypothesis H : D ≤ D0 against K : D > D0 , which rejects H when X is too large, and an analogous test for testing H  : D ≥ D0 . An important class of families of distributions that satisfy the assumptions of Theorem 3.4.1 are the one-parameter exponential families. Corollary 3.4.1 Let θ be a real parameter, and let X have probability density (with respect to some measure µ) pθ (x) = C(θ)eQ(θ)T (x) h(x), (3.19) where Q is strictly monotone. Then there exists a UMP test φ for testing H : θ ≤ θ0 against K : θ > θ0 . If Q is increasing, φ(x) = 1, γ, 0 as T (x) >, =, < C, where C and γ are determined by Eθ0 φ(X) = α. If Q is decreasing, the inequalities are reversed. A converse of Corollary 3.4.1 is given by Pfanzagl (1968), who shows under weak regularity conditions that the existence of UMP tests against one-sided alternatives for all sample sizes and one value of α implies an exponential family. As in Example 3.4.1, we shall denote the right-hand side of (3.19) by Pθ (x) instead of pθ (x) when it is a probability, that is, when X is discrete and µ is counting measure. Example 3.4.2 (Binomial) The binomial distributions b(p, n) with   n x Pp (x) = p (1 − p)n−x x satisfy (3.19) with T (x) = x, θ = p, Q(p) = log[p/(1 − p)]. The problem of testing H : p ≥ p0 arises, for instance, in the situation of Example 3.4.1 if one supposes that the production process is in statistical control, so that the various items constitute independent trials with constant probability p of being defective. The number of defectives X in a sample of size n is then sufficient statistic for the distribution of the variables Xi (i = 1, . . . , n), where Xi is 1 or 0 as the ith item drawn is defective or not, and X is distributed as b(p, n). There exists therefore a UMP test of H, which rejects H when X is too small. An alternative sampling plan which is sometimes used in binomial situations is inverse binomial sampling. Here the experiment is continued until a specified number m of successes—for example, cures effected by some new medical treatment—have been obtained. If Yi denotes the number of trials after the 68 3. Uniformly Most Powerful Tests (i − 1)st success up to but not including the ith success, the probability that Yi = y is pq y for y = 0, 1, . . . , so that the joint distribution of Y1 , . . . , Ym is Pp (y1 , . . . , ym ) = pm q  yi , yk = 0, 1, . . . , k = 1, . . . , m.  This is an exponential family with T (y) = yi and Q(p) = log(1 − p). Since Q(p) is a decreasing function of p, the UMP test of H : p ≤ p0 rejects H when T is too small. This is what one would expect, since the realization of m successes in only a few more than m trials indicates a high value of p. The test statistic T , which is the number of trials required in excess of m to get m successes, has the negative binomial distribution [Problem 1.1(i)]   m+t−1 m t P (t) = t = 0, 1, . . . . p q , m−1 Example 3.4.3 (Poisson) If X1 , . . . , Xn are independent Poisson variables with E(Xi ) = λ, their joint distribution is λx1 +···+xn −nλ . e x1 ! · · · xn !  This constitutes an exponential family with T (x) = xi , and Q(λ) = log λ. One-sided hypotheses concerning λ might arise if λ is a bacterial density and the X’s are a number of bacterial counts, or if the X’s denote the number of α-particles produced in equal time intervals by a  radioactive substance, etc. The UMP testof the hypothesis λ ≤ λ0 rejects when Xi is too large. Here the test statistic Xi has itself a Poisson distribution with parameter nλ. Instead of observing the radioactive material for given time periods or counting the number of bacteria in given areas of a slide, one can adopt an inverse sampling method. The experiment is then continued, or the area over which the bacteria are counted is enlarged, until a count of m has been obtained. The observations consist of the times T1 , . . . , Tm that it takes for the first occurrence, from the first to the second, and so on. If one is dealing with a Poisson process and the number of occurrences in a time or space interval τ has the distribution Pλ (x1 , . . . , xn ) = (λτ )x −λτ , x = 0, 1, . . . , e x! then the observed times are independently distributed, each with the exponential density λe−λt for t ≥ 0 [Problem 1.1(ii)]. The joint densities   m  ti , t1 , . . . , tm ≥ 0, pλ (t1 , . . . , tm ) = λm exp −λ P (x) = i=1  form an exponential family with T (t1 , . . . , tm ) = ti and Q(λ) = −λ. The UMP test of H : λ ≤ λ0 rejects when T = Ti is too small. Since 2λTi has density 1 −u/2 e for u ≥ 0, which is the density of a χ2 -distribution with 2 degrees of 2 freedom, 2λT has a χ2 -distribution with 2m degrees of freedom. The boundary of the rejection region can therefore be determined from a table of χ2 . The formulation of the problem of hypothesis testing given at the beginning of the chapter takes account of the losses resulting from wrong decisions only in terms of the two types of error. To obtain a more detailed description of the 3.4. Distributions with Monotone Likelihood Ratio 69 problem of testing H : θ ≤ θ0 against the alternatives θ > θ0 , one can consider it as a decision problem with the decisions d0 and d1 of accepting and rejecting H and a loss function L(θ, di ) = Li (θ). Typically, L0 (θ) will be 0 for θ ≤ θ0 and strictly increasing for θ ≥ θ0 , and L1 (θ) will be strictly decreasing for θ ≤ θ0 and equal to 0 for θ ≥ θ0 . The difference then satisfies L1 (θ) − L0 (θ) > <0 as θ < > θ0 . (3.20) The following theorem is a special case of complete class results of Karlin and Rubin (1956) and Brown, Cohen, and Strawderman (1976). Theorem 3.4.2 (i) Under the assumptions of Theorem 3.4.1, the family of tests given by (3.16) and (3.17) with 0 ≤ α ≤ 1 is essentially complete provided the loss function satisfies (3.20). (ii) This family is also minimal essentially complete if the set of points x for which pθ (x) > 0 is independent of θ. Proof. (i): The risk function of any test φ is  R(θ, φ) = pθ (x){φ(x)L1 (θ) + [1 − φ(x)]L0 (θ)} dµ(x)  = pθ (x){L0 (θ) + [L1 (θ) − L0 (θ)]φ(x)} dµ(x), and hence the difference of two risk functions is R(θ, φ ) − R(θ, φ) = [L1 (θ) − L0 (θ)] This is ≤ 0 for all θ if βφ (θ) − βφ (θ) =   = (φ − φ)pθ dµ > <0 (φ − φ)pθ dµ. for = θ > < θ0 . Given any test φ, let Eθ0 φ(X) = α. It follows from Theorem 3.4.1(i) that there exists a UMP level-α test φ for testing θ = θ0 against θ > θ0 , which satisfies (3.16) and (3.17). By Theorem 3.4.1(iv), φ also minimizes the power for θ < θ0 . Thus the two risk functions satisfy R(θ, φ ) ≤ R(θ, φ) for all θ, as was to be proved. (ii): Let φα and φα be of sizes α < α and UMP for testing θ0 against θ > θ0 . Then βφα (θ) < βφα (θ) for all θ > θ0 unless βφα (θ) = 1. By considering the problem of testing θ = θ0 against θ < θ0 it is seen analogously that this inequality also holds for all θ < θ0 unless βφα (θ) = 0. Since the exceptional possibilities are excluded by the assumptions, it follows that R(θ, φ ) < > R(θ, φ) as θ > < θ0 . Hence each of the two risk functions is better than the other for some values of θ. The class of tests previously derived as UMP at the various significance levels α is now seen to constitute an essentially complete class for a much more general decision problem, in which the loss function is only required to satisfy certain broad qualitative conditions. From this point of view, the formulation involving the specification of a level of significance can be considered a simple way of selecting a particular procedure from an essentially complete family. The property of monotone likelihood ratio defines a very strong ordering of a family of distributions. For later use, we consider also the following somewhat weaker definition. A family of cumulative distribution functions Fθ on the real line 70 3. Uniformly Most Powerful Tests is said to be stochastically increasing (and the same term is applied to random variables possessing these distributions) if the distributions are distinct and if θ < θ implies Fθ (x) ≥ Fθ (x) for all x. If then X and X  have distributions Fθ and Fθ respectively, it follows that P {X > x} ≤ P {X  > x} for all x, so that X  tends to have larger values than X. In this case the variable X  is said to be stochastically larger than X. This relationship is made more intuitive by the following characterization of the stochastic ordering of two distributions. Lemma 3.4.1 Let F0 and F1 be two cumulative distribution functions on the real line. Then F1 (x) ≤ F0 (x) for all x if and only if there exist two nondecreasing functions f0 and f1 , and a random variable V , such that (a) f0 (v) ≤ f1 (v) for all v, and (b) the distributions of f0 (V ) and f1 (V ) are F0 and F1 respectively. Proof. Suppose first that the required f0 , f1 and V exist. Then F1 (x) = P {f1 (V ) ≤ x} ≤ P {f0 (V ) ≤ x} = F0 (x) for all x. Conversely, suppose that F1 (x) ≤ F0 (x) for all x, and let fi (y) = inf{x : Fi (x − 0) ≤ y ≤ F1 (x)}, i = 0, 1. These functions are nondecreasing and for fi = f, Fi = F satisfy f [F (x)] ≤ x and F [f (y)] ≥ y for all x and y. It follows that y ≤ F (x0 ) implies f (y) ≤ f [F (x0 )] ≤ x0 and that conversely f (y) ≤ x0 , implies F [f (y)] ≤ F (x0 )] and hence y ≤ F (x0 ), so that the two inequalities f (y) ≤ x0 and y ≤ F (x0 ) are equivalent. Let V be uniformly distributed on (0,1). Then P {fi (V ) ≤ x} = P {V ≤ Fi (x)} = Fi (x). Since Fi (x) ≤ F0 (x) for all x implies f0 (y) ≤ f1 (y) for all y, this completes the proof. One of the simplest examples of a stochastically ordered family is a location parameter family, that is, a family satisfying Fθ (x) = F (x − θ). To see that this is stochastically increasing, let X be a random variable with distribution F (x). Then θ < θ implies F (x − θ) = P {x ≤ x − θ} ≥ P {X ≤ x − θ } = F (x − θ ), as was to be shown. Another example is finished by families with monotone likelihood ratio. This is seen from the following lemma, which establishes some basic properties of these families. Lemma 3.4.2 Let pθ (x) be a family of densities on the real line with monotone likelihood ratio in x. (i) If ψ is a nondecreasing function of x, then Eθ ψ(X) is a nondecreasing function of θ; if X1 , . . . , Xn are independently distributed with density pθ and ψ  is a function of x1 , . . . , xn which is nondecreasing in each of its arguments, then Eθ ψ  (X1 , . . . , Xn ) is a nondecreasing function of θ. (ii) For any θ < θ , the cumulative distribution functions of X under θ and θ satisfy Fθ (x) ≤ Fθ (x) for all x. 3.4. Distributions with Monotone Likelihood Ratio 71 (iii) Let ψ be a function with a single change of sign. More specifically, suppose there exists a value x0 such that ψ(x) ≤ 0 for x < x0 and ψ(x) ≥ 0 for x ≥ x0 . Then there exists θ0 such that Eθ ψ(X) ≤ 0 for θ < θ0 and Eθ ψ(X) ≥ 0 for θ > θ0 , unless Eθ ψ(X) is either positive for all θ or negative for all θ. (iv) Suppose that pθ (x) is positive for all θ and all x, that pθ (x)/pθ (x) is strictly increasing in x for θ < θ , and that ψ(x) is as in (iii) and is = 0 with positive probability. If Eθo ψ(X) = 0, then Eθ ψ(X) < 0 for θ < θ0 and > 0 for θ > θ0 . Proof. (i): Let θ < θ , and let A and B be the sets for which pθ (x) < pθ (x) and pθ (x) > pθ (x) respectively. If a = supA ψ(x) and b = inf B ψ(x), then b − a ≥ 0 and    ψ(pθ − pθ ) dµ ≥ a (pθ − pθ ) dµ + b (pθ − pθ ) dµ A B  = (b − a) (pθ − pθ ) dµ ≥ 0, B which proves the first assertion. The result for general n follows by induction. (ii): This follows from (i) by letting ψ(x) = 1 for x > x0 and ψ(x) = 0 otherwise. (iii): We shall show first that for any θ < θ , Eθ ψ(X) > 0 implies Eθ ψ(X) ≥ 0. If pθ (x0 )/pθ (x0 ) = ∞, then pθ (x) = 0 for x ≥ x0 and hence Eθ ψ(X) ≤ 0. Suppose therefore that pθ (x0 )/pθ (x0 ) = c < ∞. Then ψ(x) ≥ 0 on the set S = {x : pθ (x) = 0 and pθ (x) > 0}, and  Eθ ψ(X) ≥ ψ  ≥ pθ pθ dµ pθ S̃ x0 −  ∞ cψpθ dµ + −∞ cψpθ dµ = cEθ ψ(X) ≥ 0. x0 The result now follows by letting θ0 = inf{θ : Eθ ψ(X) > 0}. (iv): The proof is analogous to that of (iii). Part (ii) of the lemma shows that any family of distributions with monotone likelihood ratio in x is stochastically increasing. That the converse does not hold is shown for example by the Cauchy densities 1 1 · π 1 + (x − θ)2 The family is stochastically increasing, since θ is a location parameter; however, the likelihood ratio is not monotone. Conditions under which a location parameter family possesses monotone likelihood ratio are given in Example 8.2.1. Lemma 3.4.2 is a special case of a theorem of Karlin (1957, 1968) relating the number of sign changes of Eθ ψ(X) to those of ψ(x) when the densities pθ (x) are totally positive (defined in Problem 3.50). The application of totally positive– or equivalently, variation diminishing–distributions to statistics is discussed by Brown, Johnstone, and MacGibbon (1981); see also Problem 3.53. 72 3. Uniformly Most Powerful Tests 3.5 Confidence Bounds The theory of UMP one-sided tests can be applied to the problem of obtaining a lower or upper bound for a real-valued parameter θ. The problem of setting a lower bound arises, for example, when θ is the breaking strength of a new alloy; that of setting an upper bound, when θ is the toxicity of drug or the probability of an undesirable event. The discussion of lower and upper bounds completely parallel, and it is therefore enough to consider the case of a lower bound, say θ. Since θ = θ(X) will be a function of the observations, it cannot be required to fall below θ with certainty, but only with specified high probability. One selects a number 1 − α, the confidence level, and restricts attention to bounds θ satisfying Pθ {θ(X) ≤ θ} ≥ 1 − α for all θ. (3.21) The function θ is called a lower confidence bound for θ at confidence level 1 − α; the infimum of the left-hand side of (3.21), which in practice will be equal to 1 − α, is called the confidence coefficient of θ. Subject to (3.21), θ should underestimate θ by as little as possible. One can ask, for example, that the probability of θ falling below any θ < θ should be a minimum. A function θ for which Pθ {θ(X) ≤ θ } = minimum (3.22) for all θ < θ subject to (3.21) is a uniformly most accurate lower confidence bound for θ at confidence level 1 − α. Let L(θ, θ) be a measure of the loss resulting from underestimating θ, so that for each fixed θ the function L(θ, θ) is defined and nonnegative for θ < θ, and is nonincreasing in this second argument. One would then wish to minimize Eθ L(θ, θ) (3.23) subject to (3.21). It can be shown that a uniformly most accurate lower confidence bound θ minimizes (3.23) subject to (3.21) for every such loss function L. (See Problem 3.44.) The derivation of uniformly most accurate confidence bounds is facilitated by introducing the following more general concept, which will be considered in more detail in Chapter 5. A family of subsets S(x) of the parameter space Ω is said to constitute a family of confidence sets at confidence level 1 − α if Pθ {θ ∈ S(X)} ≥ 1 − α for all θ ∈ Ω, (3.24) that is, if the random sets S(X) covers the true parameter point with probability ≥ 1 − α. A lower confidence bound corresponds to the special case that S(x) is a one-sided interval S(x) = {θ : θ(x) ≤ θ < ∞}. Theorem 3.5.1 (i) For each θ0 ∈ Ω let A(θ0 ) be the acceptance region of a level-α test for testing H(θ0 ) : θ = θ0 , and for each sample point x let S(x) denote the set of parameter values S(x) = {θ : x ∈ A(θ), θ ∈ Ω}. Then S(x) is a family of confidence sets for θ at confidence level 1 − α. 3.5. Confidence Bounds 73 (ii) If for all θ0 , A(θ0 ) is UMP for testing H(θ0 ) at level α against the alternatives K(θ0 ), then for each θ0 ∈ / Ω, S(X) minimizes probability Pθ {θ0 ∈ S(X)} for all θ ∈ K(θ0 ) among all level 1 − α families of confidence sets for θ. Proof. (i): By definition of S(x), θ ∈ S(x) if and only if x ∈ A(θ), (3.25) and hence Pθ {θ ∈ S(X)} = Pθ {X ∈ A(θ)} ≥ 1 − α. ∗ (ii): If S (x) is any other family of confidence sets at level 1 − α, and if A∗ (θ) = {x : θ ∈ S ∗ (x)}, then Pθ {X ∈ A∗ (θ)} = Pθ {θ ∈ S ∗ (X)} ≥ 1 − α, so that A∗ (θ0 ) is the acceptance region of a level-α test of H(θ0 ). It follows from the assumed property of A(θ0 ) that for any θ ∈ K(θ0 ) Pθ {X ∈ A∗ (θ0 )} ≥ Pθ {X ∈ A(θ0 )} and hence that Pθ {θ0 ∈ S ∗ (X)} ≥ Pθ {θ0 ∈ S(X)}, as was to be proved. The equivalence (3.25) shows the structure of the confidence sets S(x) as the totality of parameter values θ for which the hypothesis H(θ) is accepted when x is observed. A confidence set can therefore be viewed as a combined statement regarding the tests of the various hypotheses H(θ), which exhibits the values for which the hypothesis is accepted [θ ∈ S(x)] and those for which it is rejected [θ ∈ S̄(x)]. Corollary 3.5.1 Let the family of densities pθ (x), θ ∈ Ω, have monotone likelihood ratio in T (x), and suppose that the cumulative distribution function Fθ (t) of T = T (X) is a continuous function in each of the variables t and θ when the other is fixed. (i) There exists a uniformly most accurate confidence bound θ for θ at each confidence level 1 − α. (ii) If x denotes the observed values of X and t = T (x), and if the equation Fθ (t) = 1 − α (3.26) has a solution θ = θ̂ in Ω then this solution is unique and θ(x) = θ̂. Proof. (i): There exists for each θ0 a constant C(θ0 ) such that Pθ0 {T > C(θ0 )} = α, and by Theorem 3.4.1, T > C(θ0 ) is a UMP level-α rejection region for testing θ = θ0 against θ > θ0 . By Corollary 3.2.1, the power of this test against any alternative θ1 > θ0 exceeds α, and hence C(θ0 ) < C(θ1 ) so that the function C is strictly increasing; it is also continuous. Let A(θ0 ) denote the acceptance region 74 3. Uniformly Most Powerful Tests T ≤ C(θ0 ), and let S(x) be defined by (3.25). If follows from the monotonicity of the function C that S(x) consists of those values θ ∈ Ω which satisfy θ ≤ θ, where θ = inf{θ : T (x) ≤ C(θ)}. By Theorem 3.5.1, the sets {θ : θ(x) ≤ θ}, restricted to possible values of the parameter, constitute a family of confidence sets at level 1 − α, which minimize Pθ {θ ≤ θ } for all θ ∈ K(θ ), that is, for all θ > θ . This shows θ to be a uniformly most accurate confidence bound for θ. (ii): It follows from Corollary 3.2.1 that Fθ (t) is a strictly decreasing function of θ at any point t for which 0 < Fθ (t) < 1, and hence that (3.26) can have at most one solution. Suppose now that t is the observed value of T and that the equation Fθ (t) = 1 − α has the solution θ̂ ∈ Ω. Then Fθ̂ (t) = 1 − α, and by definition of the function C, C(θ̂) = t. The inequality t ≤ C(θ) is then equivalent to C(θ̂) ≤ C(θ) and hence to θ̂ ≤ θ. It follows that θ = θ̂, as was to be proved. Under the same assumptions, the corresponding upper confidence bound with confidence coefficient 1 − α is the solution θ̄ of the equation Pθ {T ≥ t} = 1 − α or equivalently of Fθ (t) = α. Example 3.5.1 (Exponential waiting times) To determine an upper bound for the degree of radioactivity λ of a radioactive substance, the substance is observed until a count of m has been obtained on a Geiger counter. Under the assumptions of Example 3.4.3, the joint probability density of the times Ti (i = 1, . . . , m) elapsing between the (i − 1)st count and the ith one is  p(t1 , . . . , tm ) = λm e−λ  ti , t1 , . . . , tm ≥ 0. If T = Ti denotes the total time of observation, then 2λT has a χ2 -distribution with 2m degrees of freedom, and, as was shown in Example 3.4.3, the acceptance region of the most powerful test of H(λ0 ) : λ = λ0 against λ < λ0 is 2λ0 T ≤ C, where C is determined by the equation  C χ22m = 1 − α . 0 The set S(t1 , . . . , tm ) defined by (3.25) is then the set of values λ such that λ ≤ C/2T , and it follows from Theorem 3.5.1 that λ̄ = C/2T is a uniformly most accurate upper confidence bound for λ. This result can also be obtained through Corollary 3.5.1. If the variables X or T are discrete, Corollary 3.5.1 cannot be applied directly, since the distribution functions Fθ (t) are not continuous, and for most values θ0 the optimum test of H : θ = θ0 are randomized. However, any randomized test based on X has the following representation as a nonrandomized test depending on X and an independent variable U distributed uniformly over (0, 1). Given a critical function φ, consider the rejection region R = {(x, u) : u ≤ φ(x)}. Then P {(X, U ) ∈ R} = P {U ≤ φ(X)} = Eφ (X), 3.5. Confidence Bounds 75 whatever the distribution of X, so that R has the same power function as φ and the two tests are equivalent. The pair of variables (X, U ) has a particularly simple representation when X is integer-valued. In this case the statistic T =X +U is equivalent to the pair (X, U ), since with probability 1 X = [T ], U = T − [T ], where [T ] denotes the largest integer ≤ T . The distribution of T is continuous, and confidence bounds can be based on this statistic. Example 3.5.2 (Binomial) An upper bound is required for a binomial probability p—for example, the probability that a batch of polio vaccine manufactured according to a certain procedure contains any live virus. Let X1 , . . . , Xn denote the outcome  of n trials, Xi being 1 or 0 with probabilities p and q respectively, and let X = Xi . Then T = X + U has probability density   n [t] n−[t] , 0 ≤ t < n + 1. p q [t] This satisfies the conditions of Corollary 3.5.1, and the upper confidence bound p̄ is therefore the solution, if it exists, of the equation Pp {T < t} = α, where t is the observed value of T . A solution does exist for all values α ≤ t ≤ n + α. For n + α < t, the hypothesis H(p0 ) : p = p0 is accepted against the alternative p < p0 for all values of p0 and hence p̄ = 1. For t < α, H(p0 ) is rejected for all values of p0 and the confidence set S(t) is therefore empty. Consider instead the sets S ∗ (t) which are equal to S(t) for t ≥ α and which for t < α consist of the single point p = 0. They are also confidence sets at level 1 − α, since for all p, Pp {p ∈ S ∗ (T )} ≥ Pp {p ∈ S(T )} = 1 − α. On the other hand, Pp {p ∈ S ∗ (T )} = Pp {p ∈ S(T )} for all p > 0 and hence Pp {p ∈ S ∗ (T )} = Pp {p ∈ S(T )} for all p > p. Thus the family of sets S ∗ (t) minimizes the probability of covering p for all p > p at confidence level 1 − α. The associated confidence bound p̄∗ (t) = p̄(t) for t ≥ α and p̄∗ (t) = 0 for t < α is therefore a uniformly most accurate upper confidence bound for p at level 1 − α. In practice, so as to avoid randomization and obtain a bound not dependent on the extraneous variable U , one usually replaces T by X + 1 = [T ] + 1. Since p̄∗ (t) is a nondecreasing function of t, the resulting upper confidence bound p̄∗ ([t] + 1) is then somewhat larger than necessary; as a compensation it also gives a correspondingly higher probability of not falling below the true p. References to tables for the confidence bounds and a careful discussion of various approximations can be found in Hall (1982) and Blyth (1984). Large sample approaches will be discussed in Example 11.2.7. 76 3. Uniformly Most Powerful Tests Let θ and θ̄ be lower and upper bounds for θ with confidence coefficients 1 − α1 and 1 − α2 , and suppose that θ(x) < θ̄(x) for all x. This will be the case under the assumptions of Corollary 3.5.1 if α1 + α2 < 1. The intervals (θ, θ̄) are then confidence intervals for θ with confidence coefficient 1 − α1 − α2 ; that is, they contain the true parameter value with probability 1 − α1 − α2 , since Pθ {θ ≤ θ ≤ θ̄} = 1 − α1 − α2 for all θ. If θ and θ̄ are uniformly most accurate, they minimize Eθ L1 (θ, θ) and Eθ L2 (θ, θ̄) at their respective levels for any function L1 that is nonincreasing in θ for θ < θ and 0 for θ ≥ θ and any L2 that is nondecreasing in θ̄ for θ̄ > θ and 0 for θ̄ ≤ θ. Letting L(θ; θ, θ̄) = L1 (θ, θ) + L2 (θ, θ̄), the intervals (θ, θ̄) therefore minimize Eθ L(θ; θ, θ̄) subject to Pθ {θ > θ} ≤ α1 , An example of such a loss function is ⎧ ⎨ θ̄ − θ θ̄ − θ L(θ; θ, θ̄) = ⎩ θ−θ Pθ {θ̄ < θ} ≤ α2 . if if if θ ≤ θ ≤ θ̄, θ < θ, θ̄ < θ, which provides a natural measure of the accuracy of the intervals. Other possible measures are the actual length θ̄ − θ of the intervals, or, for example, a(θ − θ)2 + b(θ̄ − θ)2 , which gives an indication of the distance of the two end points form the true value.7 An important limiting case corresponds to the levels α1 = α2 = 12 . Under the assumptions of Corollary 3.5.1 and if the region of positive density is independent of θ so that tests of power 1 are impossible when α < 1, the upper and lower confidence bounds θ̄ and θ coincide in this case. The common bound satisfies 1 , 2 and the estimate θ of θ is therefore as likely to underestimate as to overestimate the true value. An estimate with this property is said to be median unbiased. (For the relation of this to other concepts of unbiasedness, see Problem 1.3.) It follows from the above result for arbitrary α1 and α2 that among all median unbiased estimates, θ minimizes EL(θ, θ) for any monotone loss function, that is, any loss function which for fixed θ has a minimum of 0 at θ = θ and is nondecreasing as θ moves away from θ in either direction. By taking in particular L(θ, θ) = 0 when |θ − θ| ≤  and = 1 otherwise, it is seen that among all median unbiased estimates, θ minimizes the probability of differing from θ by more than any given amount; more generally it maximizes the probability Pθ {θ ≤ θ} = Pθ {θ ≥ θ} = Pθ {−1 ≤ θ − θ < 2 } for any 1 , 2 ≥ 0. A more detailed assessment of the position of θ than that provided by confidence bounds or intervals corresponding to a fixed level γ = 1 − α is obtained by 7 Proposed by Wolfowitz (1950). 3.6. A Generalization of the Fundamental Lemma 77 stating confidence bounds for a number of levels, for example upper confidence bounds corresponding to values such as γ = .05, .1, .25, .5, .75, .9, .95. These constitute a set of standard confidence bounds,8 from which different specific intervals or bounds can be obtained in the obvious manner. 3.6 A Generalization of the Fundamental Lemma The following is useful extension of Theorem 3.2.1 to the case of more than one side condition. Theorem 3.6.1 Let f1 , . . . , fm+1 be real-valued functions defined on a Euclidean space X and integrable µ, and suppose that for given constants c1 , . . . , cm there exists a critical function φ satisfying  φfi dµ = ci , i = 1, . . . , m. (3.27) Let C be the class of critical functions φ for which (3.27) holds. (i) Among all members of C there exists one that maximizes  φfm+1 dµ. (ii) A sufficient condition for a member of C to maximize  φfm+1 dµ is the existence of constants k1 , . . . , km such that φ(x) φ(x) = = 1 0 when when fm+1 (x) > fm+1 (x) < m  i=1 m  ki fi (x), (3.28) ki fi (x). i=1 (iii) If a member of C satisfies (3.28) with k1 , . . . , km ≥ 0, then it maximizes  φfm+1 dµ among all critical functions satisfying  φfi dµ ≤ ci , i = 1, . . . , m. (3.29) (iv) The set M of points in m-dimensional space whose coordinates are    φf1 dµ, . . . , φfm dµ 8 Suggested by Tukey (1949b). 78 3. Uniformly Most Powerful Tests for some critical function φ is convex and closed. If (c1 , . . . , cm ) is an inner point9 of M , then there exist constants k1 , . . . , km and a test φ satisfying (3.27) and (3.28), and a necessary condition for a member of C to maximize  φfm+1 dµ is that (3.28) holds a.e. µ. Here the term “inner point of M ” in statement (iv) can be interpreted as meaning a point interior to M relative to m-space or relative to the smallest linear space (of dimension ≤ m) containing M . The theorem is correct with both interpretations but is stronger with respect to the latter, for which it will be proved. We also note that exactly analogous results hold for the minimization of φfm+1 dµ. Proof. (i): Let {φn } be a sequence of functions in C such that φn fm+1 dµ tends to supφ φfm+1 dµ. By the weak compactness theorem for critical functions (Theorem 3.4.2 of the Appendix), there exists a subsequence {φni } and a critical function φ such that   φni fk dµ → φfk dµ for k = 1, · · · , m + 1. It follows that φ is in C and maximizes the integral with respect to fm+1 dµ within C. (ii) and (iii) are proved exactly as was part (ii) of Theorem 3.2.1. (iv): That M is closed follows again from the weak compactness theorem, and its convexity is a consequence of the fact that if φ1 and φ2 are critical functions, so is αφ1 + (1 − α)φ2 for any 0 ≤ α ≤ 1. If N (see Figure 3.2) is the totality of points in (m + 1)-dimensional space with coordinates    φf1 dµ, . . . , φfm+1 dµ , where φ ranges over the class of all critical functions, then N is convex and closed by the same argument. Denote the coordinates of a general point in M and N by (u1 , . . . , um ) and (u1 , . . . , um+1 ) respectively. The points of N , the first m coordinates of which are c1 , . . . , cm , form a closed interval [c∗ , c∗∗ ]. Assume first that c∗ < c∗∗ . Since (c1 , . . . , cm , c∗∗ ) is a boundary point of N , there exists a hyperplane through it such that every point on N lies below or on . Let the equation of be m+1  i=1 ki u i = m  ki ci + km+1 c∗∗ . i=1 Since (c1 , . . . , cm ) is an inner point of M , the coefficient km+1 = 0. To see this, let c∗ < c < c∗∗ , so that (c1 , . . . cm , c) is an inner point of N . Then there exists a sphere with this point as center lying entirely in N and hence below . It follows 9 A discussion of the problem when this assumption is not satisfied is given by Dantzig and Wald (1951). 3.6. A Generalization of the Fundamental Lemma 79 um + 1 ⌸ (c1, …, cm, c**) N (c1, c2, … cm, c) (c1, …, cm, c*) M Figure 3.2. that the point (c1 , . . . cm , c) does not lie on and hence that km+1 = 0. We may therefore take km+1 = −1 and see that for any point of N um+1 − m  i=1 ∗∗ ki ui ≤ cm+1 − m  ki ci . i=1 That is, all critical functions φ satisfy       m m   ∗∗ ki fi dµ ≤ φ ki fi dµ, fm+1 − φ fm+1 − i=1 ∗∗ i=1 ∗∗ ∗∗ where φ is the test giving rise to the point (c1 , . . . , cm , c ). Thus φ is the critical function that maximizes the left-hand side of this inequality. Since the integral in question is maximized by putting φ equal to 1 when the integrand is ∗∗ positive and equal to 0 when it is negative, φ satisfies (3.28) a.e. µ. ∗∗ ∗   If c = c , let (c1 , . . . , cm ) be any point of M other than (c1 , . . . , cm ). We shall show now that there exists exactly one real number c such that (c1 , . . . , cm , c ) is in N . Suppose to the contrary that (c1 , . . . , cm , c )and (c1 , . . . , cm , c̄ ) are both in N , and consider any point (c1 , . . . , cm , c ) of N such that (c1 , . . . , cm ) is an interior point of the line segment joining (c1 , . . . , cm ) and (c1 , . . . , cm ). Such a point exists since (c1 , . . . , cm ) is an inner point of M . Then the convex set spanned by 80 3. Uniformly Most Powerful Tests the three points (c1 , . . . , cm , c ), (c1 , . . . , cm , c̄ ), and (c1 , . . . , cm , c ) is contained in N and contains points (c1 , . . . , cm , c) and (c1 , . . . , cm , c̄) with c < c̄, which is a contradiction. Since N is convex, contains the origin, and has at most one point on any vertical line u1 = c1 , . . . , um = cm , it is contained in a hyperplane, which passes through the origin and is not parallel to the um+1 -axis. It follows that   m  φfm+1 dµ = ki φfi dµ i=1 for all φ. This arises of course only in the trivial case that fm+1 = m  ki fi a.e. µ, i=1 and (3.28) is satisfied vacuously. Corollary 3.6.1 Let p1 , . . . , pm , pm+1 be probability densities with respect to a measure µ, and let 0 < α < 1. Then there exists a  test φ such that Ei φ(X) = α (i = 1, . . . , m) and Em+1 φ(X) > α, unless pm+1 = m i=1 ki pi , a.e. µ. Proof. The proof will be by induction over m. For m = 1 the result reduces to Corollary 3.2.1. Assume now that it has been proved for any set of m distributions, and consider the case of m + 1 densities p1 , . . . , pm+1 . If p1 , . . . , pm are linearly dependent, the number of pi can be reduced and the result follows from the induction hypothesis. Assume therefore that p1 , . . . , pm are linearly independent. Then for each j = 1, . . . , m there exist by the induction hypothesis tests φj and φj such that Ei φj (X) = Ei φj (X) = α for all i = 1, . . . , j − 1, j + 1, . . . , m and Ej φj (X) < α < Ej φj (X). It follows that the point of m-space for which all m coordinates are equal to α is an inner point of M , so that Theorem 3.6.1(iv) is applicable. The test φ(x) ≡ α is such that Ei φ(X) = α for i = 1, . . . , m. If among all tests satisfying the side conditions this one is most powerful, it has to satisfy (3.28). Since 0 < α < 1, this implies pm+1 = m  ki pi a.e.µ, i=1 as was to be proved. The most useful parts of Theorems 3.2.1 and 3.6.1 are the parts (ii), which give sufficient conditions for a critical function to maximize an integral subject to certain side conditions. These results can be derived very easily as follows by the method of undetermined multipliers. Lemma 3.6.1 Let F1 , . . . , Fm+1 be real-valued functions defined over a space U , and consider the problem of maximizing Fm+1 (u) subject to Fi (u) = ci (i = 1, . . . , m). A sufficient condition for a point u0 satisfying the side conditions to be a solution of the given problem is that among all points of U it maximizes Fm+1 (u) − m  i=1 for some k1 , . . . , km . ki Fi (u) 3.7. Two-Sided Hypotheses 81 When applying the lemma one usually carries out the maximization for arbitrary k’s, and then determines the constants so as to satisfy the side conditions. Proof. If u is any point satisfying the side conditions, then Fm+1 (u) − m  ki Fi (u) ≤ Fm+1 (u0 ) − i=1 m  ki Fi (u0 ), i=1 and hence Fm+1 (u) ≤ Fm+1 (u0 ). As an application consider the problem treated in Theorem 3.6.1. Let U be the space of critical functions φ, and let Fi (φ) = φfi dµ. Then a sufficient (φ), subject to Fi (φ) = ci , is that it maximizes condition for  φ to maximize Fm+1 Fm+1 (φ)− ki Fi (φ) = (f ki fi )φ dµ. This is achieved by setting φ(x) = m+1 −  1 or 0 as fm+1 (x) > or < ki fi (x). 3.7 Two-Sided Hypotheses UMP tests exist not only for one-sided but also for certain two-sided hypotheses of the form H : θ ≤ θ1 or θ ≥ θ2 (θ1 < θ2 ). (3.30) This problem arises when trying to demonstrate equivalence (or sometimes called bioequivalence) of treatments; for example, a new drug may be declared equivalent to the current standard drug if the difference in therapeutic effect is small, meaning θ is a small interval about 0. Such testing problems also occur when one wishes to determine whether given specifications have been met concerning the proportion of an ingredient in a drug or some other compound, or whether a measuring instrument, for example a scale, is properly balanced. One then sets up the hypothesis that θ does not lie within the required limits, so that an error of the first kind consists in declaring θ to be satisfactory when in fact it is not. In practice, the decision to accept H will typically be accompanied by a statement of whether θ is believed to be ≤ θ1 or ≥ θ2 . The implications of H are, however, frequently sufficiently important so that acceptance will in any case be followed by a more detailed investigation. If a manufacturer tests each precision instrument before releasing it and the test indicates an instrument to be out of balance, further work will be done to get it properly adjusted. If in a scientific investigation the inequalities θ ≤ θ1 and θ ≥ θ2 contradict some assumptions that have been formulated, a more complex theory may be needed and further experimentation will be required. In such situations there may be only two basic choices, to act as if θ1 < θ < θ2 or to carry out some further investigation, and the formulation of the problem as that of testing the hypothesis H may be appropriate. In the present section, the existence of a UMP test of H will be proved for one-parameter exponential families. Theorem 3.7.1 (i) For testing the hypothesis H : θ ≤ θ1 or θ ≥ θ2 (θ1 < θ2 ) against the alternatives K : θ1 < θ < θ2 in the one-parameter exponential family 82 3. Uniformly Most Powerful Tests (3.19) there exists a UMP test given by ⎧ ⎨ 1 when C1 < T (x) < C2 (C1 < C2 ), γi when T (x) = Ci , i = 1, 2, φ(x) = ⎩ 0 when T (x) < C1 or > C2 , (3.31) where the C  s and γ  s are determined by Eθ1 φ(X) = Eθ2 φ(X) = α. (3.32) (ii) This test minimizes Eθ φ(X) subject to (3.32) for all θ < θ1 and > θ2 . (iii) For 0 < α < 1 the power function of this test has a maximum at a point θ0 between θ1 and θ2 and decreases strictly as θ tends away from θ0 in either direction, unless there exist two values t1 , t2 such that Pθ {T (X) = t1 } + Pθ {T (X) = t2 } = 1 for all θ. Proof. (i): One can restrict attention to the sufficient statistic T = T (X), the distribution of which by Lemma 2.7.2 is dPθ (t) = C(θ)eQ(θ)t dν(t), where Q(θ) is assumed to be strictly increasing. Let θ1 < θ < θ2 , and consider first the problem of maximizing Eθ ψ(T ) subject to (3.32) with φ(x) = ψ[T (x)]. If M denotes the set of all points Eθ1 ψ(T ), Eθ2 ψ(T )) as ψ ranges over the totality of critical functions, then the point (α, α) is an inner point of M . This follows from the fact that by Corollary 3.2.1 the set M contains points (α, u1 ) and (α, u2 ) with u1 < α < u2 and that it contains all points (u, u) with 0 < u < 1. Hence by part (iv) of Theorem 3.6.1 there exist constants k1 , k2 and test ψ0 (t) and that φ0 (x) = ψ0 [T (x)] satisfies (3.32) and that ψ0 (t) = 1 when k1 C(θ1 )eQ(θ1 )t + k2 C(θ2 )eQ(θ2 )t < C(θ )eQ(θ  )t and therefore when a1 eb1 t + a2 eb2 t < 1 (b1 < 0 < b2 ), and ψ0 (t) = 0 when the left-hand side is > 1. Here the a’s cannot both be ≤ 0, since then the test would always reject. If one of the a’s is ≤ 0 and the other one is > 0, then the left-hand side is strictly monotone, and the test is of the one-sided type considered in Corollary 3.4.1, which has a strictly monotone power function and hence cannot satisfy (3.32). Since therefore both a’s are positive, the test satisfies (3.31). It follows from Lemma 3.7.1 below that the C’s and γ’s are uniquely determined by (3.31) and (3.32), and hence from Theorem 3.6.1(iii) that the test is UMP subject to the weaker restriction Eθi ψ(T ) ≤ α (i = 1, 2). To complete the proof that this test is UMP for testing H, it is necessary to show that it satisfies Eθ ψ(T ) ≤ α for θ ≤ θ1 and θ ≥ θ2 . This follows from (ii) by comparison with the test ψ(t) ≡ α. (ii): Let θ < θ1 , and apply Theorem 3.6.1(iv) to minimize Eθ φ(X) subject to (3.32). Dividing through by eQ(θ1 )t , the desired test is seen to have a rejection region of the form a1 eb1 t + a2 eb2 t < 1 (b1 < 0 < b2 ). Thus it coincides with the test ψ0 (t) obtained in (i). By Theorem 3.6.1(iv) the first and third conditions of (3.31) are also necessary, and the optimum test is therefore unique provided P {T = Ci } = 0. 3.8. Least Favorable Distributions 83 (iii): Without loss of generality let Q(θ) = θ. It follows from (i) and the continuity of β(θ) = Eθ φ(X) that either β(θ) satisfies (iii) or there exist three points θ < θ < θ such that β(θ ) ≤ β(θ ) = β(θ ) = c, say. Then 0 < c < 1, since β(θ ) = 0 (or 1) implies φ(t) = 0 (or 1) a.e. ν and this is excluded by (3.32). As is seen by the proof of (i), the test minimizes Eθ φ(X) subject to Eθ φ(X) = Eθ φ(X) = c for all θ < θ < θ . However, unless T takes on at most two values with probability 1 or all θ, pθ , pθ , pθ are linearly independent, which by Corollary 3.6.1 implies β(θ ) > c. In order to determine the C’s and γ’s, one will in practice start with some trial values C1∗ , γ1∗ , find C2∗ , γ2∗ such that β ∗ (θ1 ) = α, and compute β ∗ (θ2 ), which will usually be either too large or too small. For the selection of the next trial values it is then helpful to note that if β ∗ (θ2 ) < α, the correct acceptance region is to the right of the one chosen, that is, it satisfies either C1 > C1∗ or C1 = C1∗ and γ1 < γ1∗ , and that the converse holds if β ∗ (θ2 ) > α. This is a consequence of the following lemma. Lemma 3.7.1 Let pθ (x) satisfy the assumptions of Lemma 3.4.2(iv). (i) If φ and φ∗ are two tests satisfying (3.31) and Eθ1 φ(T ) = Eθ1 φ∗ (T ), and if φ∗ is to the right of φ, then β(θ) < or > β ∗ (θ) as θ > θ1 or < θ1 . (ii) If φ and φ∗ satisfy (3.31) and (3.32), then φ = φ∗ with probability one . Proof. (i): The result follows from Lemma 3.4.2(iv) with ψ = φ∗ − φ. (ii): Since Eθ1 φ(T ) = Eθ1 φ∗ (T ), φ∗ lies either to the left or the right of φ, and application of (i) completes the proof. Although a UMP test exists for testing that θ ≤ θ1 or ≥ θ2 in an exponential family, the same is not true for the dual hypothesis H : θ1 ≤ θ ≤ θ2 or for testing θ = θ0 (Problem 3.54). There do, however, exist UMP unbiased tests of these hypotheses, as will be shown in Chapter 4. 3.8 Least Favorable Distributions It is a consequence of Theorem 3.2.1 that there always exists a most powerful test for testing a simple hypothesis against a simple alternative. More generally, consider the case of a Euclidean sample space; probability densities fθ , θ ∈ ω, and g with respect to a measure µ; and the problem of testing H : fθ , θ ∈ ω, against the simple alternative K : g. The existence of a most powerful level α test then follows from the weak compactness theorem for critical functions (Theorem A.5.1 of the Appendix) as in Theorem 3.6.1(i). Theorem 3.2.1 also provides an explicit construction for the most powerful test in the case of a simple hypothesis. We shall now extend this theorem to composite hypotheses in the direction of Theorem 3.6.1 by the method of undetermined multipliers. However, in the process of extension the result becomes much less explicit. Essentially it leaves open the determination of the multipliers, which now take the form of an arbitrary distribution. In specific problems this usually still involves considerable difficulty. From another point of view the method of attack, as throughout the theory of hypothesis testing, is to reduce the composite hypothesis to a simple one. This 84 3. Uniformly Most Powerful Tests is achieved by considering weighted averages of the distributions of H. The composite hypothesis H is replaced by the simple hypothesis HΛ that the probability density of X is given by  fθ (x) dΛ(θ), hΛ (x) = ω where Λ is a probability distribution over ω. The problem of finding a suitable Λ is frequently made easier by the following consideration. Since H provides no information concerning θ and since HΛ is to be equivalent to H for the purpose of testing against g, knowledge of the distribution Λ should provide as little help for this task as possible. To make this precise suppose that θ is known to have a distribution Λ. Then the maximum power βΛ that can be attained against g is that of the most powerful test φΛ for testing HΛ against g. The distribution Λ is said to be least favorable (at level α) if for all Λ the inequality βΛ ≤ βΛ holds. Theorem 3.8.1 Let a σ-field be defined over ω such that the densities fθ (x) are jointly measurable in θ and x. Suppose that over this σ-field there exist a probability distribution Λ such that the most powerful level-α test φΛ for testing HΛ against g is of size ≤ α also with respect to the original hypothesis H. (i) The test φΛ is most powerful for testing H against g. (ii) If φΛ is the unique most powerful level-α for testing HΛ against g, it is also the unique most powerful test of H against g. (iii) The distribution Λ is least favorable. Proof. We note first that hΛ is again a density with respect to µ, since by Fubini’s theorem (Theorem 2.2.4)     hΛ (x) dµ(x) = dΛ(θ) fθ (x) dµ(x) = dΛ(θ) = 1. ω ω Suppose that φΛ is a level-α test for testing H, and let φ∗ be any other level-α test. Then since Eθ φ∗ (X) ≤ α for all θ ∈ ω, we have   φ∗ (x)hΛ (x) dµ(x) = Eθ φ∗ (X)dΛ(θ) ≤ α. ω Therefore φ∗ is a level-α test also for testing HΛ and its power cannot exceed that of φΛ . This proves (i) and (ii). If Λ is any distribution, it follows further that φΛ is a level-α test also for testing HΛ , and hence that its power against g cannot exceed that of the most powerful test, which by definition is βΛ . The conditions of this theorem can be given a somewhat different form by noting that φΛ can satisfy ω Eθ φΛ (X) dΛ(θ) = α and Eθ φΛ (X) ≤ α for all θ ∈ ω only if the set of θ s with Eθ φΛ (X) = α has Λ-measure one. Corollary 3.8.1 Suppose that Λ is is a subset of ω with Λ(ω  ) = 1. Let  1 if φΛ (x) = 0 if a probability distribution over ω and that ω  φΛ be a test such that g(x) > k g(x) < k fθ (x) dΛ(θ), fθ (x) dΛ(θ). (3.33) Then φΛ is a most powerful level-α for testing H against g provided Eθ φΛ (X) = sup Eθ φΛ (X) = α θ∈ω for θ ∈ ω . (3.34) 3.8. Least Favorable Distributions 85 Theorems 3.4.1 and 3.7.1 constitute two simple applications of Theorem 3.8.1. The set ω  over which the least favorable distribution Λ is concentrated consists of the single point θ0 in the first of these examples and of the two points θ1 and θ2 in the second. This is what one might expect, since in both cases these are the distributions of H that appear to be “closest” to K. Another example in which the least favorable distribution is concentrated is at a single point is the following. Example 3.8.1 (Sign test) The quality of items produced by a manufacturing process is measured by a characteristic X such as the tensile strength of a piece of material, or the length of life or brightness of a light bulb. For an item to be satisfactory X must exceed a given constant u, and one wishes to test the hypothesis H : p ≥ p0 , where p = P {X ≤ u} is the probability of an item being defective. Let X1 , . . . , Xn be the measurements of n sample items, so that the X’s are independently distributed with common distribution about which no knowledge is assumed. Any distribution on the real line can be characterized by the probability p together with the conditional probability distributions P− and P+ of X given X ≤ u and X > u respectively. If the distributions P− and P+ have probability densities p− and p+, for example with respect to µ = P− + P+ , then the joint density of X1 , . . . , Xn at a sample point x1 , . . . , xn satisfying xi1 , . . . , xim ≤ u < xj1 , . . . , xjn−m is pm (1 − p)n−m p− (xi1 ) · · · p− (xim )p+ (xj1 ) · · · p+ (xjn−m ). Consider now a fixed alternative to H, say (p1 , P− , P+ ), with p1 < p0 . One would then expect the least favorable distribution Λ over H to assign probability 1 to the distribution (p0 , P− , P+ ) since this appears to be closest to the selected alternative. With this choice of Λ, the test (3.33) becomes m n−m p1 q1 φΛ (x) = 1 or 0 as > or < C, p0 q0 and hence as m < or > C. The test therefore rejects when the number M of defectives is sufficiently small, or more precisely, when M < C and with probability γ when M = C, where P {M < C} + γP {M = C} = α for p = p0 . (3.35) The distribution of M is the binomial distribution b(p, n), and does not depend on P+ and P− . As a consequence, the power function of the test depends only on p and is a decreasing function of p, so that under H it takes on its maximum for p = p0 . This proves Λ to be least favorable and φΛ to be most powerful. Since the test is independent of the particular alternative chosen, it is UMP. Expressed in terms of the variables Zi = Xi − u, the test statistic M is the number of variables ≤ 0, and the test is the so-called sign test (cf. Section 4.9). It is an example of a nonparametric test, since it is derived without assuming a 86 3. Uniformly Most Powerful Tests given functional form for the distribution of the X’s such as the normal, uniform, or Poisson, in which only certain parameters are unknown . The above argument applies, with only the obvious modifications, to the case that an item satisfactory if X lies within certain limits: u < X < v. This occurs, for example, if X is the length of a metal part or the proportion of an ingredient in a chemical compound, for which certain tolerances have been specified. More generally the argument applies also to the situation in which X is vector-valued. Suppose that an item is satisfactory only when X lies in a certain set S, for example, if all the dimensions of a metal part or the proportions of several ingredients lie within specified limits. The probability of a defective is then p = P {X ∈ S c }, and P− and P+ denote the conditional distributions of X given X ∈ S and X ∈ S c respectively. As before, there exists a UMP test of H : p ≥ p0 , and it rejects H when the number M of defectives is sufficiently small, with the boundary of the test being determined by (3.35). A distribution Λ satisfying the conditions of Theorem 3.8.1 exists in most of the usual statistical problems, and in particular under the following assumptions. Let the sample space be Euclidean, let ω be a closed Borel set in s-dimensional Euclidean space, and suppose that fθ (x) is a continuous function of θ for almost all x. Then given any g there exists a distribution Λ satisfying the conditions of Theorem 3.8.1 provided  lim fθn (x) dµ(x) = 0 n→∞ S for every bounded set S in the sample space and for every sequence of vectors θn whose distance from the origin tends to infinity. From this it follows as did Corollaries 1 and 4 from Theorems 3.2.1 and 3.6.1, that if the above conditions hold and if 0 < α < 1, there exists a test of power β > α for testing H : fθ , θ ∈ ω, against g unless g = fθ dΛ(θ) for some Λ. An example of the latter possibility is obtained by letting fθ and g be the normal densities N (θ, σ02 ) and N (0, σ12 ) respectively with σ02 < σ12 . (See the following section.) The above and related results concerning the existence and structure of least favorable distributions are given in Lehmann (1952b) (with the requirement that ω be closed mistakenly omitted), in Reinhardt (1961), and in Krafft and Witting (1967), where the relation to linear programming is explored. 3.9 Applications to Normal Distributions 3.9.1 Univariate Normal Models Because of their wide applicability, the problems of testing the mean ξ and variance σ 2 of a normal distribution are of particular importance. Here and in similar problems later, the parameter not being tested is assumed to be unknown, but will not be shown explicitly in a statement of the hypothesis. We shall write, for example, σ ≤ σ0 instead of the more complete statement σ ≤ σ0 , −∞ < ξ < ∞. 3.9. Applications to Normal Distributions 87 The standard (likelihood-ratio) tests of the two hypotheses σ ≤ σ0 and ξ ≤ ξ0 are given by the rejection regions  (3.36) (xi − x̄)2 ≥ C and √ , n(x̄ − ξ0 ) ≥ C.  (xi − x̄)2 1 n−1 (3.37) The corresponding tests for the hypotheses σ ≥ σ0 and ξ ≥ ξo are obtained from the rejection regions (3.36) and (3.37) by reversing the inequalities. As will be shown in later chapters, these four tests are UMP both within the class of unbiased and within the class of invariant test (but see Section 11.3 for problems arising when the assumption of normality does not hold exactly). However, at the usual significance levels only the first of them is actually UMP. Example 3.9.1 (One-sided tests of variance.) Let X1 , . . . , Xn be a sample from N (ξ, σ 2 ), and consider first the hypotheses H1 : σ ≥ σ0 and H2 : σ ≤ σ0 , and a simple alternative K : ξ = ξ1 , σ = σ1 . It seems reasonable to suppose that the least favorable  distribution Λ in the (ξ, σ)-plane is concentrated on the line σ = σ0 . Since Y = Xi /n = X̄ and U = (Xi − X̄)2 are sufficient statistics for the parameters (ξ, σ), attention can be restricted to these variables. Their joint density under HΛ is    u n (n−3)/2 2 exp − 2 Co u exp − 2 (y − ξ) dΛ(ξ), 2σ0 2σo while under K it is C1 u(n−3)/2 exp − u 2σ12    n exp − 2 (y − ξ1 )2 . 2σ1 The choice of Λ is seen to affect only the distribution of Y . A least favorable Λ should therefore have the property that the density of Y under HΛ ,   √  n n exp − 2 (y − ξ)2 dΛ(ξ), 2σ0 2πσ02 comes as close as possible to the alternative density,   √ n n 2 exp − 2 (y − ξ1 ) . 2σ1 2πσ12 At this point one must distinguish between H1 and H2 . In the first case σ1 < σ0 . By suitable choice of Λ the mean of Y can be made equal to ξ1 , but the variance will if anything be increased over its initial value σ02 . This suggests that the least favorable distribution assigns probability 1 to the point ξ = ξ1 , since in this way the distribution of Y is normal both under H and K with the same mean in both cases and the smallest possible difference between the variances. The situation is somewhat different for H2 , for which σ0 < σ1 . If the least favorable distribution Λ has a density, say Λ , the density of Y under HΛ becomes    ∞ √ n n √ exp − 2 (y − ξ)2 Λ (ξ) dξ. 2σ0 2πσ0 −∞ 88 3. Uniformly Most Powerful Tests This is the probability density of the sum of two independent random variables, one distributed as N (0, σ02 /n) and the other with density Λ (ξ). If Λ is taken to be N (ξ1 , (σ12 − σ02 )/n), the distribution of Y under HΛ becomes N (ξ1 , σ12 /n), the same as under K. We now apply Corollary 3.8.1 with the distributions Λ suggested above. For H1 it is more convenient to work with the original variables than with Y and U . Substitution in (3.33) gives φ(x) = 1 when # $  (2πσ12 )−n/2 exp − 2σ1 2 (xi − ξ1 )2 1 # $ > C,  1 2 −n/2 (2πσ0 ) exp − 2σ2 (xi − ξ1 )2 0 that is, when  (xi − ξ1 )2 ≤ C. (3.38) To justify the choice of Λ, one must show that . - P (Xi − ξ1 )2 ≤ C|ξ, σ takes on its maximum over the half plane σ ≥ σ0 at the point ξ = ξ1 , σ = σ0 . For any fixed σ, the above is the probability of the sample point falling in a sphere radius, computed under the assumption that the X’s are independently distributed as N (ξ, σ 2 ). This probability is maximized when the center of the sphere coincides with that of the distribution that is, when ξ = ξ1 . (This follows for example from Problem 7.15.) The probability then becomes / 0 %    xi − ξ1 2 C % C P Vi2 ≤ 2 , ≤ 2 %% ξ1 , σ = P σ σ σ where V1 , . . . , Vn are independently distributed as N (0, 1). This is a decreasing function of σ and therefore takes on its maximum when σ = σ0 . In the case of H2 , application of Corollary 3.8.1 to the sufficient statistics (Y, U ) gives φ(y, u) = 1 when   # $ C1 u(n−3)/2 exp − 2σu2 exp − 2σn2 (y − ξ1 )2 1   1 # $ u n (n−3)/2 C0 u exp − 2σ2 exp − 2σ2 (y − ξ)2 Λ (ξ) dξ 0   0 u 1 1  = C exp − − 2 ≥ C, 2 σ12 σ0 that is, when  (xi − x̄)2 ≥ C. (3.39)  Since the distribution of (Xi − X̄)2 /σ 2 does not depend on ξ or σ, the probability P { (Xi − X̄)2 ≥ C | ξ, σ} is independent of ξ and increases with σ, so that the conditions of Corollary 3.8.1 are satisfied. The test (3.39), being independent of ξ1 and σ1 , is UMP for testing σ ≤ σ0 against σ > σ0 . It is also seen to coincide with the likelihood-ratio test (3.36). On the other hand, the most powerful test (3.38) for testing σ ≥ σ0 against σ < σ0 does depend on the value ξ1 of ξ under the alternative. u= 3.9. Applications to Normal Distributions 89 It has been tacitly assumed so far that n > 1. If n = 1, the argument applies without change with respect to H1 , leading to (3.38) with n = 1. However, in the discussion of H2 the statistic U now drops out, and Y coincides with the single observation X. Using the same Λ as before, one sees that X has the same distribution under HΛ as under K, and the test φΛ therefore becomes φΛ (x) ≡ α. This satisfies the conditions of Corollary 3.8.1 and is therefore the most powerful test for the given problem. It follows that a single observation is of no value for testing the hypothesis H2 , as seems intuitively obvious, but that it could be used to test H1 if the class of alternatives were sufficiently restricted. The corresponding derivation for the hypothesis ξ ≤ ξ0 is less straightforward. It turns out10 that Student’s test given by (3.37) is most powerful if the level of significance α is ≥ 12 , regardless of the alternative ξ1 > ξ0 , σ1 . This test is 1 therefore UMP for α ≥  . On the other hand, when α < 12 the most powerful 2 test of H rejects when (xi − a)2 ≤ b, where the constants a and b depend on the alternative (ξ1 , σ1 ) and on α. Thus for the significance levels that are of interest, a UMP test of H does not exist. No new problem arises for the hypothesis ξ ≥ ξ0 , since this reduces to the case just considered through the transformation Yi = ξ0 − (Xi − ξ0 ). 3.9.2 Multivariate Normal Models Let X denote a k × 1 random vector whose ith component, Xi , is a real-valued random variable. The mean of X, denoted E(X), is a vector with ith component E(Xi ) (assuming it exists). The covariance matrix of X, denoted Σ, is the k × k matrix with (i, j) entry Cov(Xi , Xj ). Σ is well-defined iff E(|X|2 ) < ∞, where | · | denotes the Euclidean norm. Note that, if A is an m × k matrix, then the m × 1 vector Y = AX has mean (vector) AE(X) and covariance matrix AΣAT , where AT is the transpose of A (Problem 3.63). The multivariate generalization of a real-valued normally distributed random variable is a random vector X = (X1 , . . . , Xk )T with the multivariate normal probability density # $  |A| 1 aij (xi − ξi )(xj − ξj ) , (3.40) 1 k exp − 2 (2π) 2 where the matrix A = (aij ) is positive definite, and |A| denotes its determinant. The means and covariance matrix of the X’s are given by E(Xi ) = ξi , E(Xi − ξi )(Xj − ξj ) = σij , (σij ) = A−1 . (3.41) The column vector ξ = (ξ1 , . . . , ξk )T is the mean vector and Σ = A−1 is the covariance matrix of X. Such a definition only applies when A is nonsingular, in which case we say that X has a nonsingular multivariate normal distribution. More generally, we say that Y has a multivariate normal distribution if Y = BX + µ for some m × k matrix of constants B and m×1 constant vector µ, where X has some nonsingular multivariate normal distribution. Then, Y is multivariate normal if and only if 10 See Lehmann and Stein (1948) 90 3. Uniformly Most Powerful Tests m 2 i=1 ci Yi is univariate normal, if we interpret N (ξ, σ ) with σ = 0 to be the distribution that is point mass at ξ. Basic properties of the multivariate normal distribution are given in Anderson (2003). Example 3.9.2 (One-sided tests of a combination of means.) Assume X is multivariate normal with unknown mean ξ = (ξ1 , . . . , ξk )T and known covariance matrix Σ. Assume a = (a1 , . . . , ak )T is a fixed vector with aT Σa > 0. The problem is to test H: k  ai ξi ≤ δ vs. i=1 K: k  ak ξi > δ . i=1  We will show that a UMP level α test exists, which rejects when i ai Xi > σz1−α , where σ 2 = aT Σa. To see why,11 we will consider four cases of increasing generality. Case 1. If k = 1 and the problem is to test the mean of X1 , the result follows by Problem 3.1. Case 2. Consider now general k, so that (X1 , . . . , Xk ) has mean (ξ1 , . . . , ξk ) and covariance matrix Σ. However, consider the special case (a1 , . . . , ak ) = (1, 0, . . . , 0). Also, assume X1 and (X2 , . . . , Xk ) are independent. Then, for any fixed alternative (ξ1 , . . . , ξk ) with ξ1 > δ, the least favorable distribution concentrates on the single point (δ, ξ2 , . . . , ξk ) (Problem 3.65). Case 3. As in case 2, consider a1 = 1 and ai = 0 if i > 1, but now allow Σ to be an arbitrary covariance matrix. We can reduce the problem to case 2 by an appropriate linear transformation. Simply let Y1 = X1 and, for i > 1, let Yi = Xi − Cov(X1 , Xi ) X1 . V ar(X1 ) Then, it is easily checked that Cov(Y1 , Yi ) = 0 if i > 1. Moreover, Y is just a 1:1 transformation of X. But, the problem of testing E(Y1 ) = E(X1 ) based on Y = (Y1 , . . . , Yk ) is in the form already studied in case 2, and the UMP test rejects for large values of Y1 = X1 . Case 4. Now, consider arbitrary (a1 , . . . , ak ) satisfying aT Σa > 0. Let Z = OX, where O is any orthogonal matrix with first row (a1 , . . . , ak ). Then, E(Z1 ) = k ) > δ reduces to i=1 ai ξi , and the problem of testing E(Z1 ) ≤ δ versus E(Z 1 case 3. Hence, the UMP test rejects for large values of Z1 = ki=1 ai Xi . Example 3.9.3 (Equivalence tests of a combination of means.) As in Example 3.9.2, assume X is multivariate normal N (ξ, Σ) with unknown mean vector ξ and known covariance matrix Σ. Fix δ > 0 and any vector a = (a1 , . . . , ak )T satisfying aT Σa > 0. Consider testing H: | k  i=1 ai ξi | ≥ δ vs K: | k  ai ξi | < δ . i=1 11 Proposition 15.2 of van der Vaart (1998) provides an alternative proof in the case Σ is invertible. 3.9. Applications to Normal Distributions 91 Then, a UMP level α test also exists and it rejects H if | k  ai Xi | < C , i=1 where C = C(α, δ, σ) satisfies Φ C −δ σ  −Φ −C − δ σ  =α (3.42) and σ 2= aT Σa. Hence, the power of this test against an alternative (ξ1 , . . . , ξk ) with | i ai ξi | = δ  < δ is   −C − δ  C − δ −Φ . Φ σ σ To see why, we again consider four cases of increasing generality. Case 1. Suppose k = 1, so that X1 = X is N (ξ, σ 2 ) and we are testing |ξ| ≥ δ versus |ξ| < δ. (This case follows by Theorem 3.7.1, but we argue independently so that the argument applies to the other cases as well.) Fix an alternative ξ = m with |m| < δ. Reduce the composite null hypothesis to a simple one via a least favorable distribution that places mass p on N (δ, σ 2 ) and mass 1−p on N (−δ, σ 2 ). The value of p will be chosen shortly so that such a distribution is least favorable (and will be seen to depend on m, α, σ and δ). By the Neyman Pearson Lemma, the MP test of pN (δ, σ 2 ) + (1 − p)N (−δ, σ 2 ) vs N (m, σ 2 ) rejects for small values of     p exp − 2σ1 2 (X − δ)2 + (1 − p) exp − 2σ1 2 (X + δ)2  1  , exp − 2σ2 (X − m)2 (3.43) or equivalently for small values of f (X), where f (x) = p exp[(δ − m)X/σ 2 ] + (1 − p) exp[−(δ + m)X/σ 2 ] . We can now choose p so that f (C) = f (−C), so that p must satisfy exp[(δ + m)C/σ 2 ] − exp[−(δ + m)C/σ 2 ] p = . 1−p exp[(δ − m)C/σ 2 ] − exp[−(δ − m)C/σ 2 ] (3.44) Since δ − m > 0 and δ + m > 0, both the numerator and denominator of the right side of (3.44) are positive, so the right side is a positive number; but, p/(1 − p) is a nondecreasing function of p with range [0, ∞) as p varies from 0 to 1. Thus, p is well-defined. Also, observe f  (x) ≥ 0 for all x. It follows that (for this special choice of C) {X : f (X) ≤ f (C)} = {X : |X| ≤ C} is the rejection region of the MP test. Such a test is easily seen to be level α for the original composite null hypothesis because its power function is symmetric and decreases away from zero. Thus, the result follows by Theorem 3.8.1. Case 2. Consider now general k, so that (X1 , . . . , Xk ) has mean (ξ1 , . . . , ξk ) and covariance matrix Σ. However, consider the special case (a1 , . . . , ak ) = 92 3. Uniformly Most Powerful Tests (1, 0, . . . , 0), so we are testing |ξ1 | ≥ δ versus |ξ1 | < δ. Also, assume X1 and (X2 , . . . , Xk ) are independent, so that the first row and first column of Σ are zero except the first entry, which is σ 2 (assumed positive). Using the same reasoning as case 1, fix an alternative m = (m1 , . . . , mk ) with |m1 | < δ and consider testing pN ((δ, m2 , . . . , mk ), Σ) + (1 − p)N ((−δ, m2 , . . . , mk ), Σ) versus N ((m1 , . . . , mk ), Σ). The likelihood ratio is in fact the same as (3.43) because each term is now multiplied by the density of (X2 , . . . , Xk ) (by independence), and these densities cancel. The UMP test from Case 1, which rejects when |X1 | ≤ C, is UMP in this situation as well. Case 3. As in Case 2, consider a1 = 1 and ai = 0 if i > 1, but now allow Σ to be an arbitrary covariance matrix. By transforming X to Y as in Case 3 of Example 3.9.2, the result follows (Problem 3.66). Case 4. Now, consider arbitrary (a1 , . . . , ak ) satisfying aT Σa > 0. As in Case 4 of Example 3.9.2), transform X to Z and the result follows (Problem 3.66). 3.10 Problems Section 3.2 Problem 3.1 Let X1 , . . . , Xn be a sample from the normal distribution N (ξ, σ 2 ). (i) If σ = σ0 (known), there exists  a UMP test for testing H : ξ ≤ ξ0 against ξ > ξ0 , which rejects when (Xi − ξ0 ) is too large. (ii) If ξ = ξ0 (known), there exists aUMP test for testing H : σ ≤ σ0 against K : σ > σ0 , which rejects when (Xi − ξ0 )2 is too large. Problem 3.2 UMP test for U (0, θ). Let X = (X1 , . . . , Xn ) be a sample from the uniform distribution on (0, θ). (i) For testing H : θ ≤ θ0 against K : θ > θ0 any test is UMP at level α for which Eθ0 φ(X) = α, Eθ φ(X) ≤ α for θ ≤ θ0 , and φ(x) = 1 when max(x1 , . . . , xn ) > θ0 . (ii) For testing H : θ = θ0 against K : θ = θ0 a unique UMP test exists, and is √ given by φ(x) = 1 when max(x1 , . . . , xn ) > θ0 or max(x1 , . . . , xn ) ≤ θ0 n α, and φ(x) = 0 otherwise. [(i): For each θ > θ0 determine the ordering established by r(x) = pθ (x)/pθ0 (x) and use the fact that many points are equivalent under this ordering. (ii): Determine the UMP tests for testing θ = θ0 against θ < θ0 and combine this result with that of part (i).] Problem 3.3 Suppose N i.i.d. random variables are generated from the same known strictly increasing absolutely continuous cdf F (·). We are told only X, the maximum of these random variables. Is there a UMP size α test of H0 : N ≤ 5 versus H1 : N > 5? 3.10. Problems 93 If so, find it. Problem 3.4 UMP test for exponential densities. Let X1 , . . . , Xn be a sample from the exponential distribution E(a, b) of Problem 1.18, and let X(1) = min(X1 , . . . , Xn ). (i) Determine the UMP test for testing H : a = a0 against K : a = a0 when b is assumed known. (ii) The power of any MP level-α test of H : a = a0 against K : a = a1 < a0 is given by β ∗ (a1 ) = 1 − (1 − α)e−n(a0 −a1 )/b . (iii) For the problem of part (i), when b is unknown, the power of any level α test which rejects when X − a0  (1) ≤ C1 or ≥ C2 [Xi − X(1) ] against any alternative (a1 , b) with a1 < a0 is equal to β ∗ (a1 ) of part (ii) (independent of the particular choice of C1 and C2 ). (iv) The test of part (iii) is a UMP level-α test of H : a = a0 against K : a = a0 (b unknown). (v) Determine the UMP test for testing H : a = a0 , b = b0 against the alternatives a < a0 , b < b0 . (vi) Explain the (very unusual) existence in this case of a UMP test in the presence of a nuisance parameter [part(iv)] and for a hypothesis specifying two parameters [part(v)]. [(i) The variables Yi = e−Xi /b are a sample from the uniform distribution on (0, e−a/b ).] Note. For more general versions of parts (ii)–(iv) see Takeuchi (1969) and Kabe and Laurent (1981). Problem 3.5 In the proof of Theorem 3.2.1(i), consider the set of c satisfying α(c) ≤ α ≤ α(c − 0). If there is only one such c, c is unique; otherwise, there is an interval of such values [c1 , c2 ]. Argue that, in this case, if α(c) is continuous at c2 , then Pi (C) = 0 for i = 0, 1, where   p1 (x) ≤ c2 . C = x : p0 (x) > 0 and c1 < p0 (x) If α(c) is not continuous at c2 , then the result is false. Problem 3.6 Let P0 , P1 , P2 be the probability distributions assigning to the integers 1, . . . , 6 the following probabilities: P0 P1 P2 1 2 3 4 5 6 .03 .06 .09 .02 .05 .05 .02 .08 .12 .01 .02 0 0 .01 .02 .92 .78 .72 94 3. Uniformly Most Powerful Tests Determine whether there exists a level-α test of H : P = P0 which is UMP against the alternatives P1 and P2 when (i) α = .01; (ii) α = .05; (iii) α = .07. Problem 3.7 Let the distribution of X be given by x 0 1 2 3 Pθ (X = x) θ 2θ .9 − 2θ .1 − θ where 0 < θ < .1. For testing H : θ = .05 against θ > .05 at level α = .05, determine which of the following tests (if any) is UMP: (i) φ(0) = 1, φ(1) = φ(2) = φ(3) = 0; (ii) φ(1) = .5, φ(0) = φ(2) = φ(3) = 0; (iii) φ(3) = 1, φ(0) = φ(1) = φ(2) = 0. Problem 3.8 A random variable X has the Pareto distribution P (c, τ ) if its density is cτ c /xc+1 , 0 < τ < x, 0 < C. (i) Show that this defines a probability density. (ii) If X has distribution P (c, τ ), then Y = log X has exponential distribution E(ξ, b) with ξ = log τ , b = 1/c. (iii) If X1 , . . . , Xn is a sample from P (c, τ ), use (ii) and Problem 3.4 to obtain UMP tests of (a) H : τ = τ0 against τ = τ0 when b is known; (b) H : c = c0 , τ = τ against c > c0 , τ < τ0 . Problem 3.9 Let X be distributed according to Pθ , θ ∈ Ω, and let T be sufficient for θ. If ϕ(X) is any test of a hypothesis concerning θ, then ψ(T ) given by ψ(t) = E[ϕ(X) | t] is a test depending on T only, an its power function is identical with that of ϕ(X). Problem 3.10 In the notation of Section 3.2, consider the problem of testing H0 : P = P0 against H1 : P = P1 , and suppose that known probabilities π0 = π and π1 = 1 − π can be assigned to H0 and H1 prior to the experiment. (i) The overall probability of an error resulting from the use of a test ϕ is πE0 ϕ(X) + (1 − π)E1 [1 − ϕ(X)]. (ii) The Bayes test minimizing this probability is given by (3.8) with k = π0 /π1 . (iii) The conditional probability of Hi given X = x, the posterior probability of Hi is πi pi (x) , π0 p0 (x) + π1 p1 (x) and the Bayes test therefore decides in favor of the hypothesis with the larger posterior probability 3.10. Problems 95 Problem 3.11 (i) For testing H0 : θ = 0 against H1 : θ = θ1 when X is N (θ, 1), given any 0 < α < 1 and any 0 < π < 1 (in the notation of the preceding problem), there exists θ1 and x such that (a) H0 is rejected when X = x but (b) P (H0 | x) is arbitrarily close to 1. (ii) The paradox of part (i) is due to the fact that α is held constant while the power against θ1 is permitted to get arbitrarily close to 1. The paradox disappears if α is determined so that the probabilities of type I and type II error are equal [but see Berger and Sellke (1987)]. [For a discussion of such paradoxes, see Lindley (1957), Bartlett (1957), Schafer (1982, 1988) and Robert (1993).] Problem 3.12 Let X1 , . . . , Xn be independently distributed, each uniformly over the integers 1, 2, . . . , θ. Determine whether there exists a UMP test for testing H : θ = θ0 , at level 1/θ0n against the alternatives (i) θ > θ0 ; (ii) θ < θ0 ; (iii) θ = θ0 . Problem 3.13 The following example shows that the power of a test can sometimes be increased by selecting a random rather than a fixed sample size even when the randomization does not depend on the observations. Let X1 , . . . , Xn be independently distributed as N (θ, 1), and consider the problem of testing H : θ = 0 against K : θ = θ1 > 0. (i) The power of the most powerful test as a function of the sample size n is not necessarily concave. (ii) In particular for α = .005, θ1 = 12 , better power is obtained by taking 2 or 16 observations with probability 12 each than by taking a fixed sample of 9 observations. (iii) The power can be increased further if the test is permitted to have different significance levels α1 and α2 for the two sample sizes and it is required only that the expected significance level be equal to α = .005. Examples are: (a) with probability 12 take n1 = 2 observations and perform the test of significance at level α1 = .001, or take n2 = 16 observations and perform the test at level α2 = .009; (b) with probability 12 take n1 = 0 or n2 = 18 observations and let the respective significance levels be α1 = 0, α2 = .01. Note. This and related examples were discussed by Kruskal in a seminar held at Columbia University in 1954. A more detailed investigation of the phenomenon has been undertaken by Cohen (1958). Problem 3.14 If the sample space X is Euclidean and P0 , P1 have densities with respect to Lebesgue measure, there exists a nonrandomized most powerful test for testing P0 against P1 at every significance level α.12 [This is a consequence of Theorem 3.2.1 and the following lemma.13 Let f ≥ 0 and A f (x) dx = a. Given any 0 ≤ b ≤ a, there exists a subset B of A such that B f (x) dx = b.] 12 For more general results concerning the possibility of dispensing with randomized procedures, see Dvoretzky, Wald, and Wolfowitz (1951). 13 For a proof of this lemma see Halmos (1974, p. 174.) The lemma is a special case of a theorem of Lyapounov (1940); see Blackwell(1951). 96 3. Uniformly Most Powerful Tests Problem 3.15 Fully informative statistics. A statistic T is fully informative if for every decision problem the decision procedures based only on T form an essentially complete class. If P is dominated and T is fully informative, then T is sufficient. [Consider any pair of distributions P0 , P1 ∈ P with densities p0 , p1 , and let gi = pi /(p0 + p1 ). Suppose that T is fully informative, and let A be the subfield induced by T . Then A contains the subfield induced by (g0 , g1 ) since it contains every rejection which is unique most powerful for testing P0 against P1 (or P1 against P0 ) at some level α. Therefore, T is sufficient for every pair of distributions (P0 , P1 ), and hence by Problem 2.11 it is sufficient for P.] Problem 3.16 Based on X with distribution indexed by θ ∈ Ω, the problem is to test θ ∈ ω versus θ ∈ ω  . Suppose there exists a test φ such that Eθ [φ(X)] ≤ β for all θ in ω, where β < α. Show there exists a level α test φ∗ (X) such that Eθ [φ(X)] ≤ Eθ [φ∗ (X)] , for all θ in ω  and this inequality is strict if Eθ [φ(X)] < 1. Problem 3.17 A counterexample. Typically, as α varies the most powerful level α tests for testing a hypothesis H against a simple alternative are nested in the sense that the associated rejection regions, say Rα , satisfy Rα ⊂ Rα , for any α < α . Even if the most powerful tests are nonrandomized, this may be false. Suppose X takes values 1, 2, and 3 with probabilities 0.85, 0.1, and 0.05 under H and probabilities 0.7, 0.2, and 0.1 under K. (i) At any level < .15, the MP test is not unique. (ii) At α = .05 and α = .1, there exist unique nonrandomized MP tests and they are not nested. (iii) At these levels there exist MP tests φ and φ that are nested in the sense that φ(x) ≤ φ (x) for all x. [This example appears as Example 10.16 in Romano and Siegel (1986).] Problem 3.18 Under the setup of Theorem 3.2.1, show there always exists MP tests that are nested in the sense of Problem 3.17(iii). Problem 3.19 Suppose X1 , . . . , Xn are i.i.d. N (ξ, σ 2 ) with σ known. For testing ξ = 0 versus ξ = 0, the average power of a test φ = φ(X1 , . . . , Xn ) is given by  ∞ Eξ (φ)dΛ(µ) , −∞ where Λ is a probability distribution on the real line. Suppose that Λ is symmetric about 0; that is, Λ{E} = Λ{−E} for all Borel sets E. Show that, among α level  tests, the one maximizing average power rejects for large values of | i Xi |. Show that this test need not maximize average power if Λ is not symmetric. Problem 3.20 Let fθ , θ ∈ Ω, denote a family of densities with respect to a measure µ. (We assume Ω is endowed with a σ-field so that the densities fθ (x) are jointly measurable in θ and x.) Consider the problem of testing a simple null hypothesis θ = θ0 against the composite alternatives ΩK = {θ : θ = θ0 }. Let Λ be a probability distribution on ΩK . 3.10. Problems 97 (i) As explicitly as possibly, find a test φ that maximizes Ω Eθ (φ)dΛ(θ), subject K to it being level α. (ii) Let h(x) = fθ (x)dΛ(θ). Consider the nonrandomized φ test that rejects if and only if h(x)/fθ0 (x) > k, and suppose µ{x : h(x) = kfθ (x)} = 0. Then, φ is admissible at level α = Eθ0 (φ) in the sense that it is impossible that there exists another level α test φ such that Eθ (φ ) ≥ Eθ (φ) for all θ. (iii) Show that the test of Problem 3.19 is admissible. Section 3.3 Problem 3.21 In Example 3.21, show that p-value is indeed given by p̂ = p̂(X) = (11 − X)/10. Also, graph the c.d.f. of p̂ under H and show that the last inequality in (3.15) is an equality if and only u is of the form 0, . . . , 10. Problem 3.22 Suppose X has a continuous distribution function F . Show that F (X) is uniformly distributed on (0, 1). [The transformation from X to F (X) is known as the probability integral transformation.] Problem 3.23 Under the setup of Lemma 3.3.1, suppose the rejection regions are defined by Sα = {X : T (X) ≥ k(α)} (3.45) for some real-valued statistic T (X) and k(α) satisfying sup Pθ {T (X) ≥ k(α)} ≤ α . θ∈ΩH Then, show p̂ = sup P {T (X) ≥ t} , θ∈ΩH where t is the observed value of T (X). Problem 3.24 Under the setup of Lemma 3.3.1, show that there exists a realvalued statistic T (X) so that the rejection region is necessarily of the form (3.45). [Hint: Let T (X) = −p̂.] Problem 3.25 (i) If p̂ is uniform on (0, 1), show that −2 log(p̂) has the Chisquared distribution with 2 degrees of freedom. (ii) Suppose p̂1 , . . . , p̂s are i.i.d. uniform on (0, 1). Let F = −2 log(p̂1 · · · p̂s ). Argue that F has the Chi-squared distribution with 2s degrees of freedom. What can you say about F if the p̂i are independent and satisfy P {p̂i ≤ u} ≤ u for all 0 ≤ u ≤ 1? [Fisher (1934a) proposed F as a means of combining p-values from independent experiments.] Section 3.4 Problem 3.26 Let X be the number of successes in a n independent trials with probability p of success, and let φ(x) be the UMP test (3.16) for testing p ≤ p0 against p > p0 at level of significance α. 98 3. Uniformly Most Powerful Tests (i) For n = 6, p0 = .25 and the levels α = .05, .1, .2 determine C and γ, and the power of the test against p1 = .3, .4, .5, .6, .7. (ii) If p0 = .2 and α = .05, and it is desired to have power β ≥ .9 against p1 = .4, determine the necessary sample size (a) by using tables of the binomial distribution, (b) by using the normal approximation.14 (iii) Use the normal approximation to determine the sample size required when α = .05, β = .9, p0 = .01, p1 = .02. Problem 3.27 (i) A necessary and sufficient condition for densities pθ (x) to have monotone likelihood ratio in x, if the mixed second derivative ∂ 2 log pθ (x)/∂θ ∂x exists, is that this derivative is ≥ 0 for all θ and x. (ii) An equivalent condition is that pθ (x) ∂ 2 pθ (x) ∂pθ (x) ∂pθ (x) ≥ ∂θ ∂x ∂θ ∂x for all θ and x. Problem 3.28 Let the probability density pθ of X have monotone likelihood ratio in T (x), and consider the problem of testing H : θ ≤ θ0 against θ > θ0 . If the distribution of T is continuous, the p-value p̂ of the UMP test is given by p̂ = Pθ0 {T ≥ t}, where t is the observed value of T . This holds also without the assumption of continuity if for randomized tests p̂ is defined as the smallest significance level at which the hypothesis is rejected with probability 1. Show that, for any θ ≤ θ0 , Pθ {p̂ ≤ u} ≤ u for any 0 ≤ u ≤ 1. Problem 3.29 Let X1 , . . . , Xn be independently distributed with density (2θ)−1 e−x/2θ , x ≥ 0, and let Y1 ≤ · · · ≤ Yn be the ordered X’s. Assume that Y1 becomes available first, then Y2 , and so on, and that observation is continued until Yr has been observed. On the basis of Y1 , . . . , Yr it is desired to test H : θ ≥ θ0 = 1000 at level α = .05 against θ < θ0 . (i) Determine the rejection region when r = 4, and find the power of the test against θ1 = 500. (ii) Find the value of r required to get power β ≥ .95 against the alternative.  [In Problem 2.15, the distribution of [ ri=1 Yi + (n − r)Yr ]/θ was found to be χ2 with 2r degrees of freedom.] Problem 3.30 When a Poisson process with rate λ is observed for a time interval of length τ , the number X of events occurring has the Poisson distribution P (λτ ). Under an alternative scheme, the process is observed until r events have occurred, and the time T of observation is then a random variable such that 2λT has a χ2 -distribution with 2r degrees of freedom. For testing H : λ ≤ λ0 at level α one can, under either design, obtain a specified power β against an alternative λ1 by choosing τ and r sufficiently large. 14 Tables and approximations are discussed, for example, in Chapter 3 of Johnson and Kotz (1969). 3.10. Problems 99 (i) The ratio of the time of observation required for this purpose under the first design to the expected time required under the second is λτ /r. (ii) Determine for which values of λ each of the two designs is preferable when λ0 = 1, λ1 = 2, α = .05, β = 9. Problem 3.31 Let X = (X1 , . . . , Xn ) be a sample from the uniform distribution U (θ, θ + 1). (i) For testing H : θ ≤ θ0 against K : θ > θ0 at level α there exists a UMP test which rejects when min(X1 , . . . , Xn ) > θ0 +C(α) or max(X1 , . . . , Xn > θ0 + 1 for suitable C(α). (ii) The family U (θ, θ +1) does not have monotone likelihood ratio. [Additional results for this family are given in Birnbaum (1954b) and Pratt (1958).] [(ii) By Theorem 3.4.1, monotone likelihood ratio implies that the family of UMP test of H : θ ≤ θ0 against K : θ > θ0 generated as α varies from 0 to 1 is independent of θ0 ]. Problem 3.32 Let X be a single observation from the Cauchy density given at the end of Section 3.4. (i) Show that no UMP test exists for testing θ = 0 against θ > 0. (ii) Determine the totality of different shapes the MP level-α rejection region for testing θ = θ0 against θ = θ1 can take on for varying α and θ1 − θ0 . Problem 3.33 Let Xi be independently distributed as N (i∆, 1), i = 1, . . . , n. Show that there exists a UMP test of H : ∆ ≤ 0 against K : ∆ > 0, and determine it as explicitly as possible. Note. The following problems (and some of the Additional Problems in later chapters) refer to the gamma, Pareto, Weibull, and inverse Gaussian distributions. For more information about these distributions, see Chapter 17, 19, 20, and 25 respectively of Johnson and Kotz (1970). Problem 3.34 Let X1 , . . . , Xn be a sample from the gamma distribution Γ(g, b) with density 1 xg−1 e−x/b , Γ(g)bg 0 < x, 0 < b, g. Show that there exist a UMP test for testing (i) H : b ≤ b0 against b > b0 when g is known; (ii) H : g ≤ g0 against g > g0 when b is known. In each case give the form of the rejection region. Problem 3.35 A random variable X has the Weibull distribution W (b, c) if its density is c  x c−1 −(x/b)c e , x > 0, b, c > 0. b b (i) Show that this defines a probability density. 100 3. Uniformly Most Powerful Tests (ii) If X1 , . . . , Xn is a sample from W (b, c), with the shape parameter c known, show that there exists a UMP test of H : b ≤ b0 against b > b0 and give its form. Problem 3.36 Consider a single observation X from W (1, c). (i) The family of distributions does not have monotone likelihood ratio in x. (ii) The most powerful test of H : c = 1 against c = 2 rejects when X < k1 and when X > k2 . Show how to determine k1 and k2 . (iii) Generalize (ii) to arbitrary alternatives c1 > 1, and show that a UMP test of H : c = 1 against c > 1 does not exist. (iv) For any c1 > 1, the power function of the MP test of H : c = 1 against c = c1 is an increasing function of c. Problem 3.37 Let X1 , . . . , Xn be a sample from the inverse Gaussian distribution I(µ, τ ) with density 1  τ τ 2 exp − (x − µ) , x > 0, τ, µ > 0. 2πx3 2xµ2 Show that there exists a UMP test for testing (i) H : µ ≤ µ0 against µ > µ0 when τ is known; (ii) H : τ ≤ τ0 against τ > τ0 when µ is known. In each case give the form of the rejection region. (iii) The distribution of V = r(Xi −µ)2 /Xi µ2 is χ21 and hence that of τ µ)2 /Xi µ2 ] is χ2n .  [(Xi − [Let Y = min(Xi , µ2 /Xi ), Z = τ (Y − µ)2 /µ2 Y . Then Z = V and Z is χ21 [Shuster (1968)].] Note. The UMP test for (ii) is discussed in Chhikara and Folks (1976). Problem 3.38 Let X1 , · · · , Xn be a sample from a location family with common density f (x−θ), where the location parameter θ ∈ R and f (·) is known. Consider testing the null hypothesis that θ = θ0 versus an alternative θ = θ1 for some θ1 > θ0 . Suppose there exists a most powerful level α test of the form: reject the null hypothesis iff T = T (X1 , · · · , Xn ) > C, where C is a constant and T (X1 , . . . , Xn ) is location equivariant, i.e. T (X1 + c, . . . , Xn + c) = T (X1 , . . . , Xn ) + c for all constants c. Is the test also most powerful level α for testing the null hypothesis θ ≤ θ0 against the alternative θ = θ1 . Prove or give a counterexample. Problem 3.39 Extension of Lemma 3.4.2. Let P0 and P1 be two distributions with densities p0 , p1 such that p1 (x)/p0 (x) is a nondecreasing function of a realvalued statistic T (x). (i) If T has probability density pi when the original distribution of Pi , then p1 (t)/p0 (t) is nondecreasing in t. (ii) E0 ψ(T ) ≤ E1 ψ(T ) for any nondecreasing function ψ. 3.10. Problems 101 (iii) If p1 (x)/p0 (x) is a strictly increasing function of t = T (x), so is p1 (t)/p0 (t), and E0 ψ(T ) < E1 ψ(T ) unless ψ[T (x)] is constant a.e. (P0 + P1 ) or E0 ψ(T ) = E1 ψ(T ) = ± ∞. (iv) For any distinct distributions with densities p0 , p1 ,     p1 (X) p1 (X) −∞ ≤ E0 log < E1 log ≤ ∞. p0 (X) p0 (X) [(i): Without loss of generality suppose that p1 (x)/p0 (x) = T (x). Then for any integrable φ,    φ(t)p1 (t) dv(t) = φ[T (x)]T (x)p0 (x) dµ(x) = φ(t)tp0 (t) dv(t), and hence p1 (t)/p0 (t) = t a.e. (iv): The possibility E0 log[p1 (X)/p0 (X)] = ∞ is excluded, since by the convexity of the function log,     p1 (X) p1 (X) E0 log < log E0 = 0. p0 (X) p0 (X) Similarly for E1 . The strict inequality now follows from (iii) with T (x) = p1 (x)/p0 (x).] Problem 3.40 F0 , F1 are two cumulative distribution functions on the real line, then Fi (x) ≤ F0 (x) for all x if and only if E0 ψ(X) ≤ E1 ψ(X) for any nondecreasing function ψ. Problem 3.41 Let F and G be two continuous, strictly increasing c.d.f.s, and let k(u) = G[F −1 (u)], 0 < u < 1. (i) Show F and G are stochastically ordered, say F (x) ≤ G(x) for all x, if and only if k(u) ≤ u for all 0 < u < 1. (ii) If F and G have densities f and g, then show they are monotone likelihood ratio ordered, say g/f nondecreasing, if and only if k is convex. (iii) Use (i) and (ii) to give an alternative proof of the fact that MLR implies stochastic ordering. Problem 3.42 Let f (x)/[1 − F (x)] be the “mortality” of a subject at time x given that it has survived to this time. A c.d.f. F is said to be smaller than G in the hazard ordering if g(x) f (x) ≤ 1 − G(x) 1 − F (x) for all x . (3.46) (i) Show that (3.46) is equivalent to 1 − F (x) 1 − G(x) is nonincreasing. (3.47) (ii) Show that (3.46) holds if and only if k is starshaped. [A function k defined on an interval I ⊂ [0, ∞) is starshaped on I if k(λx) ≤ λk(x) whenever x ∈ I, λx ∈ I, 0 ≤ λ ≤ 1. Problems 3.41 and 3.42 are based on Lehmann and Rojo (1992).] 102 3. Uniformly Most Powerful Tests Section 3.5 Problem 3.43 (i) For n = 5, 10 and 1 − α = .95, graph the upper confidence limits p̄ and p̄∗ of Example 3.5.2 as functions of t = x + u. (ii) For the same values of n and α1 = α2 = .05, graph the lower and upper confidence limits p and p̄. Problem 3.44 Confidence bounds with minimum risk. Let L(θ, θ) be nonnegative and nonincreasing in its second argument for θ < θ, and equal to 0 for θ ≥ θ. If θ and θ∗ are two lower confidence bounds for θ such that P0 {θ ≤ θ } ≤ Pθ {θ∗ ≤ θ } for all θ ≤ θ, then Eθ L(θ, θ) ≤ Eθ L(θ, θ ∗ ). [Define two cumulative distribution functions F and F ∗ by F (u) = Pθ {θ ≤ u}/Pθ {θ∗ ≤ θ}, F ∗ (u) = Pθ {θ∗ ≤ u}/Pθ {θ∗ ≤ θ} for u < θ, F (u) = F ∗ (u) = 1 for u ≥ θ. Then F (u) ≤ F ∗ (u) for all u, and it follows from Problem 3.40 that  ∗ Eθ [L(θ, θ)] = Pθ {θ ≤ θ} L(θ, u)dF (u)  ∗ ≤ Pθ {θ ≤ θ} L(θ, u)dF ∗ (u) = Eθ [L(θ, θ ∗ )].] Section 3.6 Problem 3.45 If β(θ) denotes the power function of the UMP test of Corollary 3.4.1, and if the function Q of (3.19) is differentiable, then β  (θ) > 0 for all θ for which Q (θ) > 0. [To show that β  (θ0 ) > 0, consider the problem of maximizing, subject to Eθ0 φ(X) = α, the derivative β  (θ0 ) or equivalently the quantity Eθ0 [T (X) φ(X)].] Problem 3.46 Optimum selection procedures. On each member of a population n measurements (X1 , . . . , Xn ) = X are taken, for example the scores of n aptitude tests which are administered to judge the qualifications of candidates for a certain training program. A future measurement Y such as the score in a final test at the end of the program is of interest but unavailable. The joint distribution of X and Y is assumed known. (i) One wishes to select a given proportion α of the candidates in such a way as to maximize the expectation of Y for the selected group. This is achieved by selecting the candidates for which E(Y |x) ≥ C, where C is determined by the condition that the probability of a member being selected is α. When E(Y |x) = C, it may be necessary to randomized in order to get the exact value α. (ii) If instead the problem is to maximize the probability with which in the selected population Y is greater than or equal to some preassigned score y0 , one selects the candidates for which the conditional probability P {Y ≥ y0 |x} is sufficiently large. 3.10. Problems 103 [(i): Let φ(x) denote the probability with which a candidate with measurements x is to be selected. Then the problem is that of maximizing    ypY |x (y) φ(x)dy px (x)dx subject to  φ(x)px (x)dx = α.] Problem 3.47 The following example shows that Corollary 3.6.1 does not extend to a countably infinite family of distributions. Let pn be the uniform probability density on [0, 1 + 1/n], and p0 the uniform density on (0, 1). (p1 , p2 , . . .), that is, there do not exist (i) Then p0 is linearly independent of  constants c1 , c2 , . . . such that p0 = cn pn . (ii) There does not exist a test φ such that φp0 > α. φpn = α for n = 1, 2, . . . but Problem 3.48 Let F1 , . . . , Fm+1 be real-valued functions defined over a space U . A sufficient condition for u0 to maximize Fm+1 subject to Fi (u) ≤ ci (i = 1, . . . , m) is that it satisfies these side conditions, that it maximizes Fm+1 (u) −  ki Fi (u) for some constants ki ≥ 0, and that Fi (uo ) = ci for those values i for which ki > 0. Section 3.7 Problem 3.49 For a random variable X with binomial distribution b(p, n), determine the constants Ci , γ(i = 1, 2) in the UMP test (3.31) for testing H : p ≤ .2 or ≤ .7 when α = .1 and n = 15. Find the power of the test against the alternative p = .4. Problem 3.50 Totally positive families. A family of distributions with probability densities pθ (x), θ and x real-valued and varying over Ω and X respectively, is said to be totally positive of order r(TPr ) if for all x1 < · · · < xn and θ1 < · · · < θ n % % % p (x1 ) · · · pθ1 (xn ) % %≥0 for all n = 1, 2, . . . , r. (3.48) n = %% θ1 pθn (x1 ) · · · pθn (xn ) % It is said to be strictly totally positive of order r (ST Pr ) if strict inequality holds in (3.48). The family is said to be (strictly) totally positive of infinity if (3.48) holds for all n = 1, 2, . . . . These definitions apply not only to probability densities but to any real-valued functions pθ (x) of two real variables. (i) For r = 1, (3.48) states that pθ (x) ≥ 0; for r = 2, that pθ (x) has monotone likelihood ratio in x. (ii) If a(θ) > 0, b(x) > 0, and pθ (x) is STPr then so is a(θ)b(x)pθ (x). (iii) If a and b are real-valued functions mapping Ω and X onto Ω and X  and are strictly monotone in the same direction, and if pθ (x) is (STPr , then pθ (x ) with θ = a−1 (θ) and x = b−1 (x) is (ST P )r over (Ω , X  ). 104 3. Uniformly Most Powerful Tests Problem 3.51 Exponential families. The exponential family (3.19) with T (x) = x and Q(θ) = θ is STP∞ , with Ω the natural parameter space and X = (−∞, ∞). [That the determinant |eθi xj |, i, j = 1, . . . , n, is positive can be proved by induction. Divide the ith column by eθ1 xi , i = 1, . . . , n; subtract in the resulting determinant the (n − 1)st column from the nth, the (n − 2)nd from the (n − 1)st, . . . , the 1st from the 2nd; and expand the determinant obtained in this way by the first row. Then n is seen to have the same sign as n = |eηi xj − eηi xj −1 |, i, j = 2, . . . , n, where ηi = θi −θ1 . If this determinant is expanded by the first column one obtains a sum of the form a2 (eη2 x2 − eη2 x1 ) + · · · + an (eηn x2 − eηn x1 ) = h(x2 ) − h(x1 ) = (x2 − x1 )h (y2 ), where x1 < y2 < x2 . Rewriting h (y2 ) as a determinant of which all columns but the first coincide with those of n and proceeding in the same manner with the columns, one reduces the determinant to |eηi yj |, i, j = 2, . . . , n, which is positive by the induction hypothesis.] Problem 3.52 STP3 . Let θ and x be real-valued, and suppose that the probability densities pθ (x) are such that pθ (x)/pθ (x) is strictly increasing in x for θ < θ . Then the following two conditions are equivalent: (a) For θ1 < θ2 < θ3 and k1 , k2 , k3 > 0, let g(x) = k1 pθ1 (x) − k2 pθ2 (x) + k3 pθ3 (x). If g(x1 ) − g(x3 ) = 0, then the function g is positive outside the interval (x1 , x3 ) and negative inside. (b) The determinant 3 given by (3.48) is positive for all θ1 < θ2 < θ3 , x1 < x2 < x3 . [It follows from (a) that the equation g(x) = 0 has at most two solutions.] [That (b) implies (a) can be seen for x1 , < x2 < x3 by considering the determinant % % % g(x1 ) g(x2 ) g(x3 ) %% % % pθ2 (x1 ) pθ2 (x2 ) pθ2 (x3 ) % % % % pθ (x1 ) pθ (x2 ) pθ (x3 ) % 3 3 3 Suppose conversely that (a) holds. Monotonicity of the likelihood ratios implies that the rank of 3 is at least two, so that there exist constants k1 , k2 , k3 such that g(x1 ) = g(x3 ) = 0. That the k s are positive follows again from the monotonicity of the likelihood ratios.] Problem 3.53 Extension of Theorem 3.7.1. The conclusions of Theorem 3.7.1 remain valid if the density of a sufficient statistic T (which without loss of generality will be taken to be X), say pθ (x), is STP3 and is continuous in x for each θ. [The two properties of exponential families that are used in the proof of Theorem 3.7.1 are continuity in x and (a) of the preceding problem.] Problem 3.54 For testing the hypothesis H  : θ1 ≤ θ ≤ θ2 (θ1 ≤ θ2 ) against the alternatives θ < θ1 or θ > θ2 , or the hypothesis θ = θ0 against the alternatives 3.10. Problems 105 θ = θ0 , in an exponential family or more generally in a family of distributions satisfying the assumptions of Problem 3.53, a UMP test does not exist. [This follows from a consideration of the UMP tests for the one-sided hypotheses H1 : θ ≥ θ1 and H2 : θ ≤ θ2 .] Problem 3.55 Let f , g be two probability densities with respect to µ. For testing the hypothesis H : θ ≤ θ0 or θ ≥ θ1 (0 < θ0 < θ1 < 1) against the alternatives θ0 < θ < θ1 , in the family P = {θf (x)+(1−θ)g(x), 0 ≤ θ ≤ 1}, the test ϕ(x) ≡ α is UMP at level α. Section 3.8 Problem 3.56 Let the variables Xi (i = 1, . . . , s) be independently  distributed with Poisson distribution P (λi ). For testing the hypothesis H : λj ≤ a (for example, that the combined radioactivity of a number of pieces of radioactive  material does not exceed a), there exists a UMP test, which rejects when Xj > C. [If the joint distribution of the X’s is factored into the marginal distribution of Xj (Poisson  with mean  λj ) times the conditional distribution of the variables Yi = Xj / Xj given Xj (multinomial with probabilities pi = λi / λj ), the argument is analogous to that given in Example 3.8.1.] Problem 3.57 Confidence bounds for a median. Let X1 , . . . , Xn be a sample from a continuous cumulative distribution functions F . Let ξ be the unique median of F if it exists, or more generally let ξ = inf{ξ  : F (ξ  ) = 12 }. (i) If the ordered X’s are X(1) < · · · < X(n) , a uniformly most accurate lower confidence bound for ξ is ξ = X(k) with probability ρ, ξ = X(k+1) with probability 1 − ρ, where k and ρ are determined by     n n   n 1 n 1 + (1 − ρ) = 1 − α. ρ j 2n j 2n j=k j=k+1 (ii) This bound has confidence coefficient 1 − α for any median of F . (iii) Determine most accurate lower confidence bounds for the 100p-percentile ξ of F defined by ξ = inf{ξ  : F (ξ  ) = p}. [For fixed to the problem of testing H : ξ = ξ0 to against K : ξ > ξ0 is equivalent to testing H  : p = 12 against K  : p < 12 .] Problem 3.58 A counterexample. Typically, as α varies the most powerful level α tests for testing a hypothesis H against a simple alternative are nested in the sense that the associated rejection regions, say Rα , satisfy Rα ⊂ Rα , for any α < α . The following example shows that this need not be satisfied for composite H. Let X take on the values 1, 2, 3, 4 with probabilities under distributions P0 , P1 , Q: P0 P1 Q 1 2 3 4 2 13 4 13 4 13 4 13 2 13 3 13 3 13 1 13 2 13 4 13 6 13 4 13 106 3. Uniformly Most Powerful Tests Then the most powerful test for testing the hypothesis that the distribution of 5 when X is P0 or P1 against the alternative that it is Q rejects at level α = 13 6 X = 1 or 3, and at level α = 13 when X = 1 or 2. Problem 3.59 Let X and Y be the number of successes in two sets of n binomial trials with probabilities p1 and p2 of success. (i) The most powerful test of the hypothesis H : p2 ≤ p1 against an alternative (p1 , p2 ) with p1 < p2 and p1 +p2 = 1 at level α < 12 rejects when Y −X > C and with probability γ when Y − X = C. (ii) This test is not UMP against the alternatives p1 < p2 . [(i): Take the distribution Λ assigning probability 1 to the point p1 = p2 = 12 as an a priori distribution over H. The most powerful test against (p1 , p2 ) is then the one proposed above. To see that Λ is least favorable, consider the probability of rejection β(p1 , p2 ) for p1 = p2 = p. By symmetry this is given by 2β(p, p) = P {|Y − X| > C} + γP {|Y − X| = C}. Let Xi be 1 or 0 as the ith trial in the first series is a success or failure, and let nY1 , be defined analogously with respect to the second series. Then Y1 − X = i−1 (Yi − Xi ), and the fact that 2β(p, p) attains its maximum for p = 2 can be proved by induction over n. (ii): Since β(p, p) < α for p = 1, the power β(p1 , p2 ) is < α for alternatives p1 < p2 sufficiently close to the line p1 = p2 . That the test is not UMP now follows from a comparison with φ(x, y) ≡ α.] Problem 3.60 Sufficient statistics with nuisance parameters. (i) A statistic T is said to be partially sufficient for θ in the presence of a nuisance parameter η if the parameter space is the direct product of the set of possible θ- and η-values, and if the following two conditions hold: (a) the conditional distribution given T = t depends only on η; (b) the marginal distribution of T depends only on θ. If these conditions are satisfied, there exists a UMP test for testing the composite hypothesis H : θ = θ0 against the composite class of alternatives θ = θ1 , which depends only on T . (ii) Part (i) provides an alternative proof that the test of Example 3.8.1 is UMP. [Let ψ0 (t) be the most powerful level α test for testing θ0 against θ1 that depends only on t, let φ(x) be any level-α test, and let ψ(t) = Eη1 [φ(X) | t]. Since Eθi ψ(T ) = Eθi ,η1 φ(X), it follows that ψ is a level-α test of H and its power, and therefore the power of φ, does not exceed the power of ψ0 .] Note. For further discussion of this and related concepts of partial sufficiency see Fraser (1956), Dawid (1975), Sprott (1975), Basu (1978), and BarndorffNielsen (1978). 3.11. Notes 107 Section 3.9 Problem 3.61 Let X1 , . . . , X and Y1 , . . . , Yn be independent samples from N (ξ, 1) and N (η, 1), and consider the hypothesis H : η ≤ ξ against K : η > ξ. There exists a UMP test, and it rejects the hypothesis when Ȳ − X̄ is too large. [If ξ1 < η1 , is a particular alternative, the distribution assigning probability 1 to the point η = ξ = (mξ1 + nη1 )/(m + n) is least favorable.] Problem 3.62 Let X1 , . . . , Xm ; Y1 , . . . , Yn be independently, normally distributed with means ξ and η, and variances a σ 2 and τ 2 respectively, and consider the hypothesis H : τ ≤ σ a against K : σ < τ . (i)  If ξ and η are known, there exists a UMP test given by the rejection region (Yj − η)2 / (Xi − ξ)2 ≥ C. (ii) No UMP test exists when ξ and η are unknown. Problem 3.63 Suppose X is a k × 1 random vector with E(|X|2 ) < ∞ and covariance matrix Σ. Let A be an m × k (nonrandom) matrix and let Y = AX. Show Y has mean vector AE(X) and covariance matrix AΣAT . Problem 3.64 Suppose (X1 , . . . , Xk ) has the multivariate normal distribution with unknown mean vector ξ = (ξ1 , . . . , ξk ) and known covariance matrix Σ. Suppose X1 is independent of (X2 , . . . , Xk ). Show that X1 is partially sufficient for ξ1 in the sense of Problem 3.60. Provide an alternative argument for Case 2 of Example 3.9.2. Problem 3.65 In Example 3.9.2, Case 2, verify the claim for the least favorable distribution. Problem 3.66 In Example 3.9.3, provide the details for Cases 3 and 4. 3.11 Notes Hypothesis testing developed gradually, with early instances frequently being rather vague statements of the significance or nonsignificance of a set of observations. Isolated applications are found in the 18th century [Arbuthnot (1710), Daniel Bernoulli (1734), and Laplace (1773), for example] and centuries earlier in the Royal Mint’s Trial of the Pyx [discussed by Stigler (1977)]. They became more frequent in the 19th century in the writings of such authors as Gavarret (1840), Lexis (1875, 1877), and Edgeworth (1885). A new stage began with the work of Karl Pearson, particularly his χ2 paper of 1900, followed in the decade 1915–1925 by Fisher’s normal theory and χ2 tests. Fisher presented this work systematically in his enormously influential book Statistical Methods for Research Workers (1925b). The first authors to recognize that the rational choice of a test must involve consideration not only of the hypothesis but also of the alternatives against which it is being tested were Neyman and F. S. Pearson (1928). They introduced the distinction between errors of the first and second kind, and thereby motivated their 108 3. Uniformly Most Powerful Tests proposal of the likelihood-ratio criterion as a general method of test construction. These considerations were carried to their logical conclusion by Neyman and Pearson in their paper of 1933. in which they developed the theory of UMP tests. Accounts of their collaboration can be found in Pearson’s recollections (1966), and in the biography of Neyman by Reid (1982). The Neyman–Pearson lemma has been generalized in many directions, including the results in Sections 3.6, 3.8 and 3.9. Dantzig and Wald (1951) give necessary conditions including those of Theorem 3.6.1, for a critical function which maximizes an integral subject to a number of integral side conditions, to satisfy (3.28). The role of the Neyman–Pearson lemma in hypothesis testing is surveyed in Lehmann (1985a). An extension to a selection problem, proposed by Birnbaum and Chapman (1950), is sketched in Problem 3.46. Further developments in this area are reviewed in Gibbons (1986, 1988). Grenander (1981) applies the fundamental lemma to problems in stochastic processes. Lemmas 3.4.1, 3.4.2, and 3.7.1 are due to Lehmann (1961). Complete class results for simple null hypothesis testing problems are obtained in Brown and Marden (1989). The earliest example of confidence intervals appears to occur in the work of Laplace (1812). who points out how an (approximate) probability statement concerning the difference between an observed frequency and a binomial probability p can be inverted to obtain an associated interval for p. Other examples can be found in the work of Gauss (1816), Fourier (1826), and Lexis (1875). However, in all these cases, although the statements made are formally correct, the authors appear to consider the parameter as the variable which with the stated probability falls in the fixed confidence interval. The proper interpretation seems to have been pointed out for the first time by E. B. Wilson (1927). About the same time two examples of exact confidence statements were given by Working and Hotelling (1929) and Hotelling (1931). A general method for obtaining exact confidence bounds for a real-valued parameter in a continuous distribution was proposed by Fisher (1930), who however later disavowed this interpretation of his work. For a discussion of Fisher’s controversial concept of fiducial probability, see Section 5.7. At about the same time,15 a completely general theory of confidence statements was developed by Neyman and shown by him to be intimately related to the theory of hypothesis testing. A detailed account of this work, which underlies the treatment given here, was published by Neyman in his papers of 1937 and 1938. The calculation of p-values was the standard approach to hypothesis testing throughout the 19th century and continues to be widely used today. For various questions of interpretation, extensions, and critiques, see Cox (1977), Berger and Sellke (1987), Marden (1991), Hwang, Casella, Robert, Wells and Farrell (1992), Lehmann (1993), Robert (1994), Berger, Brown and Wolpert (1994), Meng (1994), Blyth and Staudte (1995, 1997), Liu and Singh (1997), Sackrowitz and Samuel-Cahn (1999), Marden (2000), Sellke et al. (2001), and Berger (2003). Extensions of p-values to hypotheses with nuisance parameters is discussed by Berger and Boos (1994) and Bayarri and Berger (2000), and the large-sample 15 Cf. Neyman (1941b). 3.11. Notes 109 behavior of p-values in Lambert and Hall (1982) and Robins et al. (2000). An optimality theory in terms of p-values is sketched by Schweder (1988), and pvalues for the simultaneous testing of several hypotheses is treated by Schweder and Spjøtvoll (1982), Westfall and Young (1993), and by Dudoit et al. (2003). An important use of p-values occurs in meta-analysis when one is dealing with the combination of results from independent experiments. The early literature on this topic is reviewed in Hedges and Olkin (1985, Chapter 3). Additional references are Marden (1982b, 1985), Scholz (1982) and a review article by Becker (1997). Associated confidence intervals are proposed by Littell and Louv (1981). 4 Unbiasedness: Theory and First Applications 4.1 Unbiasedness For Hypothesis Testing A simple condition that one may wish to impose on tests of the hypothesis H : θ ∈ ΩH against the composite class of alternatives K : θ ∈ ΩK is that for no alternative in K should the probability of rejection be less than the size of the test. Unless this condition is satisfied, there will exist alternatives under which acceptance of the hypothesis is more likely than in some cases in which the hypothesis is true. A test φ for which the above condition holds, that is, for which the power function βφ (θ) = Eθ φ(X) satisfies βφ (θ) ≤ α βφ (θ) ≥ α if if θ ∈ ΩH , θ ∈ ΩK , (4.1) is said to be unbiased. For an appropriate loss function this was seen in Chapter 1 to be a particular case of the general definition of unbiasedness given there. Whenever a UMP test exists, it is unbiased, since its power cannot fall below that of the test φ(x) ≡ α. For a large class of problems for which a UMP test does not exist, there does exist a UMP unbiased test. This is the case in particular for certain hypotheses of the form θ ≤ θ0 or θ = θ0 , where the distribution of the random observables depends on other parameters besides θ. When βφ (θ) is a continuous function of θ, unbiasedness implies βφ (θ) = α for all θ in ω, (4.2) where ω is the common boundary of ΩH and ΩK that is, the set of points θ that are points or limit points of both ΩH and ΩK . Tests satisfying this condition are said to be similar on the boundary (of H and K). Since it is more convenient to 4.2. One-Parameter Exponential Families 111 work with (4.2) than with (4.1), the following lemma plays an important role in the determination of UMP unbiased tests. Lemma 4.1.1 If the distributions Pθ are such that the power function of every test is continuous, and if φ0 is UMP among all tests satisfying (4.2) and is a level-α test of H then φ0 is UMP unbiased. Proof. The class of tests satisfying (4.2) contains the class of unbiased tests, and hence φ0 is uniformly at least as powerful as any unbiased test. On the other hand, φ0 is unbiased, since it is uniformly at least as powerful as φ(x) ≡ α. 4.2 One-Parameter Exponential Families Let θ be a real parameter, and X = (X1 , . . . , Xn ) a random vector with probability density (with respect to some measure µ) pθ (x) = C(θ)eθT (x) h(x). It was seen in Chapter 3 that a UMP test exists when the hypothesis H and the class K of alternatives are given by (i) H : θ ≤ θ0 , K : θ > θ0 (Corollary 3.4.1) and (ii) H : θ ≤ θ1 or θ ≥ θ2 (θ1 < θ2 ), K : θ1 < θ < θ2 (Theorem 3.7.1), but not for (iii) H : θ1 ≤ θ ≤ θ2 , K : θ < θ1 or θ > θ2 . We shall now show that in case (iii) there does exist a UMP unbiased test given by ⎧ ⎨ 1 when T (x) < C1 or > C2 , γi when T (x) = Ci , i = 1, 2, (4.3) φ(x) = ⎩ 0 when C1 < T (x) < C2 , where the C’s and γ’s are determined by Eθ1 φ(X) = Eθ2 φ(X) = α. (4.4) The power function Eθ φ(X) is continuous by Theorem 2.7.1, so that Lemma 4.1.1 is applicable. The set ω consists of the two points θ1 and θ2 , and we therefore consider first the problem of maximizing Eθ φ(X) for some θ outside the interval [θ1 , θ2 ], subject to (4.4). If this problem is restated in terms of 1 − φ(x), it follows from part (ii) of Theorem 3.7.1 that its solution is given by (4.3) and (4.4). This test is therefore UMP among those satisfying (4.4), and hence UMP unbiased by Lemma 4.1.1. It further follows from part (iii) of the theorem that the power function of the test has a minimum at a point between θ1 and θ2 , and is strictly increasing as θ tends away from this minimum in either direction. A closely related problem is that of testing (iv) H : θ = θ0 against the alternatives θ = θ0 . For this there also exists a UMP unbiased test given by (4.3), but the constants are now determined by Eθ0 [φ(X)] = α (4.5) Eθ0 [T (X)φ(X)] = Eθ0 [T (X)]α. (4.6) and 112 4. Unbiasedness: Theory and First Applications To see this, let θ be any particular alternative, and restrict attention to the sufficient statistic T , the distribution of which by Lemma 2.7.2, is of the form dPθ (t) = C(θ)eθt dν(t). Unbiasedness of a test ψ(t) implies (4.5) with φ(x) = ψ[T (x)]; also that the power function β(θ) = Eθ [ψ(T )] must have a minimum at θ = θ0 . By Theorem 2.7.1, the function β(θ) is differentiable, and the derivative can be computed by differentiating Eθ ψ(T ) under the expectation sign, so that for all tests ψ(t) β  (θ) = Eθ [T ψ(T )] + C  (θ) Eθ [ψ(T )]. C(θ) For ψ(t) ≡ α, this equation becomes 0 = Eθ (T ) + C  (θ) . C(θ) Substituting this in the expression for β  (θ) gives β  (θ) = Eθ [T ψ(T )] − Eθ (T )Eθ [ψ(T )], and hence unbiasedness implies (4.6) in addition to (4.5). Let M be the set of points (Eθ0 [ψ(T )], Eθ0 [T ψ(T )]) as ψ ranges over the totality of critical functions. Then M is convex and contains all points (u, uEθ0 (T )) with 0 < u < 1. It also contains points (α, u2 ) with u2 > αEθ0 (T ). This follows from the fact that there exist tests with Eθ0 [ψ(T )] = α and β  (θ0 ) > 0 (see Problem 3.45). Since similarly M contains points (α, u1 ) with u1 < αEθ0 (T ), the point (α, αEθ0 (T )) is an inner point of M . Therefore, by Theorem 3.6.1(iv), there exist constants k1 , k2 and a test ψ(t) satisfying (4.5) and (4.6) with φ(x) = ψ[T (x)], such that ψ(t) = 1 when C(θ0 )(k1 + k2 t)eθ0 t < C(θ )eθ  t and therefore when a1 + a2 t < ebt . This region is either one-sided or the outside of an interval. By Theorem 3.4.1, a one-sided test has a strictly monotone power function and therefore cannot satisfy (4.6). Thus ψ(t) is 1 when t < C1 or > C2 , and the most powerful test subject to (4.5) and (4.6) is given by (4.3). This test is unbiased, as is seen by comparing it with φ(x) ≡ α. It is then also UMP unbiased, since the class of tests satisfying (4.5) and (4.6) includes the class of unbiased tests. A simplification of this test is possible if for θ = θ0 the distribution of T is symmetric about some point a, that is, if Pθ0 {T < a − u} = Pθ0 {T > a + u} for all real u. Any test which is symmetric about a and satisfies (4.5) must also satisfy (4.6), since Eθ0 [T ψ(T )] = Eθ0 [(T − a)ψ(T )] + aEθ0 ψ(T ) = aα = Eθ0 (T )α. The C’s and γ’s are therefore determined by Pθ0 {T < C1 } + γ1 Pθ0 {T = C1 } = C2 = 2a − C1 , α , 2 γ2 = γ1 . The above tests of the hypotheses θ1 ≤ θ ≤ θ2 and θ = θ0 are strictly unbiased in the sense that the power is > α for all alternatives θ. For the first of these 4.2. One-Parameter Exponential Families 113 tests, given by (4.3) and (4.4), strict unbiasedness is an immediate consequence of Theorem 3.7.1(iii). This states in fact that the power of the test has a minimum at a point θ0 between θ1 and θ2 and increases strictly as θ tends away from θ0 in either direction. The second of the tests, determined by (4.3), (4.5), and (4.6), has a continuous power function with a minimum of α at θ = θ0 . Thus there exist θ1 < θ0 < θ2 such that β(θ1 ) = β(θ2 ) = c where α ≤ c < 1. The test therefore coincides with the UMP unbiased level-c test of the hypothesis θ1 ≤ θ ≤ θ2 , and the power increases strictly as θ moves away from θ0 in either direction. This proves the desired result. Example 4.2.1 (Binomial) Let X be the number of successes in n binomial trials with probability p of success. A theory to be tested assigns to p the value p0 , so that one wishes to test the hypothesis H : p = p0 . When rejecting H one will usually wish to state also whether p appears to be less or greater than p0 . If, however, the conclusion that p = p0 in any case requires further investigation, the preliminary decision is essentially between the two possibilities that the data do or do not contradict the hypothesis p = p0 . The formulation of the problem as one of hypothesis testing may then be appropriate. The UMP unbiased test of H is given by (4.3) with T (X) = X. The condition (4.5) becomes     C2 −1 2  n x n−x  n i n−Ci + (1 − γi ) = 1 − α, pC p0 q0 0 q0 x C i i=1 x=C +1 1 and the left-hand side of this can be obtained from tables of the individual probabilities and cumulative distribution function of X. The condition (4.6), with the help of the identity     n − 1 x−1 (n−1)−(x−1) n x n−x = np0 q0 x p p0 q0 x−1 0 x reduces to   n − 1 x−1 (n−1)−(x−1) q0 p x−1 0 x=C1 +1   2  n−1 i −1 (n−1)−(Ci −1) + (1 − γi ) q0 =1−α pC 0 C i −1 i=1 C2 −1  the left-hand side of which can be computed from the binomial tables. For sample sizes which are not too small, and values of p0 which are not too close to 0 or 1, the distribution of X is therefore approximately symmetric. In this case, the much simpler “equal tails” test, for which the C’s and γ’s are determined by     C1 −1  n x (n−x) n 1 n−C1 + γ1 pC p0 q0 0 q0 x C 1 x=0     n  n n x n−x α C2 n−C2 = γ2 + = , p q p0 q0 2 C2 0 0 x x=C +1 2 114 4. Unbiasedness: Theory and First Applications is approximately unbiased, and constitutes a reasonable approximation to the unbiased test. Note, however, that this approximation requires large sample sizes when p0 is close to 0 or 1; in this connection, see Example 5.7.2 which discusses the corresponding problem of confidence intervals for p. The literature on this and other approximations to the binomial distribution is reviewed in Johnson, Kotz and Kemp (1992). See also the related discussion in Example 5.7.2. Example 4.2.2 (Normal variance) Let X = (X1 , . . . , Xn ) be a sample from a normal distribution with mean 0 and variance σ 2 , so that the density of the X’s is     1 1  2 √ x exp − . i 2πσ 2 2πσ  Then T (X) = Xi2 is sufficient for σ 2 , and has probability density (1/σ 2 )fn (y/σ 2 ), where 1 fn (y) = n/2 y > 0, y (n/2)−1 e(y/2) , 2 Γ(n/2) is the density of a χ2 -distribution with n degrees of freedom. For varying σ, these distributions form an exponential family, which arises also in problems of life testing (see Problem 2.15), and concerning normally distributed variables with unknown mean and variance (Section 5.3). The acceptance region of the UMP unbiased test of the hypothesis H : σ = σ0 is  x2i ≤ C2 C1 ≤ σ02 with  C2 fn (y) dy = 1 − α C1 and  C2 C1  (1 − α)Eσ0 ( Xi2 ) yfn (y) dy = = n(1 − α). σ02 For the determination of the constants from tables of the χ2 -distribution, it is convenient to use the identity yfn (y) = nfn+2 (y), to rewrite the second condition as  C2 fn+2 (y) dy = 1 − α. C1 Alternatively, one can integrate condition to  C2 C1 n/2 −C1 /2 C1 e fn (y) dy by parts to reduce the second n/2 −C2 /2 = C2 e . [For tables giving C1 and C2 see Pachares (1961).] Actually, unless n is very small or σ0 very close to 0 or ∞, the equal-tails test given by  ∞  C1 α fn (y) dy = fn (y) dy = 2 0 C2 4.3. Similarity and Completeness 115 is a good approximation to the unbiased test. This follows from the fact that T , suitably normalized, tends to be normally and hence symmetrically distributed for large n. UMP unbiased tests of the hypotheses (iii) H : θ1 ≤ θ ≤ θ2 and (iv) H : θ = θ0 against two-sided alternatives exist not only when the family pθ (x) is exponential but also more generally when it is strictly totally positive (STP∞ ). A proof of (iv) in this case is given in Brown, Johnstone, and MacGibbon (1981); the proof of (iii) follows from Problem 3.53. 4.3 Similarity and Completeness In many important testing problems, the hypothesis concerns a single real-valued parameter, but the distribution of the observable random variables depends in addition on certain nuisance parameters. For a large class of such problems a UMP unbiased test exists and can be found through the method indicated by Lemma 4.1.1. This requires the characterization of the tests φ, which satisfy Eθ φ(X) = α for all distributions of X belonging to a given family P X = {Pθ , θ ∈ ω}. Such tests are called similar with respect to P X or ω, since if φ is nonrandomized with critical region S, the latter is “similar to the sample space” X in that both the probability Pθ {X ∈ S} and Pθ {X ∈ X } are independent of θ ∈ ω. Let T be a sufficient statistic for P X , and let P T denote the family {PθT , θ ∈ ω} of distributions of T as θ ranges over ω. Then any test satisfying1 E[φ(X)|t] = α a.e. P T (4.7) is similar with respect to P , since then X Eθ [φ(X)] = Eθ {E[φ(X)|T ]} = α for all θ ∈ ω. A test satisfying (4.7) is said to have Neyman structure with respect to T . It is characterized by the fact that the conditional probability of rejection is α on each of the surfaces T = t. Since the distribution on each such surface is independent of θ for θ ∈ ω, the condition (4.7) essentially reduces the problem to that of testing a simple hypothesis for each value of t. It is frequently easy to obtain a most powerful test among those having Neyman structure, by solving the optimum problem on each surface separately. The resulting test is then most powerful among all similar tests provided every similar test has Neyman structure. A condition for this to be the case can be given in terms of the following definition. A family P of probability distributions P is complete if EP [f (X)] = 0 for all P ∈P (4.8) implies f (x) = 0 a.e. P. (4.9) 1 A statement is said to hold a.e. P if it holds except on a set N with P (N ) = 0 for all P ∈ P. 116 4. Unbiasedness: Theory and First Applications In applications, P will be the family of distributions of a sufficient statistic. Example 4.3.1 Consider n independent trials with probability p of success, and let Xi be 1 or 0 as the ith trial is a success or failure. Then T = X1 + · · · + Xn is a sufficient statistic for p, and the family of its possible distributions is P = {b(p, n), 0 < p ≤ 1}. For this family (4.8) implies that   n  n t f (t) for all 0 < ρ < ∞, ρ =0 t t=0 where ρ = p/(1 − p). The left-hand side is a polynomial in ρ, all the coefficients of which must be zero. Hence f (t) = 0 for t = 0, . . . , n and the binomial family of distributions of T is complete. Example 4.3.2 Let X1 , . . . , Xn be a sample from the uniform distribution U (0, θ), 0 < θ < ∞. Then T = max(X1 , . . . , Xn ) is a sufficient statistic for θ, and (4.8) becomes  θ  f (t) dPθT (t) = nθ−n f (t) · tn−1 dt = 0 for all θ. 0 Let f (t) = f + (t)−f − (t) where f + and f − denote the positive and negative parts of f respectively. Then   f + (t)tn−1 dt and v − (A) = f − (t)tn−1 dt v + (A) = A A are two measures over the Borel sets on (0, ∞), which agree for all intervals and hence for all A. This implies f + (t) = f − (t) except possibly on a set of Lebesgue measure zero, and hence f (t) = 0 a.e. P T . Example 4.3.3 Let X1 , . . . , Xm ; Y1 , . . . , Yn be independently normally distributed as N (ξ, σ 2 ) and N (ξ, τ 2 ) respectively. Then the joint density of the variables is  1  2 ξ  1  2 ξ  C(ξ, σ, τ ) exp − 2 xi + 2 xi − 2 yj + 2 yj . 2σ σ 2τ τ The statistic T =  Xi ,  Xi2 ,  Yj ,  Yj2    Xi /m) is identiis sufficient; it is, however, not complete, since E( Yj /n − cally zero. If the Y ’s are instead distributed with a mean E(Y ) = η which varies independently of ξ, the set of possible values of the parameters θ1 = −1/2σ 2 , θ2 = ξ/σ 2 , θ3 = −1/2τ 2 , θ4 = η/τ 2 contains a four-dimensional rectangle, and it follows from Theorem 4.3.1 below that P T is complete. Completeness of a large class of families of distributions including that of Example 4.3.1 is covered by the following theorem. Theorem 4.3.1 Let X be a random vector with probability distribution ! k "  dPθ (x) = C(θ) exp θj Tj (x) dµ(x), j=1 4.3. Similarity and Completeness 117 and let P T be the family of distributions of T = (T1 (X), . . . , Tk (X)) as θ ranges over the set ω. Then P T is complete provided ω contains a k-dimensional rectangle. Proof. By making a translation of the parameter space one can assume without loss of generality that ω contains the rectangle I = {(θ1 , . . . , θk ) : −a ≤ θj ≤ a, j = 1, . . . , k} Let f (t) = f + (t) − f − (t) be such that Eθ f (T ) = 0 for all θ ∈ ω. Then for all θ ∈ I, if ν denotes the measure induced in T -space by the measure µ,     e θj tj f + (t) dν(t) = e θj tj f − (t) dν(t) and hence in particular  f + (t) dν(t) =  f − (t) dν(t). Dividing f by a constant, one can take the common value of these two integrals to be 1, so that dP + (t) = f + (t) dν(t) and dP − (t) = f − (t) dν(t) are probability measures, and     e θj tj dP + (t) = e θj tj dP − (t) for all θ in I. Changing the point of view, consider these integrals now as functions of the complex variables θj = ξj + iηj , j = 1, . . . , k. For any fixed θ1 , . . . , θj−1 , θj+1 , . . . , θk with real parts strictly between −a and +a, they are by Theorem 2.7.1 analytic functions of θj in the strip Rj : −a < ξj < a, −∞ < ηj < ∞ of the complex plane. For θ2 , . . . , θk fixed, real, and between −a and a, equality of the integrals holds on the line segment {(ξ1 , η1 ) : −a < ξ1 < a, η1 = 0} and can therefore be extended to the strip R1 , in which the integrals are analytic. By induction the equality can be extended to the complex region {(θ1 , . . . , θk ) : (ξj , ηj ) ∈ Rj for j = 1, . . . , k}. It follows in particular that for all real (η1 , . . . , ηk )     ei ηj tj dP + (t) = ei ηj tj dP − (t). These integrals are the characteristic functions of the distributions P + and P − respectively, and by the uniqueness theorem for characteristic functions,2 the two distributions P + and P − coincide. From the definition of these distributions it then follows that f + (t) = f − (t) a.e. ν, and hence that f (t) = 0 a.e. P T , as was to be proved. 2 See for example Section 26 of Billingsley (1995). 118 4. Unbiasedness: Theory and First Applications Example 4.3.4 (Nonparametric completeness.) Let X1 , . . . , XN be independently and identically distributed with cumulative distribution function F ∈ F , where F is the family of all absolutely continuous distributions. Then the set of order statistics T (X) = (X(1) , . . . , X(N ) ) was shown to be sufficient for F in Section  2.6. We  shall now prove it to be complete. Since, by Example 2.4.1, T  (X) = ( Xi , Xi2 , . . . , XiN ) is equivalent to T (X) in the sense that both induce the same subfield of the sample space, T  (X) is also sufficient and is complete if and only if T (X) is complete. To prove the completeness of T  (X) and thereby that of T (X), consider the family of densities f (X) = C(θ1 , . . . , θN ) exp(−x2N + θ1 x + · · · + θN xN ), where C is a normalizing constant. These densities are defined for all values of the θ’s since the integral of the exponential is finite, and their distributions belong to F. The density of a sample of size N is     N C N exp − x2N + θ1 xj + . . . + θN xj j and these densities constitute an exponential family F0 . By Theorem 4.3.1, T  (X) is complete for F0 and hence also for F, as was to be proved. The same method of proof establishes also the following more general result. Let Xij , j = 1, . . . , Ni , i = 1, . . . , c, be independently distributed with abso(1) (N ) lutely continuous distributions Fi , and let Xi < · · · < Xi i denote the Ni observations Xi1 , . . . , XiNi arranged in increasing order. Then the set of order statistics (1) (N1 ) (X1 , . . . , X1 , . . . , Xc(1) , . . . , Xc(Nc ) ) is sufficient and complete for the family of distributions obtained by letting F1 , . . . , Fc range over all distributions of F . Here completeness is proved by considering the subfamily F0 of F in which the distributions Fi have densities of the form   fi (x) = Ci (θi1 , . . . , θiNi ) exp −x2Ni + θi1 x + . . . + θiNi xNi . The result remains true if F is replaced by the family F1 of continuous distributions. For a proof see Problem 4.13 or Bell, Blackwell, and Breiman (1960). For related results, see Mandelbaum and Rüschendorf (1987) and Mattner (1996). For the present purpose the slightly weaker property of bounded completeness is appropriate, a family P of probability distributions being boundedly complete if for all bounded functions f , (4.8) implies (4.9). If P is complete it is a fortiori boundedly complete. An example if which P is boundedly complete but not complete is given in Problem 4.12. For additional examples, see Hoeffding (1977), Bar-Lev and Plachky (1989) and Mattner (1993). Theorem 4.3.2 Let X be a random variable with distribution P ∈ P, and let T be a sufficient statistic for P. Then a necessary and sufficient condition for all similar tests to have Neyman structure with respect to T is that the family P T of distributions of T is boundedly complete. 4.4. UMP Unbiased Tests for Multiparameter Exponential Families 119 Proof. Suppose first that P T is boundedly complete, and let φ(X) be similar with respect to P. Then E[φ(X) − α] = 0 for all P ∈P and hence, if ψ(t) denotes the conditional expectation of φ(X) − α given t, Eψ(T ) = 0 for all P T ∈ PT . Since ψ(t) can be taken to be bounded by Lemma 2.4.1, it follows from the bounded completeness of P T that ψ(t) = 0 and hence E[φ(X)|t] = α a.e. P T , as was to be proved. Conversely suppose that P T is not boundedly complete. Then there exists a function f such that |f (t)| ≤ M for some M , that Ef (T ) = 0 for all P T ∈ P T and f (T ) = 0 with positive probability for some P T ∈ P T . Let φ(t) = cf (t) + α, where c = min(α, 1 − α)/M . Then φ is a critical function, since 0 ≤ φ(t) ≤ 1, and it is a similar test, since Eφ(T ) = α for all P T ∈ P T . But φ does not have Neyman structure, since φ(T ) = α with positive probability for at least some distribution in P T . 4.4 UMP Unbiased Tests for Multiparameter Exponential Families An important class of hypotheses concerns a real-valued parameter in an exponential family, with the remaining parameters occurring as unspecified nuisance parameters. In many of these cases, UMP unbiased tests exist and can be constructed by means of the theory of the preceding section. Let X be distributed according to ! " k  X dPθ,ϑ (x) = C(θ, ϑ) exp θU (X) + ϑi Ti (x) dµ(x), (θ, ϑ) ∈ Ω, (4.10) i=1 and let ϑ = (ϑ1 , . . . , ϑk ) and T = (T1 , . . . , Tk ). We shall consider the problems3 of testing the following hypotheses Hj against the alternatives Kj , j = 1, . . . , 4: H1 H2 H3 H4 : θ ≤ θ0 : θ ≤ θ1 or θ ≥ θ2 : θ 1 ≤ θ ≤ θ2 : θ = θ0 K1 K2 K3 K4 : θ > θ0 : θ1 < θ < θ 2 : θ < θ1 or θ > θ2 : θ = θ0 . We shall assume that the parameter space Ω is convex, and that it is not contained in a linear space of dimension < k + 1. This is the case in particular when Ω is the natural parameter space of the exponential family. We shall also assume that there are points in Ω with θ both < and > θ0 , θ1 , and θ2 respectively. 3 Such problems are also treated in Johansen (1979), which in addition discusses large sample tests of hypotheses specifying more than one parameter. 120 4. Unbiasedness: Theory and First Applications Attention can be restricted to the sufficient statistics (U, T ) which have the joint distribution   k  U,T ϑi ti dν(u, t), (θ, ϑ) ∈ Ω. (4.11) dPθ,ϑ (u, t) = C(θ, ϑ) exp θU + i=1 When T = t is given, U is the only remaining variable and, by Lemma 2.7.2, the conditional distribution of U given t constitutes an exponential family U |t dPθ (u) = Ct (θ)eθu dνt (u). In this conditional situation there exists by Corollary 3.4.1 a UMP test for testing H1 , with critical function φ1 , satisfying ⎧ when u > C0 (t), ⎨ 1 γ0 (t) when u = C0 (t), (4.12) φ(u, t) = ⎩ 0 when u < C0 (t), where the functions C0 and γ0 are determined by Eθ0 [φ1 (U, T )|t] = α For testing H2 in the conditional test with critical function ⎧ ⎨ 1 γi (t) φ(u, t) = ⎩ 0 for all t. (4.13) family there exists by Theorem 3.7.1 a UMP when C1 (t) < u < C2 (t), when u = Ci (t), i = 1, 2, when u < C1 (t) or > C2 (t), (4.14) where the C’s and γ’s are determined by Eθ1 [φ2 (U, T )|t] = Eθ2 [φ2 (U, T )|t] = α. Consider next the test φ3 satisfying ⎧ when u < C1 (t) or > C2 (t), ⎨ 1 γi (t) when u = Ci (t), i = 1, 2, φ(u, t) = ⎩ 0 when C1 (t) < u < C2 (t), (4.15) (4.16) with the C’s and γ’s determined by Eθ1 [φ3 (U, T )|t] = Eθ2 [φ3 (U, T )|t] = α. (4.17) When T = t is given, this is (by Section 4.2 of the present chapter) UMP unbiased for testing H3 and UMP among all tests satisfying (4.17). Finally, let φ4 be a critical function satisfying (4.16) with the C’s and γ’s determined by Eθ0 [φ4 (U, T )|t] = α (4.18) Eθ0 [U φ4 (U, T )|t] = αEθ0 [U |t]. (4.19) and Then given T = t, it follows again from the results of Section 4.2 that φ4 is UMP unbiased for testing H4 and UMP among all tests satisfying (4.18) and (4.19). 4.4. UMP Unbiased Tests for Multiparameter Exponential Families 121 So far, the critical functions φj have been considered as conditional tests given T = t. Reinterpreting them now as tests depending on U and T for the hypotheses concerning the distribution of X (or the joint distribution of U and T ) as originally stated, we have the following main theorem.4 Theorem 4.4.1 Define the critical functions φ1 by (4.12) and (4.13); φ2 by (4.14) and (4.15); φ3 by (4.16) and (4.17); φ4 by (4.16), (4.18), and (4.19). These constitute UMP unbiased level-α tests for testing the hypotheses H1 , . . . , H4 respectively when the joint distribution of U and T is given by (4.11). Proof. The statistic T is sufficient for ϑ if θ has any fixed value, and hence T is sufficient for each ωj = {(θ, ϑ) : (θ, ϑ) ∈ Ω, θ = θj }, j = 0, 1, 2. By Lemma 2.7.2, the associated family of distributions of T is given by  k   T dPθj ,ϑ (t) = C(θj , ϑ) exp ϑi ti dνθj (t), (θj , ϑ) ∈ ωj j = 0, 1, 2. i=1 Since by assumption Ω is convex and of dimension k + 1 and contains points on both sides of θ = θj , it follows that ωj is convex and of dimension k. Thus ωj contains a k-dimensional rectangle; by Theorem 4.3.1 the family . PjT = PθTj ,ϑ : (θ, ϑ) ∈ ωj is complete; and similarity of a test φ on ωj implies Eθj [φ(U, T )|t] = α. (1) Consider first H1 . By Theorem 2.7.1, the power function of all tests is continuous for an exponential family. It is therefore enough to prove φ1 to be UMP among all tests that are similar on ω0 (Lemma 4.1.1), and hence among those satisfying (4.13). On the other hand, the overall power of a test φ against an alternative (θ, ϑ) is    U |t T Eθ,ϑ [φ(U, T )] = φ(u, t) dPθ (u) dPθ,ϑ (t). (4.20) One therefore maximizes the overall power by maximizing the power of the conditional test, given by the expression in brackets, separately for each t. Since φ1 has the property of maximizing the conditional power against any θ > θ0 subject to (4.13), this establishes the desired result. (2) The proof for H2 and H3 is completely analogous. By Lemma 4.1.1, it is enough to prove φ2 and φ3 to be UMP among all tests that are similar on both ω1 and ω2 , and hence among all tests satisfying (4.15). For each t, φ2 and φ3 maximize the conditional power for their respective problems subject to this condition and therefore also the unconditional power. 4 A somewhat different asymptotic optimality property of these tests is established by Michel (1979). 122 4. Unbiasedness: Theory and First Applications (3) Unbiasedness of a test of H4 implies similarity on ω0 and ∂ on ω0 . [Eθ,ϑ φ(U, T )] = 0 ∂θ The differentiation on the left-hand side of this equation can be carried out under the expectation sign, and by the computation which earlier led to (4.6), the equation is seen to be equivalent to Eθ,ϑ [U φ(U, T ) − αU ] = 0 on ω0 . P0T is complete, unbiasedness implies (4.18) and (4.19). As in Therefore, since the preceding cases, the test, which in addition satisfies (4.16), is UMP among all tests satisfying these two conditions. That it is UMP unbiased now follows, as in the proof of Lemma 4.1.1, by comparison with the test φ(u, t) ≡ α. (4) The functions φ1 , . . . , φ4 were obtained above for each fixed t as a function of u. To complete the proof it is necessary to show that they are jointly measurable in u and t, so that the expectation (4.20) exists. We shall prove this here for the case of φ1 ; the proof for the other cases is sketched in Problems 4.21 and 4.22. To establish the measurability of φ1 , one needs to show that the functions C0 (t) and γ0 (t) defined by (4.12) and (4.13) are t-measurable. Omitting the subscript 0, and denoting the conditional distribution function of U given T = t and for θ = θ0 by Ft (u) = Pθ0 {U ≤ u|t}, one can rewrite (4.13) as Ft (C) − γ[Ft (C) − Ft (C − 0)] = 1 − α. Here C = C(t) is such that Ft (C − 0) ≤ 1 − α ≤ Ft (C), and hence C(t) = Ft−1 (1 − α) where Ft−1 (y) = inf{u : Ft (u) ≥ y}. It follows that C(t) and γ(t) will both be measurable provided Ft (u) and Ft (u − 0) are jointly measurable in u and t and Ft−1 (1 − α) is measurable in t. For each fixed u the function Ft (u) is a measurable function of t, and for each fixed t it is a cumulative distribution function and therefore in particular nondecreasing and continuous on the right. From the second property it follows that Ft (u) ≥ c if and only if for each n there exists a rational number r such that u ≤ r < u + 1/n and Ft (r) ≥ c. Therefore, if the rationals are denoted by r1 , r2 , . . . ,  2 1 {(u, t) : Ft (u) ≥ c} = (u, t) : 0 ≤ ri − u < , Ft (ri ) ≥ c n n i This shows that Ft (u) is jointly measurable in u and t. The proof for Ft (u − 0) is completely analogous. Since Ft−1 (y) ≤ u if and only if Ft (u) ≥ y, Ft−1 (y) is t-measurable for any fixed y and this completes the proof. The test φ1 of the above theorem is also UMP unbiased if Ω is replaced by the set Ω = Ω ∩ {(θ, ϑ) : θ ≥ θ0 }, and hence for testing H  : θ = θ0 against θ > θ0 . The assumption that Ω should contain points with θ < θ0 was in fact used only to prove that the boundary set ω0 contains a k-dimensional rectangle, and this remains valid if Ω is replaced by Ω . 4.4. UMP Unbiased Tests for Multiparameter Exponential Families 123 The remainder of this chapter as well as the next chapter will be concerned mainly with applications of the preceding theorem to various statistical problems. While this provides the most expeditious proof that the tests in all these cases are UMP unbiased, there is available also a variation of the approach, which is more elementary. The proof of Theorem 4.4.1 is quite elementary except for the following points: (i) the fact that the conditional distributions of U given T = t constitute an exponential family, (ii) that the family of distributions of T is complete, (iii) that the derivative of Eθ,ϑ φ(U, T ) exists and can be computed by differentiating under the expectation sign, (iv) that the functions φ1 , . . . , φ4 are measurable. Instead of verifying (i) through (iv) in general, as was done in the above proof, it is possible in applications of the theorem to check these conditions directly for each specific problem, which in some cases is quite easy. Through a transformation of parameters, Theorem 4.4.1 can be extended to cover hypotheses concerning parameters of the form θ ∗ = a0 θ + k  ai ϑi , a0 = 0. i=1 This transformation is formally given by the following lemma, the proof of which is immediate. Lemma 4.4.1 The exponential family of distributions (4.10) can also be written as # $  X = K(θ∗ , ϑ) exp θ∗ U ∗ (x) + ϑi Ti∗ (x) dµ(x) dPθ,ϑ where U∗ = U , a0 Ti∗ = Ti − ai U. a0 Application of Theorem 4.4.1 to the form of the distributions given in the lemma leads to UMP unbiased tests of the hypothesis H1∗ : θ∗ ≤ θ0 and the analogously defined hypotheses H2∗ , H3∗ , H4∗ . When testing one of the hypotheses Hj one is frequently interested in the power β(θ , ϑ) of φj against some alternative θ . As is indicated by the notation and is seen from (4.20), this power will usually depend on the unknown nuisance parameters ϑ. On the other hand, the power of the conditional test given T = t, β(θ |t) = Eθ [φ(U, T )|t], is independent of ϑ and therefore has a known value. The quantity β(θ |t) can be interpreted in two ways: (i) It is the probability of rejecting H when T = t. Once T has been observed to have the value t, it may be felt, at least in certain problems, that this is a more appropriate expression of the power in the given situation than β(θ , ϑ), which is obtained by averaging β(θ |t) with respect to other values of t not relevant to the situation at hand. This argument leads to difficulties, since in many cases the conditioning could be carried even further and it is not clear where the process should stop. (ii) A more clear-cut interpretation is obtained by considering β(θ |t) as an estimate of β(θ , ϑ). Since Eθ ,ϑ [β(θ |T )] = β(θ , ϑ), 124 4. Unbiasedness: Theory and First Applications this estimate is unbiased in the sense of equation (1.11). It follows further from the theory of unbiased estimation and the completeness of the exponential family that among all unbiased estimates of β(θ , ϑ) the present one has the smallest variance. (See TPE2, Chapter 2.) Regardless of the interpretation, β(θ |t) has the disadvantage compared with an unconditional power that it becomes available only after the observations have been taken. It therefore cannot be used to plan the experiment and in particular to determine the sample size, if this must be done prior to the experiment. On the other hand, a simple sequential procedure guaranteeing a specified power β against the alternatives θ = θ is obtained by continuing taking observations until the conditional power β(θ |t) is ≥ β. 4.5 Comparing Two Poisson or Binomial Populations A problem arising in many different contexts is the comparison of two treatments or of one treatment with a control situation in which no treatment is applied. If the observations consist of the number of successes in a sequence of trials for each treatment, for example the number of cures of a certain disease, the problem becomes that of testing the equality of two binomial probabilities. If the basic distributions are Poisson, for example in a comparison of the radioactivity of two substances, one will be testing the equality of two Poisson distributions. When testing whether a treatment has a beneficial effect by comparing it with the control situation of no treatment, the problem is of the one-sided type. If ξ2 and ξ1 denote the parameter values when the treatment is or is not applied, the class of alternatives is K : ξ2 > ξ1 . The hypothesis is ξ2 = ξ1 if it is known a priori that there is either no effect or a beneficial one; it is ξ2 ≤ ξ1 if the possibility is admitted that the treatment may actually be harmful. Since the test is the same for the two hypotheses, the second somewhat safer hypothesis would seem preferable in most cases. A one-sided formulation is sometimes appropriate also when a new treatment or process is being compared with a standard one, where the new treatment is of interest only if it presents an improvement. On the other hand, if the two treatments are on an equal footing, the hypothesis ξ2 = ξ1 of equality of two treatments is tested against the two-sided alternatives ξ2 = ξ1 . The formulation of this problem as one of hypothesis testing is usually quite artificial, since in case of rejection of the hypothesis one will obviously wish to know which of the treatments is better.5 Such two-sided tests do, however, have important applications to the problem of obtaining confidence limits for the extent by which one treatment is better than the other. They also arise when the parameter ξ does not measure a treatment effect but refers to an auxiliary variable which one hopes can be ignored. For example, ξ1 and ξ2 may refer to the effect of two 5 The comparison of two treatments as a three-decision problem or as the simultaneous testing of two one-sided hypotheses is discussed and the literature reviewed in Shaffer (2002). 4.5. Comparing Two Poisson or Binomial Populations 125 different hospitals in a medical investigation in which one would like to combine the patients into a single study group. (In this connection, see also Section 7.3.) To apply Theorem 4.4.1 to this comparison problem it is necessary to express the distributions in an exponential form with θ = f (ξ1 , ξ2 ), for example θ = ξ2 −ξ1 or ξ2 /ξ1 , such that the hypotheses of interest become equivalent to those of Theorem 4.4.1. In the present section the problem will be considered for Poisson and binomial distributions; the case of normal distributions will be taken up in Chapter 5. We consider first the Poisson problem in which X and Y are independently distributed according to P (λ) and P (µ), so that their joint distribution can be written as # $ e−(λ+µ) µ P {X = x, Y = y} = exp y log + (x + y) log λ . x!y! λ By Theorem 4.4.1 there exist UMP unbiased tests of the four hypotheses H1 , . . . , H4 concerning the parameter θ = log(µ/λ) or equivalently concerning the ratio ρ = µ/λ. This includes in particular the hypotheses µ ≤ λ (or µ = λ) against the alternatives µ > λ, and µ = λ against µ = λ. Comparing the distribution of (X, Y ) with (4.10), one has U = Y and T = X + Y , and by Theorem 4.4.1 the tests are performed conditionally on the integer points of the line segment X + Y = t in the positive quadrant of the (x, y) plane. The conditional distribution of Y given X + Y = t is (Problem 2.14)   y t−y t µ λ P {Y = y|X + Y = t} = , y = 0, 1, . . . , t, y λ+µ λ+µ the binomial distribution corresponding to t trials and probability p = µ/(λ + µ) of success. The original hypotheses therefore reduce to the corresponding ones about the parameter p of a binomial distribution. The hypothesis H : µ ≤ aλ, for example, becomes H : p ≤ a/(a + 1), which is rejected when Y is too large. The cutoff point depends of course, in addition to a, also on t. It can be determined from tables of the binomial, and for large t approximately from tables of the normal distribution. In many applications the ratio ρ = µ/λ is a reasonable measure of the extent to which the two Poisson populations differ, since the parameters λ and µ measure the rates (in time or space) at which two Poisson processes produce the events in question. One might therefore hope that the power of the above tests depends only on this ratio, but this is not the case. On the contrary, for each fixed value of ρ corresponding to an alternative to the hypothesis being tested, the power β(λ, µ) = β(λ, ρλ) is an increasing function of λ, which tends to 1 as λ → ∞ and to α as λ → 0. To see this consider the power β(ρ|t) of the conditional test given t. This is an increasing function of t, since it is the power of the optimum test based on t binomial trials. The conditioning variable T has a Poisson distribution with parameter λ(1 + ρ), and its distribution for varying λ forms an exponential family. It follows Lemma 3.4.2 that the overall power E[β(ρ|T )] is an increasing function of λ. As λ → 0 or ∞, T tends in probability to 0 or ∞, and the power against a fixed alternative ρ tends to α or 1. The above test is also applicable to samples X , . . . , Xm and Y1 ,  . . . , Yn from 1m n two Poisson distributions. The statistics X = X and Y = i i=1 j=1 Yj are then sufficient for λ and µ, and have Poisson distributions with parameters mλ 126 4. Unbiasedness: Theory and First Applications and nµ respectively. In planning an experiment one might wish to determine m = n so large that the test of, say, H : ρ ≤ ρ0 has power against a specified alternative ρ1 greater than or equal to some preassigned β. However, it follows from the discussion of the power function for n = 1, which applies equally to any other n, that this cannot be achieved for any fixed n, no matter how large. This is seen more directly by noting that as λ → 0, for both ρ = ρ0 and ρ = ρ1 , the probability of the event X = Y = 0 tends to 1. Therefore, the power of any level-α test against ρ = ρ1 and for varying λ cannot be bounded away from α. This difficulty can be overcome only by permitting observations to be taken sequentially. One can for example determine t0 so large that the test of the hypothesis p1 ≤ ρ0 /(1 + ρ0 ) on the basis of t0 binomial trials has power ≥ β against the alternative p1 = ρ1 /(1 + ρ1 ). By observing (X1 , Y1 ), (X2 , Y2 ), . . . and  continuing until (Xi + Yi ) ≥ t0 , one obtains a test with power ≥ β against all alternatives with ρ ≥ ρ1 .6 The corresponding comparison of two binomial probabilities is quite similar. Let X and Y be independent binomial variables with joint distribution     m x m−x n y n−y P {X = x, Y = y} = p1 q1 p q x y 2 2      m p2 p1 n m n = − log q1 q2 exp y log q2 q1 x y  p1 . +(x + y) log q1 The four hypotheses H1 , . . . , H4 , can then be tested concerning the parameter ⎛ 3 ⎞ p2 p1 ⎠ θ = log ⎝ , q2 q1 or equivalently concerning the odds ratio (also called cross-product ratio) 3 p2 p1 ρ= q2 q1 This includes in particular the problems of testing H1 : p2 ≤ p1 against p2 > p1 and H4 : p2 = p1 against p2 = p1 . As in the Poisson case, U = Y and T = X + Y , and the test is carried out in terms of the conditional distribution of Y on the line segment X + Y = t. This distribution is given by    m n y P {Y = y|X + Y = t} = Ct (ρ) y = 0, 1, . . . , t, (4.21) ρ , t−y y where Ct (ρ) = t y  =0  1 m t−y  n y ρy  . 6 A discussion of this and alternative procedures for achieving the same aim is given by Birnbaum (1954a). 4.6. Testing for Independence in a 2 × 2 Table 127 In the particular case of the hypotheses H1 and H4 , the boundary value θ0 of (4.13), (4.18), and (4.19) is 0, and the corresponding value of ρ is ρ0 = 1. The conditional distribution then reduces to  m n t−y y P {Y = y|X + Y = t} = m+n , t which is the hypergeometric distribution. Tables of critical values by Finney (1948) are reprinted in Biometrika Tables for Statisticians, Vol. 1, Table 38 and are extended in Finney, Latscha, Bennett, Hsu, and Horst (1963, 1966). Somewhat different ranges are covered in Armsen (1955), and related charts are provided by Bross and Kasten (1957). Extensive tables of the hypergeometric distributions have been computed by Lieberman and Owen (1961). Various approximations are discussed in Johnson, Kotz and Kemp (1992, Section 6.5). Critical values can also be easily computed with built-in functions of statistical packages such as R.7 The UMP unbiased test of ρ1 = ρ2 , which is based on the (conditional) hypergeometric distribution, requires randomization to obtain an exact conditional level α for each t of the sufficient statistic T . Since in practice randomization is usually unacceptable, the one-sided test is frequently performed by rejecting when Y ≥ C(T ), where C(t) is the smallest integer for which P {Y ≥ C(T )|T = t} ≤ α. This conservative test is called Fisher’s exact test [after the treatment given in Fisher (1934a)], since the probabilities are calculated from the exact hypergeometric rather than an approximate normal distribution. The resulting conditional levels (and hence the unconditional level) are often considerably smaller than α, and this results in a substantial loss of power. An approximate test whose overall level tends to be closer to α is obtained by using the normal approximation to the hypergeometric distribution without continuity correction. [For a comparison of this test with some competitors, see e.g. Garside and Mack (1976).] A nonrandomized test that provides a conservative overall level, but that is less conservative than the “exact” test, is described by Boschloo (1970) and by McDonald, Davis, and Milliken (1977). For surveys of the extensive literature on these and related aspects of 2 × 2 and more generally r × c tables, see Agresti (1992, 2002), Sahai and Khurshid (1995) and Martín and Tapia (1998). 4.6 Testing for Independence in a 2 × 2 Table Two characteristics A and B, which each member of a population may or may not possess, are to be tested for independence. The probabilities or proportion of individuals possessing properties A and B are denoted P (A) and P (B). If P (A) and P (B) are unknown, a sample from one of the categories such as A does not provide a basis for distinguishing between the hypothesis and the alternatives. This follows from the fact that the number in the sample possessing characteristic B then constitutes a binomial variable with probability p(B|A), which is completely unknown both when the hypothesis is true and when it is 7 This package can be downloaded for free from http://cran.r-project.org/. 128 4. Unbiasedness: Theory and First Applications false. The hypothesis can, however, be tested if samples are taken both from categories A and Ac , the complement of A, or both from B and B c . In the latter case, for example, if the sample sizes are m and n, the numbers of cases possessing characteristic A in the two samples constitute independent variables with binomial distributions b(p1 , m) and b(p2 , n) respectively, where p1 = P (A|B) and p2 = P (A|B c ). The hypothesis of independence of the two characteristics, P (A|B) = p(A), is then equivalent to the hypothesis p1 = p2 and the problem reduces to that treated in the preceding section. Instead of selecting samples from two of the categories, it is frequently more convenient to take the sample at random from the population as a whole. The results of such a sample can be summarized in the following 2 × 2 contingency table, the entries of which give the numbers in the various categories: B Bc A Ac X Y X Y M N T T s The joint distribution of the variables X, X  , Y , and Y  is multinomial, and is given by P {X = x, X  = x , Y = y, Y  = y  } =   s! pxAB pxAc B pyAB c pyAB c   x!x !y!y ! = s! pAB pAc B pAB c + x log + y log psAc B c exp x log   c c c c x!x !y!y ! pA B pA B pAc B c  . Lemma 4.4.1 and Theorem 4.4.1 are therefore applicable to any parameter of the form θ∗ = a0 log pAB pAc B pAB c + a1 log + a2 log . pAc B c pAc B c pAc B c ∗ Putting a1 = a2 = 1, a0 = −1, ∆ = eθ = (pAc B pAB c )/(pAB pAc B c ), and denoting the probabilities of A and B in the population by pA = pAB + pAB c , pB = pAB + pAc B , one finds pAB = pAc B = pAB c = pA c B c = 1−∆ pAc B pAB c , ∆ 1−∆ pAc pB + pAc B pAB c , ∆ 1−∆ pA pB c + pAc B pAB c , ∆ 1−∆ pAc pB c + pAc B pAB c . ∆ pA pB + 4.6. Testing for Independence in a 2 × 2 Table 129 Independence of A and B is therefore equivalent to ∆ = 1, and ∆ < 1 and ∆ > 1 correspond to positive and negative dependence respectively.8 The test of the hypothesis of independence, or any of the four hypotheses concerning ∆, is carried out in terms of the conditional distribution of X given X + X  = m, X + Y = t. Instead of computing this distribution directly, consider first the conditional distribution subject only to the condition X + X  = m, and hence Y + Y  = s − m = n. This is seen to be P {X = = x, Y = y|X + X  = m}    x m−x pAc B m n pAB pB pB x y pAB c pB c y pAc B c pB c n−y , which is the distribution of two independent binomial variables, the number of successes in m and n trials with probability p1 = pAB /pB and p2 = pAB c /pB c . Actually, this is clear without computation, since we are now dealing with samples of fixed size m and n from the subpopulations B and B c and the probability of A in these subpopulations is p1 and p2 . If now the additional restriction X + Y = t is imposed, the conditional distribution of X subject to the two conditions X + X  = m and X + Y = t is the same as that of X given X + Y = t in the case of two independent binomials considered in the previous section. It is therefore given by    m n  P {X = x|X + X = m, X + Y = t} = Ct (ρ) ρt−x , x t−x x = 0, . . . , t, that is, by (4.21) expressed in terms of x instead of y. (Here the choice of X as testing variable is quite arbitrary; we could equally well again have chosen Y .) For the parameter ρ one finds 3 pAc B pAB c p1 p2 ρ= = = ∆. q2 q1 pAB pAc B c From these considerations it follows that the conditional test given X + X  = m, X + Y = t, for testing any of the hypotheses concerning ∆ is identical with the conditional test given X + Y = t of the same hypothesis concerning ρ = ∆ in the preceding section, in which X + X  = m was given a priori. In particular, the conditional test for testing the hypothesis of independence ∆ = 1, Fisher’s exact test, is the same as that of testing the equality of two binomial p’s and is therefore given in terms of the hypergeometric distribution. At the beginning of the section it was pointed out that the hypothesis of independence can be tested on the basis of samples obtained in a number of different ways. Either samples of fixed size can be taken from A and Ac or from B and B c , or the sample can be selected at random from the population at large. Which of these designs is most efficient depends on the cost of sampling from 8 ∆ is equivalent to Yule’s measure of association. which is Q = (1 − ∆)/(1 + ∆). For a discussion of this and related measures see Goodman and Kruskal (1954, 1959), Edwards (1963), Haberman (1982) and Agresti (2002). 130 4. Unbiasedness: Theory and First Applications the various categories and from the population at large, and also on the cost of performing the necessary classification of a selected individual with respect to the characteristics in question. Suppose, however, for a moment that these considerations are neglected and that the designs are compared solely in terms of the power that the resulting tests achieve against a common alternative. Then the following results9 can be shown to hold asymptotically as the total sample size s tends to infinity: (i) If samples of size m and n (m + n = s) are taken from B and B c or from A and Ac , the best choice of m and n is m = n = s/2. (ii) It is better to select samples of equal size s/2 from B and B c than from A and Ac provided |pB − 12 | > |pA − 12 |. (iii) Selecting the sample at random from the population at large is worse than taking equal samples either from A and Ac or from B and B c . These statements, which we shall not prove here, can be established by using the normal approximation for the distribution of the binomial variables X and Y when m and n are fixed, and by noting that under random sampling from the population at large, M/s and N/s tend in probability to pB and pB c respectively. 4.7 Alternative Models for 2 × 2 Tables Conditioning of the multinomial model for the 2 × 2 table on the row (or column) totals was seen in the last section to lead to the two-binomial model of Section 4.5. Similarly, the multinomial model itself can be obtained as a conditional model in some situations in which not only the marginal totals M , N , T , and T  are random but the total sample size s is also a random variable. Suppose that the occurrence of events (e.g. patients presenting themselves for treatment) is observed over a given period of time, and that the events belonging to each of the categories AB, Ac B, AB c , Ac B c are governed by independent Poisson processes, so that by (1.2) the numbers X, X  , Y , Y  are independent Poisson variables with expectations λAB , λAc B , λAB c , λAc B c , and hence s is a Poisson variable with expectation λ = λAB + λAc B + λAB c + λAc B c . It may then be of interest to compare the ratio λAB /λAc B with λAB c /λAc B c and in particular to test the hypothesis H : λAB /λAc B ≤ λAB c /λAc B c . The joint distribution of X,X  ,Y ,Y  constitutes a four-parameter exponential family, which can be written as P (X = = x, X  = x , Y = y, Y  = y  )   λAB λAc B c 1 exp x log + (x + x) log λAc B x!x !y!y  ! λAB c λAc B +(y + x) log λAB c + (y  − x) log λAc B c 9 These χ2 .  . results were conjectured by Berkson and proved by Neyman in a course on 4.7. Alternative Models for 2 × 2 Tables 131 Thus, UMP unbiased tests exist of the usual one- and two-sided hypotheses concerning the parameter θ = λAB λAc B c /λAc B λAB c . These are carried out in terms of the conditional distribution of X given X  + X = m, Y + X = t, X + X  + Y + Y  = s, where the last condition follows from the fact that given the first two it is equivalent to Y  − X = s − t − m. By Problem 2.14, the conditional distribution of X, X  , Y given X + X  + Y + Y  = s is the multinomial distribution of Section 4.6 with pAB = λAB , λ p Ac B = λAc B , λ pAB c = λAB c , λ p Ac B c = λAc B c . λ The tests therefore reduce to those derived in Section 4.6. The three models discussed so far involve different sampling schemes. However, frequently the subjects for study are not obtained by any sampling but are the only ones readily available to the experimenter. To create a probabilistic basis for a test in such situations, suppose that B and B c are two treatments, either of which can be assigned to each subject, and that A and Ac denote success or failure (e.g. survival, relief of pain, etc.). The hypothesis of no difference in the effectiveness of the two treatments (i.e. independence of A and B) can then be tested by assigning the subjects to the treatments, say m to B and n to B c , at s random, i.e. in such a way that all possible m assignments are equally likely. It is now this random assignment which takes the place of the sampling process in creating a probability model, thus making it possible to calculate significance. Under the hypothesis H of no treatment difference, the success or failure of a subject is independent of the treatment to which it is assigned. If the numbers of subjects in categories A and Ac are t and t respectively (t + t = s), the values of t and t are therefore fixed, so that we are now dealing with a 2 × 2 table in which all four margins t, t , m, n are fixed. Then any one of the four cell counts X, X  , Y , Y  determines the other three. Under H, the distribution of Y is the hypergeometric distribution derived as the conditional null distribution of Y given X + Y = t at the end of Section 4.5. The hypothesis is rejected in favor of the alternative that treatment B c enhances success if Y is sufficiently large. Although this is the natural test under the given circumstances, no optimum property can be claimed for it, since no clear alternative model to H has been formulated.10 Consider finally the situation in which the subjects are again given rather than sampled, but B and B c are attributes (for example, male or female, smoker or nonsmoker) which cannot be assigned to the subjects at will. Then there exists no stochastic basis for answering the question whether observed differences in the rates X/M and Y /N correspond to differences between B and B c , or whether they are accidental. An approach to the testing of such hypotheses in a nonstochastic setting has been proposed by Freedman and Lane (1982). 10 The one-sided test is of course UMP against the class of alternatives defined by the right side of (4.21), but no reasonable assumptions have been proposed that would lead to this class. For suggestions of a different kind of alternative see Gokhale and Johnson (1978). 132 4. Unbiasedness: Theory and First Applications The various models for the 2 × 2 table discussed in Sections 4.6 and 4.7 may be characterized by indicating which elements are random and which fixed: (i) All margins and s random (Poisson). (ii) All margins are random, s fixed (multinomial sampling). (iii) One set of margins random, the other (and then a fortiori s) fixed (binomial sampling). (iv) All margins fixed. Sampling replaced by random assignment of subjects to treatments. (v) All aspects fixed; no element of randomness. In the first three cases there exist UMP unbiased one- and two-sided tests of the hypothesis of independence of A and B. These tests are carried out by conditioning on the values of all elements in (i)–(iii) that are random, so that in the conditional model all margins are fixed. The remaining randomness in the table can be described by any one of the four cell entries; once it is known, the others are determined by the margins. The distribution of such an entry under H has the hypergeometric distribution given at the end of Section 4.5. The models (i)–(iii) have a common feature. The subjects under observation have been obtained by sampling from a population, and the inference corresponding to acceptance or rejection of H refers to that population. This is not true in cases (iv) and (v). In (iv) the subjects are given, and a probabilistic basis is created by assigning them at random, m to B and n to B̃. Under the hypothesis H of no treatment difference, the four margins are fixed without any conditioning, and the four cell entries are again determined by any one of them, which under H has the same hypergeometric distribution as before. The present situation differs from the earlier three in that the inference cannot be extended beyond the subjects at hand.11 The situation (v) is outside the scope of this book, since it contains no basis for the type of probability calculations considered here. Problems of this kind are however of great importance, since they arise in many observational (as opposed to experimental) studies. For a related discussion, see Finch (1979). 4.8 Some Three-Factor Contingency Tables When an association between A and B exists in a 2 × 2 table, it does not follow that one of the factors has a causal influence on the other. Instead, the explanation may, for example, be in the fact that both factors are causally affected by a third factor C. If C has K possible outcomes C1 , . . . , CK , one may then be faced with the apparently paradoxical situation (known as Simpson’s paradox) that A and B are independent under each of the conditions Ck (k = 1, . . . , K) but exhibit positive (or negative) association when the tables are aggregated over C that 11 For a more detailed treatment of the distinction between population models [such as (i)–(iii)] and randomization models [such as (iv)], see Lehmann (1998). 4.8. Some Three-Factor Contingency Tables 133 is, when the K separate 2 × 2 tables are combined into a single one showing the total counts of the four categories. [An interesting example is discussed in Agresti (2002).] In order to determine whether the association of A and B in the aggregated table is indeed “spurious”, one would test the hypothesis, (which arises also in other contexts) that A and B are conditionally independent given Ck for all k = 1, . . . , K, against the alternative that there is an association for at least some k. Let Xk , Xk , Yk , Yk denote the counts in the 4K cells of the 2 × 2 × K table which extends the 2 × 2 table of Section 4.6 to the present case. Again, several sampling schemes are possible. Consider first a random sample of size s from the population at large. The joint distribution of the 4K cell counts then is multinomial with probabilities pABCk , pÃBCk , pAB̃Ck , pÃB̃Ck for the outcomes indicated by the subscripts. If ∆k denotes the AB odds ratio for Ck defined by pAB̃|Ck pÃB|Ck pAB̃Ck pÃBCk = , ∆k = pABCk pÃB̃Ck pAB|Ck pÃB̃|Ck where pAB|Ck . . . denotes the conditional probability of the indicated event given Ck , then the hypothesis to be tested is ∆k = 1 for all k. A second scheme takes samples of size sk from Ck and classifies the subjects as AB, ÃB, AB̃ or ÃB̃. This is the case of K independent 2 × 2 tables, in which one is dealing with K quadrinomial distributions of the kind considered in the preceding sections. Since the kth of these distributions is also that of the same four outcomes in the first model conditionally given Ck , we shall denote the probabilities of these outcomes in the present model again by pAB|Ck , . . .. To motivate the next sampling scheme, suppose that A and à represent success or failure of a medical treatment, B̃ and B that the treatment is applied or the subject is used as a control, and Ck the kth hospital taking part in this study. If samples of size nk and mk are obtained and are assigned to treatment and control respectively, we are dealing with K pairs of binomial distributions. Letting Yk and Xk denote the number of successes obtained by the treatment subjects and controls in the kth hospital, the joint distribution of these variables by Section 4.5 is !    "   mk   p1k nk mk nk yk log ∆k + (xk + yk ) log q1k q2k exp , xk yk q1k where p1k and q1k , (p2k and q2k ) denote the probabilities of success and failure under B (under B̃). The above three sampling schemes lead to 2×2×K tables in which respectively none, one, or two of the margins are fixed. Alternatively, in some situations a model may be appropriate in which the 4K variables Xk , Xk , Yk , Yk are independent Poisson with expectations λABCk , . . .. In this case, the total sample size s is also random. For a test of the hypothesis of conditional independence of A and B given Ck for all k (i.e. that ∆1 = · · · = ∆k = 1), see Problem 12.65. Here we shall consider the problem under the simplifying assumption that the ∆k have a common value ∆, so that the hypothesis reduces to H : ∆ = 1. Applying Theorem 4.4.1 to the third model (K pairs of binomials) and assuming the alternatives  to be ∆ > 1, we see that a UMP unbiased test exists and rejects H when Yk > C(X1 + 134 4. Unbiasedness: Theory and First Applications Y1 , . . . , XK + YK ), where C is determined so that the conditional probability of rejection, given that Xk + Yk = tk , is α for all k = 1, . . . , K. It follows from Section 4.5 that the conditional joint distribution of the Yk under H is PH [Y1 = = y1 , . . . , YK = yK |Xk + Yk = tk , k = 1, . . . , K]  n k k  t m−y y k k m +n k k tk k  The conditional distribution of Yk can now be obtained by adding the probabilities over all (y1 , . . . , yK ) whose sum has a given value. Unless the numbers are very small, this is impractical and approximations must be used [see Cox (1966) and Gart (1970)]. The assumption H  : ∆1 = · · · = ∆K = ∆ has a simple interpretation when the successes and failures of the binomial trials are obtained by dichotomizing underlying unobservable continuous response variables. In a single such trial, suppose the underlying variable is Z and that success occurs when Z > 0 and failure when Z ≤ 0. If Z is distributed as F (Z − ζ) with location parameter ζ, we have p = 1 − F (−ζ) and q = F (−ζ). Of particular interest is the logistic distribution, for which F (x) = 1/(1 + e−x ). In this case p = eζ /(1 + eζ ), q = 1/(1 + eζ ), and hence log(p/q) = ζ. Applying this fact to the success probabilities p1k = 1 − F (−ζ1k ), we find that ⎛ p2k θk = log ∆k = log ⎝ q2k p2k = 1 − F (−ζ2k ), 3 ⎞ p1k ⎠ = ζ2k − ζ1k , q1k so that ζ2k = ζ1k + θk . In this model, H  thus reduces to the assumption that ζ2k = ζ1k + θ, that is, that the treatment shifts the distribution of the underlying response by a constant amount θ. If it is assumed that F is normal rather than logistic, F (x) = Φ(x) say, then ζ = Φ−1 (p), and constancy of ζ2k − ζ1k requires the much more cumbersome condition Φ−1 (p2k ) − Φ−1 (p1k ) = constant. However, the functions log(p/q) and Φ−1 (p) agree quite well in the range .1 ≤ p ≤ .9 [see Cox (1970, p. 28)], and the assumption of constant ∆k in the logistic response model is therefore close to the corresponding assumption for an underlying normal response.12 [The socalled loglinear models, which for contingency tables correspond to the linear models to be considered in Chapter 7 but with a logistic rather than a normal response variable, provide the most widely used approach to contingency tables. See, for example, the books by Cox (1970), Haberman (1974), Bishop, Fienberg, and Holland (1975), Fienberg (1980), Plackett (1981), and Agresti (2002).] The UMP unbiased test, derived above for the case that the B- and C-margins are fixed, applies equally when any two margins, any one margin, or no margins are fixed, with the understanding that in all cases the test is carried out conditionally, given the values of all random margins. 12 The problem of discriminating between a logistic and normal response model is discussed by Chambers and Cox (1967). 4.9. The Sign Test 135 The test is also used (but no longer UMP unbiased) for testing H : ∆1 = · · · = ∆K = 1 when the ∆’s are not assumed to be equal but when the ∆k − 1 can be assumed to have the same sign, so that the departure from independence is in the same direction for all the 2 × 2 tables. A one- or two-sided version is appropriate as the alternatives do or do not specify the direction. For a discussion of this test, the Cochran–Mantel–Haenszel test, and some of its extensions see Agresti (2002, Section 7.4). Consider now the case K = 2, with mk and nk fixed, and the problem of testing H  : ∆2 = ∆1 rather than assuming it. The joint distribution of the X’s and Y ’s given earlier can then be written as ! 2    "  mk nk mk nk q q xk yk 1k 2k k=1   ∆2 p1i + (y1 + y2 ) log ∆1 + (xi + yi ) log , × exp y2 log ∆1 q1i and H  is rejected in favor of ∆2 > ∆1 if Y2 > C, where C depends on Y1 + Y2 , X1 + Y1 and X2 + Y2 , and is determined so that the conditional probability of rejection given Y1 + Y2 = w, X1 + Y1 = t1 , X2 + Y2 = t2 is α. The conditional null distribution of Y1 and Y2 , given Xk + Yk = tk (k = 1, 2), by (4.21) with ∆ in place of ρ is      n2 m1 n1 m2 Ct1 (∆)Ct2 (∆) ∆y1 +y2 , t1 − y1 y1 t 2 − y2 y2 and hence the conditional distribution of Y2 , given in addition that Y1 + Y2 = w, is of the form      m1 n1 m2 n2 k(t1 , t2 , w) . y + t1 − w w−y t2 − y y Some approximations to the critical value of this test are discussed by Birch (1964); see also Venable and Bhapkar (1978). [Optimum large-sample tests of some other hypotheses in 2 × 2 × 2 tables are obtained by Cohen, Gatsonis, and Marden (1983).] 4.9 The Sign Test To test consumer preferences between two products, a sample of n subjects are asked to state their preferences. Each subject is recorded as plus or minus as it favors product B or A. The total number Y of plus signs is then a binomial variable with distribution b(p, n). Consider the problem of testing the hypothesis p = 12 of no difference against the alternatives p = 12 (As in previous such problems, we disregard here that in case of rejection it will be necessary to decide which of the two products is preferred.) The appropriate test is the two-sided sign test, which rejects when |Y − 12 n| is too large. This is UMP unbiased (Section 4.2). Sometimes the subjects are also given the possibility of declaring themselves as undecided. If p− , p+ , and p0 denote the probabilities of preference for product A, product B, and of no preference respectively, the numbers X, Y , and Z of 136 4. Unbiasedness: Theory and First Applications decisions in favor of these three possibilities are distributed according to the multinomial distribution n! x y z p− p+ p0 x!y!z! (x + y + z = n), (4.22) and the hypothesis to be tested is H : p+ = p− . The distribution (4.22) can also be written as y z p0 p+ n! (1 − p0 − p+ )n , (4.23) x!y!z! 1 − p0 − p+ 1 − p0 − p+ and is then seen to constitute an exponential family with U = Y , T = Z, θ = log[p+ /(1 − p0 − p+ )], ϑ = log[p0 /(1 − p0 − p+ )]. Rewriting the hypothesis H as p+ = 1 − p0 − p+ it is seen to be equivalent to θ = 0. There exists therefore a UMP unbiased test of H, which is obtained by considering z as fixed and determining the best unbiased conditional test of H given Z = z. Since the conditional distribution of Y given z is a binomial distribution b(p, n − z) with p = p+ /(p+ + p− ), the problem reduces to that of testing the hypothesis p = 1 in a binomial distribution with n − z trials, for which the rejection region 2 is |Y − 12 (n − z)| > C(z). The UMP unbiased test is therefore obtained by disregarding the number of cases in which no preference is expressed (the number of ties), and applying the sign test to the remaining data. The power of the test depends strongly on p0 , which governs the distribution of Z. For large p0 , the number n−z of trials in the conditional binomial distribution can be expected to be small, and the test will thus have little power. This may be an advantage in the present case, since a sufficiently high value of p0 , regardless of the value of p+ /p− , implies that the population as a whole is largely indifferent with respect to the products. The above conditional sign test applies to any situation in which the observations are the result of n independent trials, each of which is either a success (+), a failure (−), or a tie. As an alternative treatment of ties, it is sometimes proposed to assign each tie at random (with probability 12 each) to either plus or minus. The total number Y  of plus signs after the ties have been broken is then a binomial variable with distribution b(π, n), where π = p+ + 12 p0 . The hypothesis H becomes π = 12 , and is rejected when |Y  − 12 n| > C, where the probability of rejection is α when π = 12 . This test can be viewed also as a randomized test based on X, Y , and Z, and it is unbiased for testing H in its original form, since p+ is = or = p− as π is = or = 1. Since the test involves randomization other than on the boundaries of the rejection region, it is less powerful than the UMP unbiased test for this situation, so that the random breaking of ties results in a loss of power. This remark might be thought to throw some light on the question of whether in the determination of consumer preferences it is better to permit the subject to remain undecided or to force an expression of preference. However, here the assumption of a completely random assignment in case of a tie does not apply. Even when the subject is not conscious of a definite preference, there will usually be a slight inclination toward one of the two possibilities, which in a majority of the cases will be brought out by a forced decision. This will be balanced in part by the fact that such forced decisions are more variable than those reached 4.9. The Sign Test 137 voluntarily. Which of these two factors dominates depends on the strength of the preference. Frequently, the question of preference arises between a standard product and a possible modification or a new product. If each subject is required to express a definite preference, the hypothesis of interest is usually the one sided hypothesis p+ ≤ p− , where + denotes a preference for the modification. However, if an expression of indifference is permitted the hypothesis to be tested is not p+ ≤ p− but rather p+ ≤ p0 + p− , since typically the modification is of interest only if it is actually preferred. As was shown in Example 3.8.1, the one-sided sign test which rejects when the number of plus signs is too large is UMP for this problem. In some investigations, the subject is asked not only to express a preference but to give a more detailed evaluation, such as a score on some numerical scale. Depending on the situation, the hypothesis can then take on one of two forms. One may be interested in the hypothesis that there is no difference in the consumer’s reaction to the two products. Formally, this states that the distribution of the scores X1 , . . . , Xn expressing the degree of preference of the n subjects for the modified product is symmetric about the origin. This problem, for which a UMP unbiased test does not exist without further assumptions, will be considered in Section 6.10. Alternatively, the hypothesis of interest may continue to be H : p+ = p− . Since p− = P {X < 0} and p+ = P {X > 0}, this now becomes H : P {X > 0} = P {X < 0}. Here symmetry of X is no longer assumed even when P {X < 0} = P {X > 0}. If no assumptions are made concerning the distribution of X beyond the fact that the set of its possible values is given, the sign test based on the number of X’s that are positive and negative continues to be UMP unbiased. To see this, note that any distribution of X can be specified by the probabilities p− = P {X < 0}, p+ = P {X > 0}, p0 = P {X = 0}, and the conditional distributions F− and F+ of X given X < 0 and X > 0 respectively. Consider any fixed distributions F− , F+ , and denote by F0 the family of all distributions with F− = F− , F+ = F+ and arbitrary p− , p+ , p0 . Any test that is unbiased for testing H in the original family of distributions F in which F− and F+ are unknown is also unbiased for testing H in the smaller family F0 . We shall show below that there exists a UMP unbiased test φ0 of H in F0 . It turns out that φ0 is also unbiased for testing H in F and is independent of F− , F+ . Let φ be any other unbiased test of H in F, and consider any fixed alternative, which without loss of generality can be assumed to be in F0 . Since φ is unbiased for F , it is unbiased for testing p+ = p− in F0 ; the power of φ0 against the particular alternative is therefore at least as good as that of φ. Hence φ0 is UMP unbiased. To determine the UMP unbiased test of H in F0 , let the densities of F− and   F+ with respect to some measure µ be f− and f+ . The joint density of the X’s at a point (x1 , . . . , xn ) with xi1 , . . . , xir < 0 = xj1 = · · · = xjs < xki , . . . , xkm is     pr− ps0 pm + f− (xi1 ) . . . f− (xir )f+ (xk1 ) . . . f+ (xkm ). 138 4. Unbiasedness: Theory and First Applications The set of statistics (r, s, m) is sufficient for (p− , p0 , p+ ), and its distribution is given by (4.22) with x = r, y = m, z = s. The sign test is therefore seen to be UMP unbiased as before. A different application of the sign test arises in the context of a 2 × 2 table for matched pairs. In Section 4.5, success probabilities for two treatments were compared on the basis of two independent random samples. Unless the population of subjects from which these samples are drawn is fairly homogeneous, a more powerful test can often be obtained by using a sample of matched pairs (for example, twins or the same subject given the treatments at different times). For each pair there are then four possible outcomes: (0, 0), (0, 1), (1, 0), and (1, 1), where 1 and 0 stand for success and failure, and the first and second number in each pair of responses refer to the subject receiving treatment 1 or 2 respectively. The results of such a study are sometimes displayed in a 2 × 2 table, 1st 2nd 0 1 0 X Y 1 X Y which despite the formal similarity differs from that considered in Section 4.6. If a sample of s pairs is drawn, the joint distribution of X, Y , X  , Y  as before is multinomial, with probabilities p00 , p01 , p10 ,p11 . The success probabilities of the two treatments are π1 = p10 + p11 for the first and π2 = p01 + p11 for the second treatment, and the hypothesis to be tested is H : π1 = π2 or equivalently p10 = p01 rather than p10 p01 = p00 p11 as it was earlier. In exponential form, the joint distribution can be written as s!ps11 p10 p00 p01 + (x + y) log + x log exp y log x!x !y!y  ! p10 p11 p11  . (4.24) There exists a UMP unbiased test, McNemar’s test, which rejects H in favor of the alternatives p10 < p01 when Y > C(X  + Y, X), where the conditional probability of rejection given X  + Y = d and X = x is α for all d and x. Under this condition, the numbers of pairs (0, 0) and (1, 1) are fixed, and the only remaining variables are Y and X  = d − Y which specify the division of the d cases with mixed response between the outcomes (0, 1) and (1, 0). Conditionally, one is dealing with d binomial trials with success probability p = p01 /(p01 + p10 ), H becomes p = 12 , and the UMP unbiased test reduces to the sign test. [The issue of conditional versus unconditional power for this test is discussed by Frisén (1980).] The situation is completely analogous to that of the sign test in the presence of undecided opinions, with the only difference that there are now two types of ties, (0, 0) and (1, 1), both of which are disregarded in performing the test. 4.10. Problems 139 4.10 Problems Section 4.1 Problem 4.1 Admissibility. Any UMP unbiased test φ0 , is admissible in the sense that there cannot exist another test φ1 which is at least as powerful as φ0 against all alternatives and more powerful against some. [If φ is unbiased and φ is uniformly at least as powerful as φ, then φ is also unbiased.] Problem 4.2 p-values. Consider a family of tests of H : θ = θ0 (or θ ≤ θ0 ), with level-α rejection regions Sα , such that (a) Pθ0 {X ∈ Sα } for all 0 < α < 1, and (b) Sα ⊂ Sα for α < α . If the tests Sα are unbiased, the distribution of α̂ under any alternative θ satisfies Pθ {α̂ ≤ α} ≥ Pθ0 {α̂ ≤ α} = α so that it is shifted toward the origin. Section 4.2 Problem 4.3 Let X have the binomial distribution b(p, n), and consider the hypothesis H : p = p0 at level of significance α. Determine the boundary values of the UMP unbiased test for n = 10 with α = .1, p0 = .2 and with α = .05, p0 = .4, and in each case graph the power functions of both the unbiased and the equal-tails test. Problem 4.4 Let X have the Poisson distribution P (τ ), and consider the hypothesis H : τ = τ0 . Then condition (4.6) reduces to C2 −1  x=C1 +1 τ Ci −1 −τ0 τ0x−1 −τ0  + (1 − γi ) 0 = 1 − α, e e (x − 1)! (Ci − 1)! i=1 2 provided C1 > 1. Problem 4.5 Let Tn /θ have a χ2 -distribution with n degrees of freedom. For testing H : θ = 1 at level of significance α = .05, find n so large that the power of the UMP unbiased test is ≥ .9 against both θ ≥ 2 and θ ≤ 12 . How large does n have to be if the test is not required to be unbiased? Problem 4.6 Suppose X has density (with respect to some measure µ) pθ (x) = C(θ) exp[θT (x)]h(x) , for some real-valued θ. Assume the distribution of T (X) is continuous under θ (for any θ). Consider the problem of testing θ = θ0 versus θ = θ0 . If the null hypothesis is rejected, then a decision is to be made as to whether θ > θ0 or θ < θ0 . We say that a Type 3 (or directional) error is made when it is declared that θ > θ0 when in fact θ < θ0 (or vice-versa). Consider a level α test that rejects the null hypothesis if T < C1 or T > C2 for constants C1 < C2 . Further suppose that it is declared that θ < θ0 if T < C1 and θ > θ0 if T > C2 . 140 4. Unbiasedness: Theory and First Applications (i) If the constants are chosen so that the test is UMPU, show that the Type 3 error is controlled in the sense that sup Pθ {Type 3 error is made} ≤ α . (4.25) θ=θ0 (ii) If the constants are chosen so that the test is equi-tailed in the sense Pθ0 {T (X) < C1 } = Pθ0 {T (X) > C2 } = α/2 , then show (4.25) holds with α replaced by α/2. (iii) Give an example where the UMPU level α test has the left side of (4.25) strictly > α/2. [Confidence intervals for θ after rejection of a two-sided test are discussed in Finner (1994).] Problem 4.7 Let X and Y be independently distributed according to oneparameter exponential families, so that their joint distribution is given by dPθ1 ,θ2 (x, y) = C(θ1 )eθ1 T (x) dµ(x)K(θ2 )eθ2 U (y) dν(y). Suppose that with probability 1 the statistics T and U each take on at least three values and that (a, b) is an interior point of the natural parameter space. Then a UMP unbiased test does not exist for testing H : θ1 = a, θ2 = b against the alternatives θ1 = a or θ2 = b.13 [The most powerful unbiased tests against the alternatives θ1 = a, θ2 = b have acceptance regions C1 < T (x) < C2 and K1 < U (y) < K2 respectively. These tests are also unbiased against the wider class of alternatives K : θ1 = a or θ2 = b or both.] Problem 4.8 Let (X, Y ) be distributed according to the exponential family dPθ1 ,θ2 (x, y) = C(θ1 , θ2 )eθ1 x+θ2 y dµ(x, y) . The only unbiased test for testing H : θ1 ≤ a, θ2 ≤ b against K : θ1 > a or θ2 > b or both is φ(x, y) ≡ α. [Take a = b = 0, and let β(θ1 , θ2 ) be the power function of any level-α test. Unbiasedness implies β(0, θ2 ) = α for θ2 < 0 and hence for all θ2 , since β(0, θ2 ) is an analytic function of θ2 . For fixed θ2 > 0, β(θ1 , θ2 ) considered as a function of θ1 therefore has a minimum at θ1 = 0, so that ∂β(θ1 , θ2 )/∂θ1 vanishes at θ1 = 0 for all positive θ2 , and hence for all θ2 . By considering alternatively positive and negative values of θ2 and using the fact that the partial derivatives of all orders of β(θ1 , θ2 ) with respect to θ1 are analytic, one finds that for each fixed θ2 these derivatives all vanish at θ1 = 0 and hence that the function β must be a constant. Because of the completeness of (X, Y ), β(θ1 , θ2 ) ≡ α implies φ(x, y) ≡ α.] Problem 4.9 For testing the hypothesis H : θ = θ0 , (θ0 an interior point of Ω) in the one-parameter exponential family of Section 4.2, let C be the totality of tests satisfying (4.3) and (4.5) for some −∞ ≤ C1 ≤ C2 ≤ ∞ and 0 ≤ γ1 , γ2 ≤ 1. 13 For counterexamples when the conditions of the problem are not satisfied, see Kallenberg et al. (1984). 4.10. Problems 141 (i) C is complete in the sense that given any level-α test φ0 of H there exists φ ∈ C such that φ is uniformly at least as powerful as φ0 . (ii) If φ1 , φ2 ∈ C, then neither of the two tests is uniformly more powerful than the other. (iii) Let the problem be considered as a two-decision problem, with decisions d0 and d1 corresponding to acceptance and rejection of H and with loss function L(θ, di ) = Li (θ), i = 0, 1. Then C is minimal essentially complete provided L1 (θ) < L0 (θ) for all θ = θ0 . (iv) Extend the result of part (iii) to the hypothesis H  : θ1 ≤ θ ≤ θ2 . (For more general complete class results for exponential families and beyond, see Brown and Marden (1989).) [(i): Let the derivative of the power function of φ0 at θ0 be βφ 0 (θ0 ) = ρ. Then there exists φ ∈ C such that βφ (θ0 ) = ρ and φ is UMP among all tests satisfying this condition. (ii): See the end of Section 3.7. (iii): See the proof of Theorem 3.4.2.] Section 4.3 Problem 4.10 Let X1 , . . . , Xn be a sample from (i) the normal distribution N (aσ, σ 2 ), with a fixed and 0 < σ < ∞; (ii) the uniform distribution U (θ − 12 , θ + 1 ), −∞ < θ < ∞; (iii) the uniform distribution U (θ1 , θ2 ), ∞ < θ1 < θ2 < ∞. 2 For these  three  families of distributions the following statistics are sufficient: (i), T = ( Xi , Xi2 ); (ii) and (iii), T = (min(X1 , . . . , Xn ), max(X1 , . . . , Xn )). The family of distributions of T is complete for case (iii), but for (i) and (ii) it is not complete or even boundedly  complete.  2 [(i): The distribution of Xi / Xi does not depend on σ.] , Xm and . . . , Yn . be samples from N (ξ, σ 2 ) and Problem 4.11 Let X 1 , . . .  Y1 , N (ξ, τ 2 ). Then T = ( Xi , Yj , Xi2 , Yj2 ), which in Example 4.3.3 was seen not to be complete, is also not boundedly complete. [Let f (t) be 1 or −1 as ȳ − x̄ is positive or not.] Problem 4.12 Counterexample. Let X be a random variable taking on the values −1, 0, 1, 2, . . . with probabilities Pθ {X = −1} = θ; Pθ {X = x} = (1 − θ)2 θx , x = 0, 1, . . . . Then P = {Pθ , 0 < θ < 1} is boundedly complete but not complete. [Girschick et al. (1946)] Problem 4.13 The completeness of the order statistics in Example 4.3.4 remains true if the family F is replaced by the family F1 of all continuous distributions. [Due to Fraser (1956). To show that for any integrable symmetric function φ, φ(x1 , . . . , xn ) dF (x1 ) . . . dF (xn ) = 0 for all continuous F implies φ = 0 a.e., replace F by α1 F1 +· · ·+αn Fn , 142 4. Unbiasedness: Theory and First Applications  where 0 < αi < 1, αi = 1. By considering the left side of the resulting identity as a polynomial in the α’s one sees that φ(x1 , . . . , xn ) dF1 (x1 ) . . . dFn (xn ) = 0 for all continuous Fi . This last equation remains valid if the Fi are replaced by Iai (x)F (x), where Iai (x) = 1 if x ≤ ai and = 0 otherwise. This implies that φ = 0 except on a set which has measure 0 under F × . . . × F for all continuous F .] Problem 4.14 Determine whether T is complete for each of the following situations: (i) X1 , . . . , Xn are independently distributed according to the uniform distribution over the integers 1, 2, . . . , θ and T = max(X1 , . . . , Xn ). (ii) X takes on the values 1,2,3,4 with probabilities pq, p2 q, pq 2 , 1 − 2pq respectively, and T = X. Problem 4.15 Let X, Y be independent binomial b(p, m) and b(p2 , n) respectively. Determine whether (X, Y ) is complete when (i) m = n = 1, (ii) m = 2, n = 1. Problem 4.16 Let X1 , . . . , Xn be a sample from the uniform distribution over the integers 1, . . . , θ and let a be a positive integer. (i) The sufficient statistic X(n) is complete when the parameter space is Ω = {θ : θ ≤ a}. (ii) Show that X(n) is not complete when Ω = {θ : θ ≥ a}, a ≥ 2, and find a complete sufficient statistic in this case. Section 4.4 Problem 4.17 Let Xi (i = 1, 2) be independently distributed according to distributions from the exponential families (3.19) with C, Q, T , and h replaced by Ci , Qi , Ti , and hi . Then there exists a UMP unbiased test of (i) H : Q2 (θ2 ) − Q1 (θ1 ) ≤ c and hence in particular of Q2 (θ2 ) ≤ Q1 (θ1 ); (ii) H : Q2 (θ2 ) + Q1 (θ1 ) ≤ c. Problem 4.18 Let X, Y , Z be independent Poisson variables with means λ, µ, v. Then there exists a UMP unbiased test of H : λµ ≤ v 2 . Problem 4.19 Random sample size. Let N be a random variable with a powerseries distribution a(n)λn P (N = n) = , n = 0, 1, . . . (λ > 0, unknown). C(λ) When N = n, a sample X1 , . . . , Xn from the exponential family (3.19) is observed. On the basis of (N, X1 , . . . , XN ) there exists a UMP unbiased test of H : Q(θ) ≤ c. 4.10. Problems 143 Problem 4.20 Suppose P {I = 1} = p = 1 − P {I = 2}. Given I = i, X ∼ N (θ, σi2 ), where σ12 < σ22 are known. If p = 1/2, show that, based on the data (X, I), there does not exist a UMP test of θ = 0 vs θ > 0. However, if p is also unknown, show a UMPU test exists. [See Examples 10.20-21 in Romano and Siegel (1986).] Problem 4.21 Measurability of tests of Theorem 4.4.1. The function φ3 defined by (4.16) and (4.17) is jointly measurable in u and t. [With C1 = v and C2 = w, the determining equations for v, w, γ1 , γ2 are Ft (v−) + [1 − Ft (w)] + γ1 [Ft (v) − Ft (v−)] (4.26) +γ2 [Ft (w) − Ft (w−)] = α and Gt (v−) + [1 − Gt (w)] + γ1 [Gt (v) − Gt (v−)] (4.27) +γ2 [Gt (w) − Gt (w−)] = α where   u Ft (u) = −∞ Ct (θ1 )eθ1 y dvt (y), Gt (u) = u −∞ Ct (θ2 )eθ2 y dvt (y), (4.28) denote the conditional cumulative distribution function of U given t when θ = θ1 and θ = θ2 respectively. (1) For each 0 ≤ y ≤ α let v(y, t) = Ft−1 (y) and w(y, t) = Ft−1 (1 − α + y), where the inverse function is defined as in the proof of Theorem 4.4.1. Define γ1 (y, t) and γ2 (y, t) so that for v = v(y, t) and w = w(y, t), Ft (v−) + γ1 [Ft (v) − Ft (v−)] = y, 1 − Ft (w) + γ2 [Ft (w) − Ft (w−)] = α − y. (2) Let H(y, t) denote the left-hand side of (4.27), with v = v(y, t), etc. Then H(0, t) > α and H(α, t) < α. This follows by Theorem 3.4.1 from the fact that v(0, t) = −∞ and w(α, t) = ∞ (which shows the conditional tests corresponding to y = 0 and y = α to be one-sided), and that the left-hand side of (4.27) for any y is the power of this conditional test. (3) For fixed t, the functions H1 (y, t) = Gt (v−) + γ1 [Gt (v) − Gt (v−)] and H2 (y, t) = 1 − Gt (w) + γ2 [Gt (w) − Gt (w−)] are continuous functions of y. This is a consequence of the fact, which follows from (4.28), that a.e. P T the discontinuities and flat stretches of Ft and Gt coincide. (4) The function H(y, t) is jointly measurable in y and t. This follows from the continuity of H by an argument similar to the proof of measurability of Ft (u) in the text. Define y(t) = inf{y : H(y, t) < α}, and let v(t) = v[y(t), t], etc. Then (4.26) and (4.27) are satisfied for all t. The measurability of v(t), w(t), γ1 (t), and γ2 (t) defined in this manner will follow from 144 4. Unbiasedness: Theory and First Applications measurability in t of y(t) and Ft−1 [y(t)]. This is a consequence of the relations, which hold for all real c,  {t : H(r, t) < α}, {t : y(t) < c} = r λ0 . Problem 4.28 Positive dependence. Two random variables (X, Y ) with c.d.f. F (x, y) are said to be positively quadrant dependent if F (x, y) ≥ F (x, ∞)F (∞, y) for all x, y.14 For the case that (X, Y ) takes on the four pairs of values (0, 0), (0, 1), (1, 0), (1, 1) with probabilities p00 , p01 , p10 , p11 , (X, Y ) are positively quadrant dependent if and only if the odds ratio ∆ = p01 p10 /p00 p11 ≤ 1. Problem 4.29 Runs. Consider a sequence of N dependent trials, and let Xi be 1 or 0 as the i th trial is a success or failure. Suppose that the sequence has the Markov property15 P {Xi = 1|xi , . . . , xi−1 } = P {Xi = 1|xi−1 } and the property of stationarity according to which P {Xi = 1} and P {Xi = 1|xi−1 } are independent of i. The distribution of the X’s is then specified by the 14 For a systematic discussion of this and other concepts of dependence, see Tong (1980, Chapter 5), Kotz, Wang and Hung (1990) and Yanagimoto (1990). 15 Statistical inference in these and more general Markov chains is discussed, for example, in Bhat and Miller (2002); they provide references at the end of Chapter 5. 146 4. Unbiasedness: Theory and First Applications probabilities p1 = P {Xi = 1|xi−1 = 1} and p0 = P {Xi = 1|xi−1 = 0} and by the initial probabilities π1 = P {X1 = 1} and π0 = 1 − π1 = P {X1 = 0} (i) Stationarity implies that π1 = p0 , p0 + q1 q1 . p0 + q1 π0 = (ii) A set of successive outcomes xi , xi+1 , . . . , xi+j is said to form a run of zeros if xi = xi+1 = · · · = xi+j = 0, and xi−1 = 1 or i = 1, and xi+j+1 = 1 or i + j = N . A run of ones is defined analogously. The probability of any particular sequence of outcomes (x1 , . . . , xN ) is 1 pv0 p1n−v q1u q0m−u , p0 + q1 where m and n denote the numbers of zeros and ones, and u and v the numbers of runs of zeros and ones in the sequence. Problem 4.30 Continuation. For testing the hypothesis of independence of the X’s, H : p0 = p1 , against the alternatives K : p0 < p1 , consider the run test, which rejects H when the total number of runs R = U + V is less than a constant C(m) depending on the number m of zeros in the sequence. When R = C(m), the hypothesis is rejected with probability γ(m), where C and γ are determined by PH {R < C(m)|m} + γ(m)PH {R = C(m)|m} = α. (i) Against any alternative of K the most powerful similar test (which is at least as powerful as the most powerful unbiased test) coincides with the run test in that it rejects H when R < C(m). Only the supplementary rule for bringing the conditional probability of rejection (given m) up to α depends on the specific alternative under consideration. (ii) The run test is unbiased against the alternatives K. (iii) The conditional distribution of R given m, when H is true, is16  n−1 2 m−1 r−1 m+nr−1 , P {R = 2r} = m m−1 n−1 P {R = 2r + 1} = r−1 r + m−1 n−1 m+n r r−1 , m [(i): Unbiasedness implies that the conditional probability of rejection given m is α for all m. The most powerful conditional level-α test rejects H for those sample 16 This distribution is tabled by Swed and Eisenhart (1943) and Gibbons and Chakraborti (1992); it can be obtained from the hypergeometric distribution [Guenther (1978)]. For further discussion of the run test, see Lou (1996). 4.10. Problems 147 sequences for which ∆(u, v) = (p0 /p1 )v (q1 /q0 )u is too large. Since p0 < p1 and q1 < q0 and since |v − u| can only take on the values 0 and 1, it follows that ∆(1, 1) > ∆(1, 2), ∆(2, 1) > ∆(2, 2) > ∆(2, 3), ∆(3, 2) > · · · . Thus only the relation between ∆(i, i + 1) and ∆(i + 1, i) depends on the specific alternative, and this establishes the desired result. (ii): That the above conditional test is unbiased for each m is seen by writing its power as β(p0 , p1 |m) = (1 − γ)P {R < C(m)|m} + γP {R ≤ C(m)|m}, since by (i) the rejection regions R < C(m) and R < C(m) + 1 are both UMP at their respective conditional levels. (iii): When H is true, the conditional probability given m of any set of m zeros and n ones is 1/ m+n . The number of ways of dividing n ones into r groups is m  n−1 , and that of dividing m zeros into r + 1 groups is m−1 . The conditional r−1 r probability of getting r + 1 runs of zeros and r runs of ones is therefore m−1 n−1 m+nr−1 . r m To complete the proof, note that the total number of runs is 2r + 1 if and only if there are either r + 1 runs of zeros and r runs of ones or r runs of zeros and r + 1 runs of ones.] Problem 4.31 (i) Based on the conditional distribution of X2 , . . . , Xn given X1 = x1 in the model of Problem 4.29, there exists a UMP unbiased test of H : p0 = p1 against p0 > p1 for every α. (ii) For the same testing problem, without conditioning on X1 there exists a UMP unbiased test if the initial probability π1 is assumed to be completely unknown instead of being given by the value stated in (i) of Problem 4.29. [The conditional distribution of X2 , . . . , Xn given x1 is of the form C(x1 ; p0 , p1 , q0 , q1 )py11 py00 q1z1 q0z0 (y1 , y2 , z1 , z2 ), where y1 is the number of times a 1 follows a 1, y0 the number of times a 1 follows a 0, and so on, in the sequence x1 , X2 , . . . , Xn . [See Billingsley (1961, p. 14).] Problem 4.32 Rank-sum test. Let Y1 , . . . , YN be independently distributed according to the binomial distributions b(pi , ni ), i = 1, . . . , N where pi = 1 . 1 + e−(α+βxi ) This is the model frequently assumed in bioassay, where xi denotes the dose, or some function of the dose such as its logarithm, of a drug given to ni experimental subjects, and where Yi is the number among these subjects which respond to the drug at level xi . Here the xi are known, and α and β are unknown parameters. (i) The joint distribution of the Y ’s constitutes an exponential family, and UMP unbiased tests exist for the four hypotheses of Theorem 4.4.1, concern both α and β. 148 4. Unbiasedness: Theory and First Applications (ii) Suppose in particular that xi = ∆i, where ∆ is known, and that ni = 1 for all i. Let n be the number of successes in the N trials, and let these successes occur in the s1 st, s2 nd,. . . , sn th trial, where s1 < s2 < · · · < sn . Then the UMP unbiased test for testing H : β = 0 against the alternatives β 0 is carried out conditionally, given n, and rejects when the rank sum > n i−1 si is too large. (iii) Let Y1 , . . . , YM and Z1 , . . . , ZN . be two independent sets of experiments of the type described at the beginning of the problem, corresponding, say, to two different drugs. If Yi is distributed as b(pi , mi ) and Zj as b(πj , nj ), with 1 1 , πj = , pi = 1 + e−(α+βui ) a + e−(γ+βvj ) then UMP unbiased tests exist for the four hypotheses concerning γ − α and δ − β. Section 4.8 Problem 4.33 In a 2 × 2 × 2 table with m1 = 3, n1 = 4; m2 = 4, n2 = 4; and t1 = 3, t1 = 4, t2 = t2 = 4, determine the probabilities that P (Y1 + Y2 ≤ K|Xi + Yi = ti , i = 1, 2) for k = 0, 1, 2, 3. Problem 4.34 In a 2 × 2 × K table with ∆k = ∆, the test derived in the text as UMP unbiased for the case that the B and C margins are fixed has the same property when any two, one, or no margins are fixed. Problem 4.35 The UMP unbiased test of H : ∆ = 1 derived in Section 4.8 for the case that the B- and C-margins are fixed (where the conditioning now extends to all random margins) is also UMP unbiased when (i) only one of the margins is fixed; (ii) the entries in the 4K cells are independent Poisson variables with means λABC , . . ., and ∆ is replaced by the corresponding cross-ratio of the λ’s. Problem 4.36 Let Xijkl (i, j, k = 0, 1, l = 1, . . . , L) denote the entries in a 2 × 2 × 2 × L table with factors A, B, C, and D, and let Γl = PAB c CDl PÃBCDl PAB̃ C̃Dl PÃB C̃Dl PABCDl PÃB̃CDl PAB C̃Dl PÃB̃ C̃Dl . Then (i) under the assumption Γl = Γ there exists a UMP unbiased test of the hypothesis Γ ≤ Γ0 to for any fixed Γ0 ; (ii) When l = 2, there exists a UMP unbiased test of the hypothesis Γ1 = Γ2 —in both cases regardless of whether 0, 1, 2 or 3 of the sets of margins are fixed. 4.11. Notes 149 Section 4.9 Problem 4.37 In the 2×2 table for matched pairs, show by formal computation that the conditional distribution of Y given X  + Y = d and X = x is binomial with the indicated p. Problem 4.38 Consider the comparison of two success probabilities in (a) the two-binomial situation of Section 4.5 with m = n, and (b) the matched-pairs situation of Section 4.9. Suppose the matching is completely at random, that is, a random sample of 2n subjects, obtained from a population of size N (2n ≤ N ), is divided at random into n pairs, and the two treatments B and B c are assigned at random within each pair. (i) The UMP unbiased test for design (a) (Fisher’s exact test) is always more powerful than the UMP unbiased test for design (b) (McNemar’s test). (ii) Let Xi (respectively Yi ) be 1 or 0 as the 1st (respectively 2nd) member of the i th pair is a success or failure. Then the correlation coefficient of Xi and Yi can be positive or negative and tends to zero as N → ∞. [(ii): Assume that the kth member of the population has probability of success (k) (k) PA under treatment A and Pà under Ã.] Problem 4.39 In the 2 × 2 table for matched pairs, in the notation of Section 4.9, the correlation between the responses of the two members of a pair is p11 − π1 π2 ρ= . π1 (1 − π1 )π2 (1 − π2 ) For any given values of π1 < π2 , the power of the one-sided McNemar test of H : π1 = π2 is an increasing function of ρ. [The conditional power of the test given X + Y = d, X = x is an increasing function p = p0l /(p01 + p10 ).] Note. The correlation ρ increases with the effectiveness of the matching, and McNemar’s test under (b) of Problem 4.38 soon becomes more powerful than Fisher’s test under (a). For detailed numerical comparisons see Wacholder and Weinberg (1982) and the references given there. 4.11 Notes The closely related properties of similarity (on the boundary) and unbiasedness are due to Neyman and Pearson (1933, 1936), who applied them to a variety of examples. It was pointed out by Neyman (1937) that similar tests could be obtained through the construction method now called Neyman structure. Theorem 4.3.1 is due to Ghosh (1948) and Hoel (1948). The concepts of completeness and bounded completeness, and the application of the latter to Theorem 4.4.1, were developed by Lehmann and Scheffé (1950). The sign test, proposed by Arbuthnot (1710) to test that the probability of a male birth is 1/2, may be the first significance test in the literature. The exact test for independence in 2 by 2 table is due to Fisher (1934). 5 Unbiasedness: Applications to Normal Distributions; Confidence Intervals 5.1 Statistics Independent of a Sufficient Statistic A general expression for the UMP unbiased tests of the hypotheses H1 : θ ≤ θ0 and H4 : θ = θ0 in the exponential family # $  ϑi Ti (x) dµ(x) dPθ,ϑ (x) = C(θ, ϑ) exp θU (x) + (5.1) was given in Theorem 4.4.1 of the preceding chapter. However, this turns out to be inconvenient in the applications to normal and certain other families of continuous distributions, with which we shall be concerned in the present chapter. In these applications, the tests can be given a more convenient form, in which they no longer appear as conditional tests in terms of U given t, but are expressed unconditionally in terms of a single test statistic. The following are three general methods of achieving this. (i) In many of the problems to be considered below, the UMP unbiased test φ0 , is also UMP invariant, as will be shown in Chapter 6. From Theorem 6.5.3, it is then possible to conclude that φ0 is UMP unbiased. This approach, in which the latter property must be taken on faith during the discussion of the test in the present chapter, is the most economical of the three, and has the additional advantage that it derives the test instead of verifying a guessed solution as is the case with methods (ii) and (iii). (ii) The conditional descriptions (4.12), (4.14), and (4.16) can be replaced by equivalent unconditional ones, and it is then enough to find an unbiased test which has the indicated structure. This approach is discussed in Pratt (1962). (iii) Finally, it is often possible to show the equivalence of the test given by Theorem 4.4.1 to a test suspected to be optimal, by means of Theorem 5.1.2 5.1. Statistics Independent of a Sufficient Statistic 151 below. This is the course we shall follow here; the alternative derivation (i) will be discussed in Chapter 6. The reduction by method (iii) depends on the existence of a statistic V = h(U, T ), which is independent of T when θ = θ0 , and which for each fixed t is monotone in U for H1 and linear in U for H4 . The critical function φ1 , for testing H1 then satisfies ⎧ 1 when v > C0 , ⎪ ⎪ ⎪ ⎨ γ0 when v = C0 , (5.2) φ(v) = ⎪ ⎪ ⎪ ⎩ 0 when v < C0 , where C0 and γ0 are no longer dependent on t, and are determined by Eθ0 φ1 (V ) = α. Similarly the test φ4 of H4 reduces to ⎧ 1 when ⎪ ⎪ ⎪ ⎨ γi when φ(v) = ⎪ ⎪ ⎪ ⎩ 0 when (5.3) v < C1 or v > C2 , v = Ci , i = 1, 2, (5.4) C1 < v < C2 , where the C’s and γ’s are determined by Eθ0 [φ4 (V )] = α (5.5) Eθ0 [V φ4 (V )] = αEθ0 (V ). (5.6) and The corresponding reduction for the hypotheses H2 : θ ≤ θ1 , or θ ≥ θ2 and H3 : θ1 ≤ θ ≤ θ2 requires that V be monotone in U for each fixed t, and be independent of T when θ = θ1 and θ = θ2 . The test φ3 is then given by (5.4) with the C’s and γ’s determined by Eθ1 φ3 (V ) = Eθ2 φ3 (V ) = α. (5.7) The test for H2 as before has the critical function φ2 (v; α) = 1 − φ3 (v; 1 − α). This is summarized in the following theorem. Theorem 5.1.1 Suppose that the distribution of X is given by (5.1) and that V = h(U, T ) is independent of T when θ = θ0 . Then φ1 is UMP unbiased for testing H1 provided the function h is increasing in u for each t, and φ4 is UMP unbiased for H4 provided h(u, t) = a(t)u + b(t) with a(t) > 0. The tests φ2 and φ3 , are UMP unbiased for H2 and H3 if V is independent of T when θ = θ1 and θ2 , and if h is increasing in u for each t. Proof. The test of H1 defined by (4.12) and (4.13) is equivalent to that given by (5.2), with the constants determined by Pθ0 {V > C0 (t) | t} + γ0 (t)Pθ0 {V = C0 (t) | t} = α. 152 5. Unbiasedness: Applications to Normal Distributions By assumption, V is independent of T when θ = θ0 , and C0 and γ0 therefore do not depend on t. This completes the proof for H1 , and that for H2 and H3 is quite analogous. The test of H4 given in Section 4.4 is equivalent to that defined by (5.4) with the constants Ci and γi determined by Eθ0 [φ4 (V, t) | t] = α and %  %    V − b(t) %% V − b(t) %% Eθ0 φ4 (V, t) t = αEθ0 t , a(t) % a(t) % which reduces to Eθ0 [V φ4 (V, t) | t] = αEθ0 [V | t]. Since V is independent of T for θ = θ0 , so are the C’s and γ’s as was to be proved. To prove the required independence of V and T in applications of Theorem 5.1.1 to special cases, the standard methods of distribution theory are available: transformation of variables, characteristic functions, and the geometric method. Alternatively, for a given model {Pϑ , ϑ ∈ ω}, suppose V is any statistic whose distribution does not depend on ϑ; such a statistic is said to be ancillary. Then, the following theorem gives sufficient conditions to show V and T are independent. Theorem 5.1.2 (Basu) Let the family of possible distributions of X be P = {Pϑ , ϑ ∈ ω}, let T be sufficient for P, and suppose that the family P T of distributions of T is boundedly complete. If V is any ancillary statistic for P, then V is independent of T . Proof. For any critical function φ, the expectation Eϑ φ(V ) is by assumption independent of ϑ. It therefore follows from Theorem 4.3.2 that E[φ(V ) | t] is constant (a.e. P T ) for every critical function φ, and hence that V is independent of T . Corollary 5.1.1 Let P be the exponential family obtained from (5.1) by letting θ have some fixed value. Then a statistic V is independent of T for all ϑ provided the distribution of V does not depend on ϑ. Proof. It follows from Theorem 4.3.1 that P T is complete and hence boundedly complete, and the preceding theorem is therefore applicable. Example 5.1.1 Let X1 , . . . , Xn , be independently, normally distributed with 2 2 mean ξ and variance σ 2 . Suppose first that  σ is fixed at σ0 . Then the assumptions of Corollary 5.1.1 hold with T = X̄ = Xi /n and ϑ proportional to ξ. Let f be any function satisfying f (x1 + c, . . . , xn + c) = f (x1 , . . . , xn ) for all real c. If V = f (X1 , . . . , Xn ), then also V = f (X1 − ξ, . . . , Xn − ξ). Since the variables Xi − ξ are distributed as N (0, σ02 ), which does not involve ξ, the distribution of V does not depend on ξ. It follows  from Corollary 5.1.1 that any such statistic V , and therefore in particular V = (Xi − X̄)2 , is independent of X̄. This is true for all σ. 5.2. Testing the Parameters of a Normal Distribution 153 Suppose, on the other hand, that ξ is fixed at ξ0 . Then Corollary 5.1.1 applies with T = (Xi − ξ0 )2 and ϑ = −1/2σ 2 . Let f be any function such that f (cx1 , . . . , cxn ) = f (x1 , . . . , xn ) for all c > 0, and let V = f (X1 − ξ0 , . . . , Xn − ξ0 ). Then V is unchanged if each Xi − ξ0 is replaced by (Xi − ξ0 )/σ, and since these variables are normally distributed with zero mean and unit variance, the distribution of V does not depend on σ. It follows that all such statistics V , and hence for example X̄ − ξ0  (Xi − X̄)2 are independent of when ξ = ξ0 . and X̄ − ξ0 ,  (Xi − ξ0 )2  (Xi − ξ0 )2 . This, however, does not hold for all ξ, but only Example 5.1.2 Let U1 /σ12 and U2 /σ22 be independently distributed according to χ2 -distributions with f1 and f2 degrees of freedom respectively, and suppose that σ22 /σ12 = a. The joint density of the U ’s is then   1 (f /2)−1 (f2 /2)−1 u2 exp − 2 (au1 + u2 ) Cu1 1 2σ2 so that Corollary 5.1.1 is applicable with T = aU1 + U2 and ϑ = −1/2σ22 . Since the distribution of V = U2 /σ22 U2 =a U1 U1 /σ12 does not depend on σ2 , V is independent of aU1 + U2 . For the particular case that σ2 = σ1 , this proves the independence of U2 /U1 and U1 + U2 . Example 5.1.3 Let (X1 , . . . , Xn ) and (Y1 , . . . , Yn ) be samples from normal  distributions N (ξ, σ 2 ) and N (η, τ 2 ) respectively. Then T = (X̄, Xi2 , Ȳ , Yi2 ) is sufficient for (ξ, σ 2 , η, τ 2 ) and the family of distributions of T is complete. Since  (Xi − X̄)(Yi − Ȳ ) V =  (Xi − X̄)2 (Yi − Ȳ )2 is unchanged when Xi and Yi are replaced by (Xi − ξ)/σ and (Yi − η)/τ , the distribution of V does not depend on any of the parameters, and Theorem 5.1.2 shows V to be independent of T . 5.2 Testing the Parameters of a Normal Distribution The four hypotheses σ ≤ σ0 , σ ≥ σ0 , ξ ≤ ξ0 , ξ ≥ ξ0 concerning the variance σ 2 and mean ξ of a normal distribution were discussed in Section 3.9, and it was 154 5. Unbiasedness: Applications to Normal Distributions pointed out there that at the usual significance levels there exists a UMP test only for the first one. We shall now show that the standard (likelihood-ratio) tests are UMP unbiased for the above four hypotheses as well as for some of the corresponding two-sided problems. For varying ξ and σ, the densities   nξ 2 ξ  1  2 (2πσ 2 )−n/2 exp − 2 exp − 2 xi + 2 xi (5.8) 2σ 2σ σ of a sample X1 , . . . , Xn from N (ξ, σ 2 ) constitute a two-parameter exponential family, which coincides with (5.1) for   2 xi 1 nξ θ = − 2 , ϑ = 2 , U (X) = xi , T (x) = x̄ = . 2σ σ n By Theorem 4.4.1, there exists therefore a UMP unbiased test of the hypothesis θ ≥ θ0 , which for θ0 = −1/2σ02 is equivalent to H : σ ≥ σ0 . The rejection region of this test can be obtained from (4.12), with the inequalities reversed because the hypothesis is now θ ≥ θ0 . In the present case this becomes  2 xi ≤ C0 (x̄) where pσ0 If this is written as -  . Xi2 ≤ C0 (x̄) | x̄ = α. x2i − nx̄2 < C0 (x̄)  2  it follows from the independence of X1 − nX̄ 2 = (Xi − X̄)2 and X̄ (Example   5.1.1) that C0 (x) does not depend on x̄. The test therefore rejects when (xi − 2  x̄) ≤ C0 , or equivalently when  (xi − x̄)2 ≤ C0 , (5.9) σ02   with C0 determined by Pσ0 { (Xi − X̄)2 /σ02 ≤ C0 } = α. Since (Xi − X̄)2 /σ02 has a χ2 -distribution with n − 1 degrees of freedom, the determining condition for C0 is  C0 χ2n−1 (y) dy = α , (5.10) 0 χ2n−1 where denotes the density of a χ2 variable with n − 1 degrees of freedom. The same result can be obtained through Theorem 5.1.1. A statistic V = h(U, T ) of the kind required by the theorem – that is, independent of X̄ for σ = σ0 , and all ξ – is  V = (Xi − X̄)2 = U − nT 2 . This is in fact independent of X̄ for all ξ and σ 2 . Since h(u, t) is an increasing function of u for each t, it follows that the UMP unbiased test has a rejection region of the form V ≤ C0 . 5.2. Testing the Parameters of a Normal Distribution 155 This derivation also shows that the UMP unbiased rejection region for H : σ ≤ σ1 or σ ≥ σ2 is  (xi − x̄)2 < C2 (5.11) C1 < where the C’s are given by  C2 /σ12  χ2n−1 (y) dy = 2 C1 /σ1 2 C2 /σ2 2 C1 /σ2 χ2n−1 (y) dy = α. (5.12) Since h(u, t) is linear in u, it is further seen that the UMP unbiased test of H : σ = σ0 , has the acceptance region  (xi − x̄)2 C1 < < C2 (5.13) σ02 with the constants determined by  C2  C1 1 χ2n−1 (y) dy = yχ2n−1 (y) dy = 1 − α. (5.14) n − 1 C2 C1  is just the test obtained in Example 4.2.2 with (xi − x̄)2 in place of This x2i and n − 1 degrees of freedom instead of n, as could have been foreseen. Theorem 5.1.1 shows for this and the other hypotheses considered that the UMP unbiased test depends only on V . Since the distributions of V do not depend on ξ, and constitute an exponential family in σ, the problems are thereby reduced to the corresponding ones for a one-parameter exponential family, which were solved previously. The power of the above tests can be obtained explicitly in terms of the χ2 distribution. In the case of the one-sided test (5.9) for example, it is given by    C0 σ2 /σ2 0 (Xi − X̄)2 C0 σ02 β(σ) = Pσ ≤ χ2n−1 (y) dy. = 2 2 σ σ 0 The same method can be applied to the problems of testing the hypotheses ξ ≤ ξ0 against ξ > ξ0 and ξ = ξ0 against ξ = ξ0 . As is seen by transforming to the variables Xi − ξ0 , there is no loss of generality in assuming that ξ0 = 0. It is convenient here to make the identification of (5.8) with (5.1) through the correspondence  2 nξ 1 θ = 2 , ϑ = − 2 , U (x) = x̄, T (x) = xi . σ 2σ Theorem 4.4.1 then shows that UMP unbiased tests exist for the hypotheses θ ≤ 0 and θ = 0, which are equivalent to ξ ≤ 0 and ξ = 0. Since V = X̄ U = √  2 T − nU 2 (Xi − X̄)  2 is independent of T = Xi when ξ = 0 (Example 5.1.1), it follows from Theorem 5.1.1 that the UMP unbiased rejection region for H : ξ ≤ 0 is V ≥ C0 or equivalently t(x) ≥ C0 , (5.15) 156 5. Unbiasedness: Applications to Normal Distributions where √ t(x) = , 1 n−1 nx̄ .  (xi − x̄)2 (5.16)  2 Xi . This is In order to apply the theorem to H  : ξ = 0, let W = X̄/ 2 also independent of Xi when ξ = 0, and in addition is linear in U = X̄. The distribution of W is symmetric about 0 when ξ = 0, and conditions (5.4), (5.5), (5.6) with W in place of V are therefore satisfied for the rejection region |w| ≥ C  with Pξ=0 {|W | ≥ C  } = α. Since t(x) = (n − 1)nW (x) 1 − nW 2 (x) , the absolute value of t(x) is an increasing function of |W (x)|, and the rejection region is equivalent to |t(x)| ≥ C. (5.17) From (5.16) it is seen that t(X) is the ratio of the two independent random  √ (Xi − X̄)2 /(n − 1)σ 2 . The denominator is distributed as the nX̄/σ and square root of a χ2 -variable with n − 1 degrees of freedom, divided by n − 1; the distribution of the numerator, when ξ = 0, is the normal distribution N (0, 1). The distribution of such a ratio is Student’s t-distribution with n − 1 degrees of freedom, which has probability density (Problem 5.3) tn−1 (y) = Γ( 1 n) 1 1 # 2 $ 1n . 2 π(n − 1) Γ 1 (n − 1) y2 2 1 + n−1 (5.18) The distribution is symmetric about 0, and the constants C0 and C of the oneand two-sided tests are determined by  ∞  ∞ α tn−1 (y) dy = α and tn−1 (y) dy = . (5.19) 2 C0 C For ξ = 0, the distribution of t(X) is the so-called noncentral t-distribution, which is derived in Problem 5.3. Some properties of the power function of the oneand two-sided t-test are given in Problems 5.1, 5.2, and 5.4. We note here that the distribution of t(X), and therefore the power of the above tests, depends only on √ the noncentrality parameter δ = nξ/σ. This is seen from the expression of the probability density given in Problem 5.3, but can also be shown by the following direct argument. Suppose that ξ  /σ  = ξ/σ = 0, and denote the common value of ξ  /ξ and σ  /σ by c, which is then also different from zero. If Xi = cXi and the Xi are distributed as N (ξ, σ 2 ), the variables Xi have distribution N (ξ  , σ 2 ). Also t(X) = t(X  ), and hence t(X  ) has the same distribution as t(X), as was to be proved. [Tables of the power of the t-test are discussed, for example, in Chapter 31, Section 7 of Johnson, Kotz and Balakrishnan (1995, Vol. 2).] If ξ1 denotes any alternative value to ξ = 0, the power β(ξ, σ) = f (δ) depends on σ. As σ → ∞, δ → 0, and β(ξ1 , σ) → f (0) = β(0, σ) = α, since f is continuous by Theorem 2.7.1. Therefore, regardless of the sample size, the probability of detecting the hypothesis to be false when ξ ≥ ξ1 > 0 cannot be 5.3. Comparing the Means and Variances of Two Normal Distributions 157 made ≥ β > α for all σ. This is not surprising, since the distributions N (0, σ 2 ) and N (ξ1 , σ 2 ) become practically indistinguishable when σ is sufficiently large. To obtain a procedure with guaranteed power for ξ ≥ ξ1 , the sample size must be made to depend on σ. This can be achieved by a sequential procedure, with the stopping rule depending on an estimate of σ, but not with a procedure of fixed sample size. (See Problems 5.23 and 5.25.) The tests of the more general hypotheses ξ ≤ ξ0 and ξ = ξ0 are reduced to those above by transforming to the variables Xi − ξ0 . The rejection regions for these hypotheses are given as before by (5.15), (5.17), and (5.19), but now with √ n(x̄ − ξ0 ) . t(x) = ,  1 (xi − x̄)2 n−1 It is seen from the representation of (5.8) as an exponential family with θ = nξ/σ 2 that there exists a UMP unbiased test of the hypothesis a ≤ ξ/σ 2 ≤ b, but the method does not apply to the more interesting hypothesis a ≤ ξ ≤ b;1 nor is it applicable to the corresponding hypothesis for the mean expressed in σ-units: a ≤ ξ/σ ≤ b, which will be discussed in Chapter 6. The dual equivalence problem of testing ξ/σ ∈ / [a, b] is treated in Brown, Casella and Hwang (1995), Brown, Hwang, and Munk (1997) and Perlman and Wu (1999). When testing the mean ξ of a normal distribution, one may from extensive past experience believe σ to be essentially known. If in fact σ is known to be equal to σ0 , it follows from Problem 3.1 that there exists a UMP test φ0 of H : ξ ≤ ξ0 , against K : ξ > ξ0 , which rejects when (X̄ − ξ0 )/σ0 is sufficiently large, and this test is then uniformly more powerful than the t-test (5.15). On the other hand, if the assumption σ = σ0 is in error the size of φ0 will differ from α and may greatly exceed it. Whether to take such a risk depends on one’s confidence in the assumption and the gain resulting from the use of φ0 when σ is equal to σ0 . A measure of this gain is the deficiency d of the t-test with respect to φ0 , the number of additional observations required by the t-test to match the power of φ0 when σ = σ0 . Except for very small n, d is essentially independent of sample size and for typical values of α is of the order of 1 to 3 additional observations. [For details see Hodges and Lehmann (1970). Other approaches to such comparisons are reviewed, for example, in Rothenberg (1984).] 5.3 Comparing the Means and Variances of Two Normal Distributions The problem of comparing the parameters of two normal distributions arises in the comparison of two treatments, products, etc., under conditions similar to those discussed at the beginning of Section 4.5. We consider first the comparison of two variances σ 2 and τ 2 , which occurs for example when one is concerned with the variability of analyses made by two different laboratories or by two different methods, and specifically the hypotheses H : τ 2 /σ 2 ≤ ∆0 and H  : τ 2 /σ 2 = ∆0 . 1 This problem is discussed in Section 3 of Hodges and Lehmann (1954). 158 5. Unbiasedness: Applications to Normal Distributions Let X = (X1 , . . . , Xm ) and Y = (Y1 , . . . , Yn ) be samples from the normal distributions N (ξ, σ 2 ) and N (η, τ 2 ) with joint density  1  2 1  2 mξ nη C(ξ, η, σ, τ ) exp − 2 xi − 2 yj + 2 x̄ + 2 ȳ . 2σ 2τ σ τ This is an exponential family with the four parameters 1 nη mξ 1 , ϑ1 = − 2 , ϑ2 = 2 , ϑ3 = 2 2τ 2 2σ τ σ and the sufficient statistics  2  2 U= Yj , T1 = Xi , T2 = Ȳ , T3 = X̄. θ=− It can be expressed equivalently (see Lemma 4.4.1) in terms of the parameters θ∗ = − and the statistics  2 U∗ = Yj , 1 1 + , 2τ 2 2∆0 σ 2 T1∗ =  Xi2 + ϑ∗i = ϑi (i = 1, 2, 3) 1  2 Yj , ∆0 T2∗ = Ȳ , T3∗ = X̄. The hypotheses θ∗ ≤ 0 and θ∗ = 0, which are equivalent to H and H  respectively, therefore possess UMP unbiased tests by Theorem 4.4.1. When τ 2 = ∆0 σ 2 , the distribution of the statistic   (Yj − Ȳ )2 /∆0 (Yj − Ȳ )2 /τ 2  V =  = (Xi − X̄)2 (Xi − X̄)2 /σ 2 does not depend on σ, ξ, or η, and it follows from Corollary 5.1.1 that V is independent of (T1∗ , T2∗ , T3∗ ). The UMP unbiased test of H is therefore given by (5.2) and (5.3), so that the rejection region can be written as  (Yj − Ȳ )2 /∆0 (n − 1)  (5.20) ≥ C0 . (Xi − X̄)2 /(m − 1) When τ 2 = ∆0 σ 2 , the statisticon the left-hand side  of (5.20) is the ratio of the two independent χ2 variables (Yj − Ȳ )2 /τ 2 and (Xi − X̄)2 /σ 2 , each divided by the number of its degrees of freedom. The distribution of such a ratio is the F-distribution with n − 1 and m − 1 degrees of freedom, which has the density # $  1 (n−1) Γ 12 (m + n − 2) n−1 2 # $ # $ (5.21) Fn−1,m−1 (y) = m−1 Γ 12 (m − 1) Γ 12 (n − 1) 1 × 1+ y 2 (n−1)−1  1 (m+n−2) . 2 n−1 y m−1 The constant C0 of (5.20) is then determined by  ∞ Fn−1,m−1 (y) dy = α. C0 In order to apply Theorem 5.1.1 to H  let  (Yj − Ȳ )2 /∆0  . W =  2 (Xi − X̄) + (1/∆0 ) (Yj − Ȳ )2 (5.22) 5.3. Comparing the Means and Variances of Two Normal Distributions 159 This is also independent of T ∗ = (T1∗ , T2∗ , T3∗ ) when τ 2 = ∆0 σ 2 , and is linear in U ∗ . The UMP unbiased acceptance region of H  is therefore C1 ≤ W ≤ C2 (5.23) with the constants determined by (5.5) and (5.6) where V is replaced by W . On dividing numerator and denominator of W by σ 2 it is seen that for τ 2 = ∆0 σ 2 , the statistic W is a ratio of the form W1 /(W1 + W2 ), where W1 and W2 are independent χ2 variables with n − 1 and m − 1 degrees of freedom respectively. Equivalently, W = Y /(1 + Y ), where Y = W1 /W2 and where (m − 1)Y /(n − 1) has the distribution Fn−1,m−1 . The distribution of W is the beta-distribution2 with density # $ Γ 12 (m + n − 2) 1 1 $ # $ w 2 (n−3) (1 − w) 2 (m−3) , B 1 (n−1), 1 (m−1) (w) = # 2 2 1 1 Γ 2 (m − 1) Γ 2 (n − 1) (5.24) 0 < w < 1. The conditions (5.5) and (5.6), by means of the relations E(W ) = n−1 m+n−2 and wB 1 (n−1), 1 m−1) (w) = 2 2 become  C2 C1  B 1 (n−1), 1 (m−1) (w) dw = 2 2 n−1 (w), B1 1 m + n − 2 2 (n+1), 2 (m−1) C2 C1 B 1 (n+1), 1 (m−1) (w) dw = 1 − α. 2 2 (5.25) The definition of V shows that its distribution depends only on the ratio τ 2 /σ 2 , and so does the distribution of W . The power of the tests (5.20) and (5.23) is therefore also a function only of the variable ∆ = τ 2 /σ 2 ; it can be expressed explicitly in terms of the F -distribution, for example in the first case by    (Yj − Ȳ )2 /τ 2 (n − 1) C0 ∆0 β(∆) = P  ≥ ∆ (Xi − X̄)2 /σ 2 (m − 1)  ∞ = Fn−1,m−1 (y) dy. C0 ∆0 /∆ The hypothesis of equality of the means ξ, η of two normal distributions with unknown variances σ 2 and τ 2 , the so-called Behrens-Fisher problem, is not accessible by the present method. (See Example 4.3.3; for a discussion of this problem, Section 6.6, Section 11.3.1 and Example 13.5.4.) We shall therefore consider only 2 The relationship W = Y /(1 + Y ) shows the F - and beta-distributions to be equivalent. Tables of these distributions are discussed in Chapters 24 and 26 of Johnson, Kotz and Balakrishnan (1995. Vol. 2). Critical values of F are tabled by Mardia and Zemroch (1978), who also provide algorithms for the associated computations. 160 5. Unbiasedness: Applications to Normal Distributions the simpler case in which the two variances are assumed to be equal. The joint density of the X’s and Y ’s is then   η  ξ  1  2  2  yj + 2 xi + 2 yj , xi + (5.26) C(ξ, η, σ) exp − 2 2σ σ σ which is an exponential family with parameters θ= η , σ2 ϑ1 = ξ , σ2 and the sufficient statistics   U= Yj , T1 = Xi ϑ2 = − T2 =  1 2σ 2 Xi2 +  Yj2 . For testing the hypotheses H :η−ξ ≤0 H : η − ξ = 0 and it is more convenient to represent the densities as an exponential family with the parameters η−ξ , + n1 σ 2 m θ∗ =  1 ϑ∗1 = mξ + nη , (m + n)σ 2 and the sufficient statistics U ∗ = Ȳ − X̄, T1∗ = mX̄ + nȲ , T2∗ =  ϑ∗2 = ϑ2 Xi2 +  Yj2 . That this is possible is seen from the identity mξ x̄ + nη ȳ = (ȳ − x̄)(η − ξ) (mx̄ + nȳ)(mξ + nη) + . 1 1 m+n + m n It follows from Theorem 4.4.1 that UMP unbiased tests exist for the hypotheses θ∗ ≤ 0 and θ∗ = 0, and hence for H and H  . When η = ξ, the distribution of V = = Ȳ − X̄   (Xi − X̄)2 + (Yj − Ȳ )2 , T2∗ − U∗ 1 T ∗2 m+n 1 − mn U ∗2 m+n does not depend on the common mean ξ or on σ, as is seen by replacing Xi with (Xi − ξ)/σ and Yj with (Yj − ξ)/σ in the expression for V , and V is independent of (T1∗ , T2∗ ). The rejection region of the UMP unbiased test of H can therefore be written as V ≥ C0 or t(X, Y ) ≥ C0 , where 5, 1 (Ȳ − X̄) + m (5.27) 1 n t(X, Y ) = , .   (Xi − X̄)2 + (Yj − Ȳ )2 /(m + n − 2) (5.28) 5.4. Confidence Intervals and Families of Tests 161 The statistic t(X, Y ) is the ratio of the two independent variables 6  (Xi − X̄)2 + (Yj − Ȳ )2 Ȳ − X̄ , and . (m + n − 2)σ 2 1 + n1 σ 2 m √ The numerator is normally distributed with mean (η −ξ)/ m−1 + n−1 σ and unit variance; the square of the denominator as a χ2 variable with m + n − 2 degrees of freedom, divided by m + n − 2. Hence t(X, Y ) has a noncentral t-distribution with m + n − 2 degrees of freedom and noncentrality parameter δ= , η−ξ 1 m + . 1 σ n When in particular η − ξ = 0, the distribution of t(X, Y ) is Student’s t-distribution, and the constant C0 is determined by  ∞ tm+n−2 (y) dy = α. (5.29) C0 As before, the assumptions required by Theorem 5.1.1 for H  are not satisfied by V itself but by a function of V , W = 1  Ȳ − X̄ Xi2 +  Yj2 −  (  2 Xi + Yj ) m+n which is related to V through V = , 1− W mn W2 m+n . Since W is a function of V , it is also independent of (T1∗ , T2∗ ) when η = ξ; in addition it is a linear function of U ∗ with coefficients dependent only on T ∗ . The distribution of W being symmetric about 0 when η = ξ, it follows, as in the derivation of the corresponding rejection region (5.17) for the one-sample problem, that the UMP unbiased test of H  rejects when |W | is too large, or equivalently when |t(X, Y )| > C. (5.30) The constant C is determined by  ∞ α tm+n−2 (y) dy = . 2 C The power of the tests (5.27) and (5.30) depends only on (η − ξ)/σ and is given in terms of the noncentral t-distribution. Its properties are analogous to those of the one-sample t-test (Problems 5.1, 5.2, and 5.4). 5.4 Confidence Intervals and Families of Tests Confidence bounds for a parameter θ corresponding to a confidence level 1 − α were defined in Section 3.5, for the case that the distribution of the random 162 5. Unbiasedness: Applications to Normal Distributions variable X depends only on θ. When nuisance parameters ϑ are present the defining condition for a lower confidence bound θ becomes Pθ,ϑ {θ(X) ≤ θ} ≥ 1 − α for all θ, ϑ. (5.31) Similarly, confidence intervals for θ at confidence level 1 − α are defined as a set of random intervals with end points θ(X), θ̄(X) such that Pθ,ϑ {θ(X) ≤ θ ≤ θ̄(X)} ≥ 1 − α for all θ, ϑ. (5.32) The infimum over (θ, ϑ) of the left-hand side of (5.31) and (5.32) is the confidence coefficient associated with these statements. As was already indicated in Chapter 3, confidence statements permit a dual interpretation. Directly, they provide bounds for the unknown parameter θ and thereby a solution to the problem of estimating θ. The statement θ ≤ θ ≤ θ̄ is not as precise as a point estimate, but it has the advantage that the probability of it being correct can be guaranteed to be at least 1 − α. Similarly, a lower confidence bound can be thought of as an estimate θ which overestimates the true parameter value with probability ≤ α. In particular for α = 12 , if θ satisfies 1 , 2 the estimate is as likely to underestimate as to overestimate and is then said to be median unbiased. (See Problem 1.3, for the relation of this property to a more general concept of unbiasedness.) For an exponential family given by (4.10) there exists an estimator of θ which among all median unbiased estimators uniformly minimizes the risk for any loss function L(θ, d) that is monotone in the sense of the last paragraph of Section 3.5. A full treatment of this result including some probabilistic and measure-theoretic complications, is given by Pfanzagl (1979). Alternatively, as was shown in Chapter 3, confidence statements can be viewed as equivalent to a family of tests. The following is essentially a review of the discussion of this relationship in Chapter 3, made slightly more specific by restricting attention to the two-sided case. For each θ0 , let A(θ0 ) denote the acceptance region of a level-α test (assumed for the moment to be nonrandomized) of the hypothesis H(θ0 ) : θ = θ0 . If Pθ,ϑ {θ ≤ θ} = Pθ,ϑ {θ ≥ θ} = S(x) = {θ : x ∈ A(θ)} then θ ∈ S(x) if and only if x ∈ A(θ), (5.33) and hence Pθ,ϑ {θ ∈ S(X)} ≥ 1 − α for all θ, ϑ. (5.34) Thus any family of level-α acceptance regions, through the correspondence (5.33), leads to a family of confidence sets at confidence level 1 − α. Conversely, given any class of confidence sets S(x) satisfying (5.34), let A(θ) = {x : θ ∈ S(x)}. (5.35) Then the sets A(θ0 ) are level-α acceptance regions for testing the hypotheses H(θ0 ) : θ = θ0 , and the confidence sets S(x) show for each θ0 whether for the particular x observed the hypothesis θ = θ0 is accepted or rejected at level α. 5.4. Confidence Intervals and Families of Tests 163 Exactly the same arguments apply if the sets A(θ0 ) are acceptance regions for the hypotheses θ ≤ θ0 . As will be seen below, one- and two-sided tests typically, although not always, lead to one-sided confidence bounds and to confidence intervals respectively. Example 5.4.1 (Normal mean) Confidence intervals for the mean ξ of a normal distribution with unknown variance can be obtained from the acceptance regions A(ξ0 ) of the hypothesis H : ξ = ξ0 . These are given by √ | n(x̄ − ξ0 )| ≤ C,  (xi − x̄)2 /(n − 1) where C is determined from the t-distribution so that the probability of this inequality is 1 − α when ξ = ξ0 . [See (5.17) and (5.19) of Section 5.2.] The set S(x) is then the set of ξ’s satisfying this inequality with ξ = ξ0 , that is, the interval 1 1 1  1  C C 2 x̄ − √ (5.36) (xi − x̄) ≤ ξ ≤ x̄ + √ (xi − x̄)2 . n n−1 n n−1 The class of these intervals therefore constitutes confidence intervals for ξ with confidence coefficient 1 − α.  The length of the intervals (5.36) is proportional to (xi − x̄)2 and their expected length to σ. For large σ, the intervals will therefore provide little information concerning the unknown ξ. This is a consequence of the fact, which led to similar difficulties for the corresponding testing problem, that two normal distributions N (ξ0 , σ 2 ) and N (ξ1 , σ 2 ) with fixed difference of means become indistinguishable as a tends to infinity. In order to obtain confidence intervals for ξ whose length does not tend to infinity with σ, it is necessary to determine the number of observations sequentially so that it can be adjusted to σ. A sequential procedure leading to confidence intervals of prescribed length is given in Problems 5.23 and 5.24. However, even such a sequential procedure does not really dispose of the difficulty, but only shifts the lack of control from the length of the interval to the number of observations, As σ → ∞, the number of observations required to obtain confidence intervals of bounded length also tends to infinity. Actually, in practice one will frequently have an idea of the order of magnitude of σ. With a sample either of fixed size or obtained sequentially, it is then necessary to establish a balance between the desired confidence 1 − α, the accuracy given by the length l of the interval, and the number of observations n one is willing to expend. In such an arrangement two of the three quantities 1 − α, l, and n will be fixed, while the third is a random variable whose distribution depends on σ, so that it will be less well controlled than the others. If 1 − α is taken as fixed, the choice between a sequential scheme and one of fixed sample size thus depends essentially on whether it is more important to control l or n. To obtain lower confidence limits for ξ, consider the acceptance regions √ n(x̄ − ξ0 ) ≤ C0  (xi − x̄)2 /(n − 1) 164 5. Unbiasedness: Applications to Normal Distributions for testing ξ ≤ ξ0 to against ξ > ξ0 . The sets S(x) arc then the one-sided intervals 1 1  C0 x̄ − √ (xi − x̄)2 ≤ ξ, n n−1 the left-hand sides of which therefore constitute the desired lower bounds ξ. If α = 12 , the constant C0 is 0; the resulting confidence bound ξ = X̄ is a median unbiased estimate of ξ, and among all such estimates it uniformly maximizes P {−∆1 ≤ ξ − ξ ≤ ∆2 } for all ∆1 , ∆2 ≥ 0. (For a proof see Section 3.5.) 5.5 Unbiased Confidence Sets Confidence sets can be viewed as a family of tests of the hypotheses θ ∈ H(θ ) against alternatives θ ∈ K(θ ) for varying θ . A confidence level of 1 − α then simply expresses the fact that all the tests are to be at level α, and the condition therefore becomes Pθ,ϑ {θ ∈ S(X)} ≥ 1 − α for all θ ∈ H(θ ) and all ϑ. (5.37) In the case that H(θ ) is the hypothesis θ = θ and S(X) is the interval [θ(X), θ̄(X)], this agrees with (5.32). In the one-sided case in which H(θ ) is the hypothesis θ ≤ θ and S(X) = {θ : θ(X) ≤ θ}, the condition reduces to Pθ,ϑ {θ(X) ≤ θ } ≥ 1 − α for all θ ≥ θ, and this is seen to be equivalent to (5.31). With this interpretation of confidence sets, the probabilities Pθ,ϑ {θ ∈ S(X)}, θ ∈ K(θ ), (5.38)  are the probabilities of false acceptance of H(θ ) (error of the second kind). The smaller these probabilities are, the more desirable are the tests. From the point of view of estimation, on the other hand, (5.38) is the probability of covering the wrong value θ . With a controlled probability of covering the true value, the confidence sets will be more informative the less likely they are to cover false values of the parameter. In this sense the probabilities (5.38) provide a measure of the accuracy of the confidence sets. A justification of (5.38) in terms of loss functions was given for the one-sided case in Section 3.5. In the presence of nuisance parameters, UMP tests usually do not exist, and this implies the nonexistence of confidence sets that are uniformly most accurate in the sense of minimizing (5.38) for all θ such that θ ∈ K(θ ) and for all ϑ. This suggests restricting attention to confidence sets which in a suitable sense are unbiased. In analogy with the corresponding definition for tests, a family of confidence sets at confidence level 1 − α is said to be unbiased if Pθ,ϑ {θ ∈ S(X)} ≤ 1 − α (5.39) for all θ such that θ ∈ K(θ ) and for all ϑ and θ, so that the probability of covering these false values does not exceed the confidence level. 5.5. Unbiased Confidence Sets 165 In the two- and one-sided cases mentioned above, the condition (5.39) reduces to Pθ,ϑ {θ ≤ θ ≤ θ̄} ≤ 1 − α for all θ = θ and all ϑ and Pθ,ϑ {θ ≤ θ } ≤ 1 − α for all θ < θ and all ϑ. With this definition of unbiasedness, unbiased families of tests lead to unbiased confidence sets and conversely. A family of confidence sets is uniformly most accurate unbiased at confidence level 1 − α if it minimizes the probabilities Pθ,ϑ {θ ∈ S(X)} for all θ such that θ ∈ K(θ ) and for all ϑ and θ, subject to (5.37) and (5.39). The confidence sets obtained on the basis of the UMP unbiased tests of the present and preceding chapter are therefore uniformly most accurate unbiased. This applies in particular to the confidence intervals obtained in the preceding sections. Some further examples are the following. Example 5.5.1 (Normal variance) If X1 , . . . , Xn is a sample from N (ξ, σ 2 ), the UMP unbiased test of the hypothesis σ = σ0 is given by the acceptance region (5.13)  (xi − x̄)2  C1 ≤ ≤ C2 , σ02 where C1 and C2 are determined by (5.14). The most accurate unbiased confidence intervals for σ 2 are therefore 1  1  (xi − x̄)2 ≤ σ 2 ≤  (xi − x̄)2 . C2 C1 [Tables of C1 and C2 are provided by Tate and Klett (1959).] Similarly, from (5.9) and (5.10) the most accurate unbiased upper confidence limits for σ 2 are 1  (xi − x̄)2 , σ2 ≤ C0 where  ∞ χ2n−1 (y) dy = 1 − α. C0 The corresponding lower confidence limits are uniformly most accurate (without the restriction of unbiasedness) by Section 3.9. Example 5.5.2 (Difference of means) Confidence intervals for the difference ∆ = η − ξ of the means of two normal distributions with common variance are obtained from tests of the hypothesis η−ξ = ∆0 . If X1 , . . . , Xm and Y1 , . . . , Yn are distributed as N (ξ, σ 2 ) and N (η, σ 2 ) respectively, and if Yj = Yj −∆0 , η  = η−∆0 , the hypothesis can be expressed in terms of the variables Xi and Yj as η  − ξ = 0. From (5.28) and (5.30) the UMP unbiased acceptance region is then seen to be 5, 1 |(ȳ − x̄ − ∆0 )| + n1 m 6 ≤ C, 5   [ (xi − x̄)2 + (yj − ȳ)2 ] (m + n − 2) 166 5. Unbiasedness: Applications to Normal Distributions where C is determined by the equation following (5.30). The most accurate unbiased confidence intervals for η − ξ are therefore (ȳ − x̄) − CS ≤ η − ξ ≤ (ȳ − x̄) + CS where S2 = 1 1 + m n (5.40)   (xi − x̄)2 + (yj − ȳ)2 m+n−2 The one-sided intervals are obtained analogously. Example 5.5.3 (Ratio of variances) If X1 , . . . , Xm and Y1 , . . . , Yn are samples from N (ξ, σ 2 ) and N (η, τ 2 ), most accurate unbiased confidence intervals for ∆ = τ 2 /σ 2 are derived from the acceptance region (5.23) as   1 − C2 (yj − ȳ)2 τ2 1 − C1 (yj − ȳ)2   ≤ ≤ , (5.41) C2 σ2 C1 (xi − x̄)2 (xi − x̄)2 where C1 and C2 are determined from (5.25).3 In the particular case that m = n, the intervals take on the simpler form   (yj − ȳ)2 1 (yj − ȳ)2 τ2   ≤ ≤ k , (5.42) (xi − x̄)2 k (xi − x̄)2 σ2 where k is determined from the F -distribution. Most accurate unbiased lower confidence limits for the variance ratio are  (y − ȳ)2 /(n − 1) 1 τ2  j ∆= (5.43) ≤ 2 2 C0 (xi − x̄) /(m − 1) σ with C0 given by (5.22). If in (5.22) α is taken to be 12 , this lower confidence limit ∆ becomes a median unbiased estimate of τ 2 /σ 2 . Among all such estimates it uniformly minimizes   τ2 P −∆1 ≤ 2 − ∆ ≤ ∆2 for all ∆1 , ∆2 ≥ 0. σ (For a proof see Section 3.5). So far it has been assumed that the tests from which the confidence sets are obtained are nonrandomized. The modifications that are necessary when this assumption is not satisfied were discussed in Chapter 3. The randomized tests can then be interpreted as being nonrandomized in the space of X and an auxiliary variable V which is uniformly distributed on the unit interval. If in particular X is integer-valued as in the binomial or Poisson case, the tests can be represented in terms of the continuous variable X + V . In this way, most accurate unbiased confidence intervals can be obtained, for example, for a binomial probability p from the UMP unbiased tests of H : p = p0 (Example 4.2.1). It is not clear a priori that the resulting confidence sets for p will necessarily by intervals. This is, however, a consequence of the following Lemma. 3 A comparison of these limits with those obtained from the equal-tails test is given by Scheffé (1942); some values of C 1 and C 2 are provided by Ramachandran (1958). 5.5. Unbiased Confidence Sets 167 Lemma 5.5.1 Let X be a real-valued random variable with probability density pθ (x) which has monotone likelihood ratio in x. Suppose that UMP unbiased tests of the hypotheses H(θ0 ) : θ = θ0 exist and are given by the acceptance regions C1 (θ0 ) ≤ x ≤ C2 (θ0 ) and that they are strictly unbiased. Then the functions Ci (θ) are strictly increasing in θ, and the most accurate unbiased confidence intervals for θ are C2−1 (x) ≤ θ ≤ C1−1 (x). Proof. Let θ0 < θ1 , and let β0 (θ) and β1 (θ) denote the power functions of the above tests φ0 and φ1 , for testing θ = θ0 and θ = θ1 . It follows from the strict unbiasedness of the tests that Eθ0 [φ1 (X) − φ0 (X)] = β1 (θ0 ) − α > 0 > α − β0 (θ1 ) = Eθ1 [φ1 (X) − φ0 (X)] . Thus neither of the two intervals [C1 (θi ), C2 (θi )] (i = 0, 1) contains the other, and it is seen from Lemma 3.4.2(iii) that Ci (θ0 ) < Ci (θ1 ) for i = 1, 2. The functions Ci therefore have inverses, and the inequalities defining the acceptance region for H(θ) are equivalent to C2−1 (x) ≤ θ ≤ C1−1 (x), as was to be proved. The situation is indicated in Figure 5.1. From the boundaries x = C1 (θ) and x = C2 (θ) of the acceptance regions A(θ) one obtains for each fixed value of x the confidence set S(x) as the interval of θ’s for which C1 (θ) ≤ x ≤ C2 (θ). C2(␪) ␪ S(x) C1(␪) ␪0 A(␪0) x x Figure 5.1. By Section 4.2, the conditions of the lemma are satisfied in particular for a one-parameter exponential family, provided the tests are nonrandomized. In cases such as that of binomial or Poisson distributions, where the family is exponential but X is integer-valued so that randomization is required, the intervals can be obtained by applying the lemma to the variable X + V instead of X, where V is independent of X and uniformly distributed over (0, 1). Example 5.5.4 In the binomial case, a table of the (randomized) uniformly most accurate unbiased confidence intervals is given by Blyth and Hutchinson (1960). The best choice of nonrandomized intervals and some approximations 168 5. Unbiasedness: Applications to Normal Distributions are discussed (and tables provided) by Blyth and Still (1983) and Blyth (1984). Recent approximations and comparisons are provided by Agresti and Coull (1998) and Brown, Cai and DasGupta (2001, 2002). A large sample approach will be considered in Example 11.2.7. In Lemma 5.5.1, the distribution of X was assumed to depend only on θ. Consider now the exponential family (5.1) in which nuisance parameters are present in addition to θ. The UMP unbiased tests of θ = θ0 , are then performed as conditional tests given T = t, and the confidence intervals for θ will as a consequence also be obtained conditionally. If the conditional distributions are continuous, the acceptance regions will be of the form C1 (θ; t) ≤ u ≤ C2 (θ; t), where for each t the functions Ci are increasing by Lemma 5.5.1. The confidence intervals are then C2−1 (u; t) ≤ θ ≤ C1−1 (u; 1). If the conditional distributions are discrete, continuity can be obtained as before through addition of a uniform variable. Example 5.5.5 (Poisson ratio) Let X and Y be independent Poisson variables with means λ and µ, and let ρ = µ/λ. The conditional distribution of Y given X + Y = t is the binomial distribution b(p, t) with ρ p= . 1+ρ The UMP unbiased test φ(y, t) of the hypothesis ρ = ρ0 is defined for each t as the UMP unbiased conditional test of the hypothesis ρ = ρ0 /(1 + ρ0 ). If p(t) ≤ p ≤ p̄(t) are the associated most accurate unbiased confidence intervals for p given t, it follows that the most accurate unbiased confidence intervals for µ/λ are p(t) p̄(t) µ ≤ ≤ . 1 − p(t) λ 1 − p̄(t) The binomial tests which determine the functions p(t) and p̄(t) are discussed in Example 4.2.1. 5.6 Regression The relation between two variables X and Y can be studied by drawing an unrestricted sample and observing the two variables for each subject, obtaining n pairs of measurements (X1 , Y1 ), . . . , (Xn , Yn ) (see Section 5.13 and Problem 5.13). Alternatively, it is frequently possible to control one of the variables such as the age of a subject, the temperature at which an experiment is performed, or the strength of the treatment that is being applied. Observations Y1 , . . . , Yn of Y can then be obtained at a number of predetermined levels x1 , . . . , xn of x. Suppose that for fixed x the distribution of Y is normal with constant variance 5.6. Regression 169 σ 2 and a mean which is a function of x, the regression of Y on x, and which is assumed to be linear,4 If we put vi = (xi − x̄)/  vi2 = 1, and α=γ−δ E[Y |x] = α + βx.   (xj − x̄)2 and γ + δvi = α + βxi , so that vi = 0, x̄ ,  (xj − x̄)2 β= δ ,  (xj − x̄)2 the joint density of Y1 , . . . , Yn is   1 1  2 √ exp − 2 (yi − γ − δvi ) . 2σ ( 2πσ)n These densities constitute an exponential family (5.1) with   2  U= vi Yi , T1 = Yi , T2 = Yi θ= ϑ1 = − 2σ1 2 , δ , σ2 ϑ2 = γ . σ2 This representation implies the existence of UMP unbiased tests of the hypotheses aγ + bδ = c where a, b, and c are given constants, and therefore of most accurate unbiased confidence intervals for the parameter ρ = aγ + bδ. To obtain these confidence intervals explicitly, one requires the UMP unbiased test of H : ρ = ρ0 , which is given by the acceptance region 5  |b vi Yi + aȲ − ρ0 | (a2 /n) + b2 6 ≤C (5.44) 5    2 (Yi − Ȳ )2 − ( vi Yi ) (n − 2) where  C −C tn−2 (y) dy = 1 − α ; see  Problem 5.33. The resulting confidence intervals for ρ are centered at b vi Yi + aȲ , and their length is 6   (Yi − Ȳ )2 − ( vi Yi )2 a2 L = 2C + b2 . n n−2 # It follows from the transformations given in Problem 5.33 that (Yi − Ȳ )2 − $  ( vi Yi )2 /σ 2 has a χ2 -distribution with n−2 degrees of freedom and hence that 4 The literature on regression is enormous and we treat the simplest model. Some texts on the subject include Weisberg (1985), Atkinson and Riani (2000) and Chatterjee, Hadi and Price (2000). 170 5. Unbiasedness: Applications to Normal Distributions the expected length of the intervals is 1 E(L) = 2Cn σ a2 + b2 . n In particular applications, a and b typically are functions of the x’s. If these are at the disposal of the experimenter and there is therefore some choice with respect to a and b, the expected length of L is minimized by minimizing (a2 /n) + b2 . Actually, it is not clear that the expected length is a good criterion for the accuracy of confidence intervals, since short intervals are desirable when they cover the true parameter value but not necessarily otherwise. However, the same result holds for other criteria such as the expected value of (ρ̄ − ρ)2 + (ρ − ρ)2 or more generally of f1 (|ρ̄−ρ|)+f2 (|ρ−ρ|), where f1 and f2 are increasing functions of their arguments. (See Problem 5.33.) Furthermore, the same choice of a and b also minimizes the probability of the intervals covering any false value of the parameter. We shall therefore consider (a2 /n) + b2 as an inverse measure of the accuracy of the intervals. Example 5.6.1 (Slope of regression line) Confidence levels for the slope β = δ/ (xj − x̄)2 are obtained from the above intervals by letting a = 0   and b = 1/ (xj − x̄)2 . Here the accuracy increases with (xj − x̄)2 , and if the xj must be chosen from an interval [C0 , C1 ], it is maximized by putting half of the values at each end point. However, from a practical point of view, this is frequently not a good design, since it permits no check of the linearity of the regression. Example 5.6.2 (Ordinate of regression line) Another parameter of interest is the value α + βx0 to be expected from an observation Y at x = x0 . Since α + βx0 = γ + δ(x0 − x̄) ,  (xj − x̄)2  (xj − x̄)2 . The maximum the constants a and b are a = 1, b = (x0 − x̄)/ accuracy is obtained by minimizing |x̄ − x | and, if x̄ = x0 cannot be achieved 0  exactly, also maximizing (xj − x̄)2 . Example 5.6.3 (Intercept of regression line) Frequently it is of interest to estimate the point x at which α+βx has a preassigned value. One may for example wish to find the dosage at which E(Y | x) = 0, or equivalently the  x = −α/β value v = (x − x̄)/ (xj − x̄)2 at which γ + δv = 0. Most accurate unbiased confidence sets for the solution −γ/δ of this equation can be obtained from the UMP unbiased tests of the hypotheses −γ/δ = v0 . The acceptance regions of these tests are given by (5.44) with a = 1, b = v0 , and ρ0 = 0, and the resulting confidence sets for v are the sets of values v satisfying   2   1  v2 C 2 S 2 − vi Yi − 2v Ȳ vi Yi + (C 2 S 2 − nȲ 2 ) ≥ 0. n 5.7. Bayesian Confidence Sets 171   where S 2 = [ (Yi − Ȳ )2 ( vi Yi )2 ]/(n − 2). If the associated quadratic equation in v has roots v, v̄, the confidence statement becomes % % % % % vi Yi % v ≤ v ≤ v̄ when >C S and % % % % % vi Yi % when < C. v ≤ v or v ≥ v̄ S The somewhat surprising possibility that the confidence sets may be the outside of an interval actually is quite appropriate here. When the line y = γ +δv is nearly parallel to the v-axis, the intercept with the v-axis will be large in absolute value, but its sign can be changed by a very small change in angle. There is the further possibility that the discriminant of the quadratic polynomial is negative,  2 nȲ 2 + vi Yi < C 2 S 2 , in which case the associated quadratic equation has no solutions. This condition implies that the leading coefficient of the quadratic polynomial is positive, so that the confidence set in this case becomes the whole real axis. The fact that the confidence sets are not necessarily finite intervals has led to the suggestion that their use be restricted to the cases in which they do have this form. Such usage will however affect the probability with which the sets cover the true value and hence the validity of the reported confidence coefficient.5 5.7 Bayesian Confidence Sets The left side of the confidence statement (5.34) denotes the probability that the random set S(X) will contain the constant point θ. The interpretation of this probability statement, before X is observed, is clear: it refers to the frequency with which this random event will occur. Suppose for example that X is distributed as N (θ, 1), and consider the confidence interval X − 1.96 < θ < X + 1.96 corresponding to confidence coefficient γ = .95. Then the random interval (X − 1.96, X +1.96) will contain θ with probability .95. Suppose now that X is observed to be 2.14. At this point, the earlier statement reduces to the inequality 0.18 < θ < 4.10, which no longer involves any random element. Since the only unknown quantity is θ, it is tempting (but not justified) to say that θ lies between 0.18 and 4.10 with probability .95. To attach a meaningful probability to the event θ ∈ S(x) when x is fixed requires that θ be random. Inferences made under the assumption that the parameter θ is itself a random (though unobservable) quantity with a known 5 A method for obtaining the size of this effect was developed by Neyman, and tables have been computed on its basis by Fix. This work is reported by Bennett (1957). 172 5. Unbiasedness: Applications to Normal Distributions distribution are called Bayesian, and the distribution Λ of θ before any observations are taken its prior distribution. After X = x has been observed, inferences concerning θ can be based on its conditional distribution given x, the posterior distribution. In particular, any set S(x) with the property P [θ ∈ S(x) | X = x] ≥ γ for all x is a 100γ% Bayesian confidence set or credible region for θ. In the rest of this section, the random variable with prior distribution Λ will be denoted by Θ, with θ being the value taken on by Θ in the experiment at hand. Example 5.7.1 (Normal mean) Suppose that Θ has a normal prior distribution N (µ, b2 ) and that given Θ = θ, the variables X1 , . . . , Xn . are independent N (θ, σ 2 ), σ known. Then the posterior distribution of Θ given x1 , . . . , xn is normal with mean (Problem 5.34) ηx = E[Θ | x] = nx̄/σ 2 + µ/b2 n/σ 2 + 1/b2 and variance τx2 = V ar[Θ | x] = 1 n/σ 2 + 1/b2 Since [Θ − ηx ]/τx then has a standard normal distribution, the interval I(x) with endpoints nx̄/σ 2 + µ/b2 ± n/σ 2 + 1/b2 1.96 n/σ 2 + 1/b2 satisfies P [Θ ∈ I(x) | X = x] = .95 and is thus a 95% credible region. For n = 1, µ = 0, σ = 1, the interval reduces to x 1.96 ±, 1 + b12 1 + b12 which for large b is very close to the confidence interval for θ stated at the beginning of the section. But now the statement that θ lies between these limits with probability .95 is justified, since it is a probability statement concerning the random variable Θ. The distribution N (µ, b2 ) assigns higher probability to θ-values near µ than to those further away. Suppose instead that no information whatever about θ is available, so that one wishes to model a state of complete ignorance. This could be done by assigning a constant density to all values of θ, that is, by assigning to Θ the density π(θ) ≡ c, −∞ < θ < ∞. Unfortunately, the resulting π is not a ∞ probability density, since −∞ π(θ) dθ = ∞. However, if this fact is ignored and the posterior distribution of Θ given x is calculated in the usual way, it turns out (Problem 5.35) that π(θ | x) is the density of a genuine probability distribution, namely N (µ, σ 2 /n), the limit of the earlier posterior distribution as b → ∞. The improper (since it integrates to infinity), noninformative prior density π(θ) ≡ c thus leads approximately to the same results as the normal prior N (µ, b2 ) for large b, and can be viewed as an approximation to the latter. 5.7. Bayesian Confidence Sets 173 Unlike confidence sets, Bayesian credible regions provide exactly the desired kind of probability statement even after the observations are known. They do so, however, at the cost of an additional assumption: that θ is random and has a known prior distribution. Detailed accounts of the Bayesian approach, its application to credible regions, and comparison of the two approaches can be found in Berger (1985a) and Robert (1994). The following examples provide a few illustrations and additional comments. Example 5.7.2 Let X be binomial b(p, n), and suppose that the prior distribution for p is the beta distribution6 B(a, b) with density Cpa−1 (1−p)b−1 , 0 < p < 1, 0 < a, b. Then the posterior distribution of p given X = x is the beta distribution B(a+x, b+n−x) (Problem 5.36). There are of course many sets S(x) whose probability under this distribution is equal to the prescribed coefficient γ. A choice that is frequently recommended is the HPD (highest probability density) region, defined by the requirement that the posterior density of p given x be ≥ k. With a beta prior, only the following possibilities can occur: for fixed x, (a) π(p | x) is decreasing, (b) π(p | x) is increasing, (c) π(p | x) is increasing in (0, p0 ) and decreasing in (p0 , 1) for some p0 , (d) π(p | x) is U-shaped, i.e. decreasing in (0, p0 ) and increasing in (p0 , 1) for some p0 . The HPD region then is of the form (a) p < K(−x), (b) p > K(x), (c) K1 (x) < p < K2 (x), (d) p < K1 (x) or p > K2 (x), where the K’s are determined by the requirement that the posterior probability of the region, given x, be γ; in cases (c) and (d) this condition must be supplemented by π[K1 (x) | x] = π[K2 (x) | x]. In general, if π(θ | x) denotes the posterior density of θ, the HPD region is defined by π(θ | x) ≥ k with C determined by the size condition P [π(θ) | x) ≥ k] = γ. 6 This is the so-called conjugate of the binomial distribution; for a more general discussion of conjugate distributions, see Chapter 4 of TPE2 and Robert (1994), Section 3.2. 174 5. Unbiasedness: Applications to Normal Distributions Example 5.7.3 (Two-parameter normal mean) Let X1 , . . . , Xn be independent N (ξ, σ 2 ), and for the sake of simplicity suppose that (ξ, σ) has the joint improper prior density given by 1 dσ for all − ∞ < ξ < ∞, 0 < σ, σ which is frequently used to model absence of information concerning the parameters. Then the joint posterior density of (ξ, σ) given x = (x1 , . . . , xn ) is of the form   n 1 1  π(ξ, σ | x) dξ dσ = C(x) n+1 exp − 2 (ξ − xi )2 dξ dσ. σ 2σ i=1 π(ξ, σ) dξ dσ = dξ Determination of a credible region for ξ requires the marginal posterior density of given x, which is obtained by integrating the joint posterior density with respect to σ. These densities depend only on the sufficient statistics x̄ and S 2 =  (xi − x̄)2 , and the posterior density of ξ is of the form (Problem 5.37)  n/2 1 A(x) 2 1 + n(ξ−x̄) S2 Here x̄ and S enter only as location and scale parameters, and the linear function √ n(ξ − x̄) √ t= S/ n − 1 of ξ has the t-distribution with n−1 degrees of freedom. Since this agrees with the distribution of t for fixed ξ and σ given in Section 5.2, the credible 100(1 − α)% region √   n(ξ − x̄)   √   S/ n − 1  ≤ C is formally identical with the confidence intervals (5.36). However, they are derived under different assumptions, and their interpretation differs accordingly. The relationship between Bayesian intervals and classical intervals is further explored in Nicolaou (1993) and Severini (1993). Example 5.7.4 (Two-parameter normal: estimating σ) Under the assumptions of the preceding example, credible regions for σ are based on the posterior distribution of σ given x, obtained by integrating the joint posteriordensity of  (ξ, σ) with respect to ξ. Using the fact that (ξ − xi )2 = n(ξ − x̄)2 + (xi − x̄)2 , it is 5.38) that given x, the conditional (posterior) distribution seen (Problem of (xi − x̄)2 /σ 2 is χ2 with n − 1 degrees of freedom. As in the case of the mean, this agrees with the sampling distribution of the same quantity when a is a (constant) parameter, given in Section 5.2. (The agreement in both cases of two distributions derived under such different assumptions is a consequence of the particular choice of the prior distribution and the fact that it is invariant in the sense of TPE2, Section 4.4.) A change of variables now gives the posterior density of σ and shows that π(σ | x) is of the form (c) of Example 5.7.2, so that the HPD region is of the form K1 (x) < σ < K2 (x) with 0 < K1 (x) < K2 (x) < ∞. Suppose that a credible region is required, not for σ, but for σ r for some r > 0. For consistency, this should then be given by [K1 (x)]r < σ r < [K2 (x)]r , but this 5.7. Bayesian Confidence Sets 175 is not the case, since the relative height of the density of a random variable at two points is not invariant under monotone transformations of the variable. In fact, in the present case, the HPD region for σ r will become one-sided for sufficiently large r although it is two-sided for r = 1 (Problem 5.38). Such inconsistencies do not occur if the HPD region is replaced by the equaltails interval (C1 (x), C2 (x)) for which P [Θ < C1 (x) | X = x] = P [Θ > C2 (x) | X = x] = (1 − γ)/2.7 More generally inconsistencies under transformations of Θ are avoided when the posterior distribution of Θ is summarized by a number of its percentiles corresponding to the standard confidence points mentioned in Section 3.5. Such a set is a compromise between providing the complete posterior distribution and providing a single interval corresponding to only two percentiles. Both the confidence and the Bayes approach present difficulties: the first, the problem of postdata interpretation; the second, the choice of a prior distribution and the interpretation of the posterior coverage probabilities if there is no clear basis for this choice. It is therefore not surprising that efforts have been made to find an approach without these drawbacks. The first such attempt, from which most later ones derive, is due to Fisher [1930; for his final account see Fisher (1973)]. To discuss Fisher’s concept of fiducial probability, consider once more the example at the beginning of the section, in which X is distributed as N (θ, 1). Since then X − θ is distributed as N (0, 1), so is θ − X, and hence P (θ − X ≤ y) = Φ(y) for all y. For fixed X = x, this is the formal statement that a random variable θ has distribution N (x, 1). Without assuming θ to be random, Fisher calls N (x, 1) the fiducial distribution of θ. Since this distribution is to embody the information about θ provided by the data, it should be unique, and Fisher imposes conditions which he hopes will ensure uniqueness. This leads to some technical difficulties, but more basic is the question of how to interpret fiducial probability. In a series of independent repetitions of the experiment with arbitrarily varying θi , the quantities θ1 − X1 , θ2 − X2 , . . . will constitute a sequence of independent standard normal variables. From this fact, Fisher attempts to derive the fiducial distribution N (x, 1) of θ as a frequency distribution with respect to an appropriate reference set. However, this argument is difficult to follow and unconvincing. For summaries of the fiducial literature and of later related developments by Dempster, Fraser, and others, see Buehler (1983), Edwards (1983), Seidenfeld (1992), Zabell (1992), Barnard (1995, 1996) and Fraser (1996). Fisher’s effort to define a suitable frame of reference led him to the important concept of relevant subsets, which will be discussed in Chapter 10. To appreciate the differences between the frequentist, Bayesian and Fisherian points of view, see Lehmann (1993), Robert (1994), Berger, Boukai and Wang (1997), Berger (2003) and Bayarri and Berger (2004). 7 They also do not occur when the posterior distribution of Θ is discrete. 176 5. Unbiasedness: Applications to Normal Distributions 5.8 Permutation Tests For the comparison of a treatment with a control situation in which no treatment is given, it was shown in Section 5.3 that the one-sided t-test is UMP unbiased for testing H : η = ξ against η − ξ = ∆ > 0 when the measurements X1 , . . . , Xm and Y1 , . . . , Yn are samples from normal populations N (ξ, σ 2 ) and N (η, σ 2 ). It will be shown in Section 11.3 that the level of this test is (asymptotically) robust against nonnormality – that is, that except for small m or n the level of the test is approximately equal to the nominal level α when the X’s and Y ’s are samples from any distributions with densities f (x) and f (y − ∆) with finite variance. If such an approximate level is not satisfactory, one may prefer to try to obtain an exact level-α unbiased test (valid for all f ) by replacing the original normal model with the nonparametric model for which the joint density of the variables is f (x1 ) . . . f (xm )f (y1 − ∆) . . . f (yn − ∆), f ∈ F, (5.45) where we shall take F to be the family of all probability densities that are continuous a.e. If there is much variation in the population being sampled, the sensitivity of the experiment can frequently be increased by dividing the population into more homogeneous subgroups, defined for example by some characteristic such as age or sex. A sample of size Ni (i = 1, . . . , c) is then taken from the ith subpopulation: mi to serve as controls, and the other ni = Ni − mi , to receive the treatment. If the observations in the ith subgroup of such a stratified sample are denoted by (Xi1 , . . . , Ximi ; Yi1 , . . . , Yini ) = (Zi1 , . . . , ZiNi ), the density of Z = (Z11 , . . . , ZcNc ) is p∆ (z) = c  [fi (xi1 ) . . . fi (ximi )fi (yi1 − ∆) . . . fi (yini − ∆)] . (5.46) i=1 Unbiasedness of a test φ for testing ∆ = 0 against ∆ > 0 implies that for all f1 , . . . , fc ,  φ(z)p0 (z) dz = α (dz = dz11 . . . dzcNc ). (5.47) Theorem 5.8.1 If F is the family of all probability densities f that are continuous a.e., then (5.47) holds for all f1 , . . . , fc ∈ F if and only if  1 φ(z  ) = α a.e., (5.48) N1 ! . . . Nc !  z ∈S(z) where S(z) is the set of points obtained from z by permuting for each i = 1, . . . , c the coordinates zij (j = 1, . . . , Ni ) within the ith subgroup in all N1 ! . . . Nc ! possible ways. Proof. To prove the result for the case c = 1, note that the set of order statistics T (Z) = (Z(1) , . . . , Z(N ) ) is a complete sufficient statistic for F (Example 4.3.4). A necessary and sufficient condition for (5.47) is therefore E[φ(Z) | T (z)] = α a.e. (5.49) 5.9. Most Powerful Permutation Tests 177 The set S(z) in the present case (c = 1) consists of the N points obtained from z through permutation of coordinates, so that S(z) = {z  : T (z  ) = T (z)}. It follows from Section 2.4 that the conditional distribution of Z given T (z) assigns probability 1/N ! to each of the N ! points of S(z). Thus (5.49) is equivalent to 1  φ(z  ) = α a.e., (5.50) N!  z ∈S(z) as was to be proved. The proof for general c is completely analogous and is left as an exercise (Problem 5.44.) The tests satisfying (5.48) are called permutation tests. An extension of this definition is given in Problem 5.54. 5.9 Most Powerful Permutation Tests For the problem of testing the hypothesis H : ∆ = 0 of no treatment effect on the basis of a stratified sample with density (5.46) it was shown in the preceding section that unbiasedness implies (5.48). We shall now determine the test which, subject to (5.48), maximizes the power against a fixed alternative (5.46) or more generally against an alternative with arbitrary fixed density h(z). The power of a test φ against an alternative h is   φ(z)h(z) dz = E[φ(Z) | t] dpT (t). Let t = T (z) = (z(1) , . . . , z(N ) ), so that S(z) = S(t). As was seen in Example 2.4.1 and Problem 2.6, the conditional expectation of φ(Z) given T (Z) = t is  φ(z)h(z) ψ(t) = z∈S(t)  h(z) . z∈S(t) To maximize the power of φ subject to (5.48) it is therefore necessary to maximize ψ(t) for each t subject to this condition. The problem thus reduces to the determination of a function φ which subject to  1 φ(z) = α, N1 ! . . . Nc ! z∈S(t) maximizes  z∈S(t) h(z)  . h(z  ) φ(z) z  ∈X(t) By the Neyman–Pearson fundamental lemma, this is achieved by rejecting H for those points z of S(t) for which the ratio h(z)N1 ! . . . Nc !  h(z  ) z  ∈S(t) 178 5. Unbiasedness: Applications to Normal Distributions is too large. Thus the most powerful test is given by the critical function ⎧ when h(z) > C[T (z)], ⎨ 1 γ when h(z) = C[T (z)], (5.51) φ(z) = ⎩ 0 when h(z) < C[T (z)]. To carry out the test, the N1 ! . . . Nc ! points of each set S(z) are ordered according to the values of the density h. The hypothesis is rejected for the k largest values and with probability γ for the (k + 1)st value, where k and γ are defined by k + γ = αN1 ! . . . Nc !. Consider now in particular the alternatives (5.46). The most powerful permutation test is seen to depend on ∆ and the fi , and is therefore not UMP. Of special interest is the class of normal alternatives with common variance: fi = N (ξi , σ 2 ). The most powerful test against these alternatives, which turns out to be independent of the ξi , σ 2 , and ∆, is appropriate when approximate normality is suspected but the assumption is not felt to be reliable. It may then be desirable to control the size of the test at level α regardless of the form of the densities fi and to have the test unbiased against all alternatives (5.46). However, among the class of tests satisfying these broad restrictions it is natural to make the selection so as to maximize the power against the type of alternative one expects to encounter, that is, against the normal alternatives. With the above choice of fi , (5.46) becomes −N √ 2πσ × h(z) = ⎛ ⎞⎤ Nj mi c    1 2 2 ⎝ (zij − ξi ) + (zij − ξi − ∆) ⎠⎦ . exp ⎣− 2 2σ i=1 j=1 j=m +1 ⎡ (5.52) i   i 2 2 Since the factor exp[− i N j=1 (zij − ξi ) /2σ ] is constant over S(t), the test  Ni (5.51) therefore rejects H when exp(∆ i j=mi +1 zij ) > C[T (z)] and hence when nj c   i=1 j=1 yij = Ni c   zij > C[T (z)]. (5.53) i=1 j=mi +1 Of the N1 ! . . . Nc ! values that the test statistic takes on over S(t), only     N1 Nc ... n1 nc are distinct, since the value of the statistic is the same for any two points z  and     z  for which (zi1 , . . . , zim ) and (zi1 , . . . , zim ) are permutations of each other for i i each i. It is therefore enough to compare these distinct values, and to reject H for the k largest ones and with probability γ  for the (k + 1)st, where     N1 Nc ... . k + γ  = α n1 nc 5.9. Most Powerful Permutation Tests 179 The test (5.53) is most powerful against the normal alternatives under consideration among all tests which are unbiased and of level α for testing H : ∆ = 0 in the original family (5.46) with f1 , . . . , fc ∈ F .8 To complete the proof of this statement it is still necessary to prove the test unbiased against the alternatives (5.46). We shall show more generally that it is unbiased against all alternatives for which Xij (j = 1, . . . , mi ), Yik (k = 1, . . . , ni ) are independently distributed with cumulative distribution functions Fi , Gi respectively such that Yik is stochastically larger than Xij , that is, such that Gi (z) ≤ Fi (z) for all z. This is a consequence of the following lemma. Lemma 5.9.1 X1 , . . . , Xm , Y1 , . . . , Yn be samples from continuous distributions F , G, and let φ(x1 , . . . , xm ; y1 , . . . , yn ) be a critical function such that (a) its expectation is α whenever G = F , and (b) yi ≤ yi for i = 1, . . . , n implies φ(x1 , . . . , xm ; y1 , . . . , yn ) ≤ φ(x1 , . . . , xm ; y1 , . . . , yn ). Then the expectation β = β(F, G) of φ is ≥ α for all pairs of distributions for which Y is stochastically larger than X; it is ≤ α if X is stochastically larger than Y . Proof. By Lemma 3.4.1, there exist functions f , g and independent random variables V1 , . . . , Vm+n such that the distributions of f (Vi ) and g(Vi ) are F and G respectively and that f (z) ≤ g(z) for all z. Then Eφ[f (V1 ), . . . , f (Vm ); f (Vm+1 ), . . . , f (Vm+n )] = α and Eφ[f (V1 ), . . . , f (Vm ); g(Vm+1 ), . . . , g(Vm+n )] = β. Since for all (v1 , . . . , vm+n ), φ[f (v1 ), . . . , f (vm ); f (vm+1 ), . . . , f (vm+n )] ≤ φ[f (v1 ), . . . , f (vm ); g(vm+1 ), . . . , g(vm+n )], the same inequality holds for the expectations of both sides, and hence α ≤ β. The proof for the case that X is stochastically larger than Y is completely analogous. The lemma also generalizes to the case of c vectors (Xi1 , . . . , Ximi ; Yi1 , . . . , Yini ) with distributions (Fi , Gi ). If the expectation of a function φ is α when Fi = Gi and φ is nondecreasing in each yij when all other variables are held fixed, then it follows as before that the expectation of φ is ≥ α when the random variables with distribution Gi are stochastically larger than those with distribution Fi . In applying the lemma to the permutation test (5.53) it is enough to consider the case c = 1, the argument in the more general case being completely analogous. Since the rejection probability of the test (5.53) is α whenever F = G, it is only necessary satisfies (b). Now φ = 1 m+n to show that the critical function φ of the test m+n if i=m+1 zi exceeds sufficiently many of the sums i=m+1 zji , and hence if 8 For a closely related result. see Odén and Wedel (1975). 180 5. Unbiasedness: Applications to Normal Distributions sufficiently many of the differences m+n  m+n  zi − i=m+1 zji i=m+1 are positive. For a particular permutation (j1 , . . . , jm+n ) m+n  zi − i=m+1 m+n  zji = i=m+1 p  zsi − i=1 p  zri , i=1 where r1 < · · · < rp denote those of the integers jm+1 , . . . , jm+n that are ≤ m, and s1 < · · · < sp those integers m + 1, . . . , m + n not included in the set  of the   (jm+1 , . . . , jm+n ). If zsi − zri is positive   and  yi ≤ yi , that is, zi ≤ zi for i = m + 1, . . . , m + n, then the difference zsi − zri is also positive and hence φ satisfies (b). The same argument also shows that the rejection probability of the test is ≤ α when the density of the variables is given by (5.46) with ∆ ≤ 0. The test is therefore equally appropriate if the hypothesis ∆ = 0 is replaced by ∆ ≤ 0. Except for small values of the sample sizes Ni , the amount of computation required to carry out the permutation test (5.53) is large. Computational methods are discussed by Green (1977), John and Robinson (1983b), Diaconis and Holmes (1994) and Chapter 13 of Good (1994), who has an extensive bibliography. One can relate the permutation test to the corresponding normal theory t-test as follows. On multiplying both sides of the inequality  yj > C[T (z)]   by (1/m) + (1/n) and subtracting ( x1 , + y j )/m, the rejection region for n 2 c = 1 becomes ȳ − x̄ > C[T (z)] or W = (ȳ − x̄)/ i=1 (zi − z̄) > C[T (z)], since the denominator of W is constant over S(z) and hence depends only on T (z). As was seen at the end of Section 5.3, this is equivalent to 7, 1 (ȳ − x̄) + m 1 n 1# > C[T (z)]. $   (xi − x̄)2 + (yj − ȳ)2 /(m + n − 2) (5.54) The rejection region therefore has the form of a t-test in which the constant cutoff point C0 of (5.27) has been replaced by a random one. It turns out that when the hypothesis is true, so that the Z  s are identically and independently distributed, and m/n is bounded away from zero and infinity as m and n tend to infinity, the difference between the random cutoff point C[T (Z)] and C0 is small in an appropriate asymptotic sense, and so the permutation test and the t-test given by (5.27) − (5.29) behave similarly in large samples. Such results will be developed in Section 15.2. the permutation test can be approximated for large samples by the standard t-test. Exactly analogous results hold for c > 1; the appropriate generalization of the two-sample t-test is provided in Problem 7.9. 5.10. Randomization As A Basis For Inference 181 5.10 Randomization As A Basis For Inference The problem of testing for the effect of a treatment was considered in Section 5.3 under the assumption that the treatment and control measurements X1 , . . . , Xm , and Y1 , . . . , Yn constitute samples from normal distributions, and in Sections 5.8 and 5.9 without relying on the assumption of normality. We shall now consider in somewhat more detail the structure of the experiment from which the data are obtained, resuming for the moment the assumption that the distributions involved are normal. Suppose that the experimental material consists of m + n patients, plants, pieces of material, or the like, drawn at random from the population to which the treatment could be applied. The treatment is given to n of these while the other m serve as controls. The characteristic that is to be influenced by the treatment is then measured in each case, leading to observations X1 , . . . , Xm ; Y1 , . . . , Yn . To be specific, suppose that the treatment is carried out by injecting a drug and that m + n ampules are assigned to the m + n patients. The ith measurement can be considered as the sum of two components. One, say Ui , is associated with the ith patient; the other, Vi , with the ith ampule and the circumstances under which it is administered and under which the measurements are taken. The variables Ui and Vi are assumed to be independently distributed, the V ’s with normal distribution N (η, σ 2 ) or N (ξ, σ 2 ) as the ampule contains the drug or is one of those used for control. If in addition the U ’s are assumed to constitute a random sample from N (µ, σ12 ), it follows that the X’s and Y ’s are independently normally distributed with common variance σ 2 + σ12 and means E(X) = µ + ξ, E(Y ) = µ + η. Except for a change of notation their joint distribution is then given by (5.26), and the hypothesis η = ξ can be tested by the standard t-test Unfortunately, under actual experimental conditions, it is frequently not possible to ensure that the patients or other experimental units constitute a random sample from the population of such units. They may be patients in a certain hospital at a given time, or volunteers for an experiment, and may constitute a haphazard rather than a random sample. In this case the U ’s would have to be considered as unknown constants, since they are not obtained by any definite sampling procedure. This assumption is appropriate also in a different context. Suppose that the experimental units are all the machines in a shop or fields on a farm. If the experiment is performed only to determine the best method for this particular shop or farm, these experimental units are the only relevant ones; that is, a replication of the experiment would consist in comparing the two treatments again for the same machines or fields rather than for a new batch drawn at random from a large population. In this case the units themselves, and therefore the u’s, are constant. Under the above assumptions the joint density of the m + n measurements is ! m " n   1 1 √ exp − 2 (xi − ui − ξ)2 + (yj − um+j − η)2 . 2σ ( 2πσ)m+n i=1 j=1 Since the u’s are completely arbitrary, it is clearly impossible to distinguish between H : η = ξ and the alternatives K : η > ξ. In fact, every distribution of K 182 5. Unbiasedness: Applications to Normal Distributions also belongs to H and vice versa, and the most powerful level-α test for testing H against any simple alternative specifying ξ, η, σ, and the u’s rejects H with probability α regardless of the observations. Data which could serve as a basis for testing whether or not the treatment has an effect can be obtained through the fundamental device of randomization. Suppose that the N = m + n patients are assigned to the N ampules at random, that is, in such a way that each of the N ! possible assignments has probability 1/N ! of being chosen. Then for a given assignment the N measurements are independently normally distributed with variance σ 2 and means ξ + uji (i = 1, . . . , m) and η + uji (i = m + 1, . . . , m + n). The overall joint density of the variables (Z1 , . . . , ZN ) = (X1 , . . . , Xm ; Y1 , . . . , Yn ) is therefore 1 N!  1 √ (5.55) ( 2πσ)N (j1 ,...,jN ) m " ! n   1 2 2 (xi − uji − ξ) + (yi − ujm+i − η) × exp − 2 2σ i=1 i=1 where the outer summation extends over all N ! permutations (j1 , . . . , jN ) of (1, . . . , N ). Under the hypothesis η = ξ this density can be written as ! " N  1 1 1  √ exp − 2 (zi − ζji )2 , (5.56) N! 2σ i=1 ( 2πσ)N (j1 ,...,jN ) where ζji = uji + ξ = uji + η. Without randomization a set of y’s which is large relative to the x-values could be explained entirely in terms of the unit effects ui . However, if these are assigned to the y’s at random, they will on the average balance those assigned to the x’s. As a consequence, a marked superiority of the second sample becomes very unlikely under the hypothesis, and must therefore be attributed to the effectiveness of the treatment. The method of assigning the treatments to the experimental units completely at random permits the construction of a level-α test of the hypothesis η = ξ, whose power exceeds α against all alternatives η − ξ > 0. The actual power of such a test will however depend not only on the alternative value of η − ξ, which measures the effect of the treatment, but also on the unit effects ui . In particular, if there is excessive variation among the u’s this will swamp the treatment effect (much in the same way as an increase in the variance σ 2 would), and the test will accordingly have little power to detect any given alternative η − ξ. In such cases the sensitivity of the experiment can be increased by an approach exactly analogous to the method of stratified sampling discussed in Section 5.8. In the present case this means replacing the process of complete randomization described above by a more restricted randomization procedure. The experimental material is divided into subgroups, which are more homogeneous than the material as a whole, so that within each group the differences among the u’s are small. In animal experiments, for example, this can frequently be achieved by a division into litters. Randomization is then applied only within each group. If the ith group 5.10. Randomization As A Basis For Inference 183 and contains Ni units, ni of these are selected at random  to receive  the treatment,  the remaining mi = Ni − ni serve as controls ( Ni = N, mi = m, ni = n). An example of this approach is the method of matched pairs. Here the experimental units are divided into pairs, which are as like each other as possible with respect to all relevant properties, so that within each pair the difference of the u’s will be as small as possible. Suppose that the material consists of n such pairs, and denote the associated unit effects (the U ’s of the previous discussion) by U1 , U1 ; . . . ; Un , Un . Let the first and second member of each pair receive the treatment or serve as control respectively, and let the observations for the ith pair be Xi and Yi . If the matching is completely successful, as may be the case, for example, when the same patient is used twice in the investigation of a sleeping drug, or when identical twins are used, then Ui = Ui for all i, and the density of the X’s and Y ’s is  $  1 1 # 2 2 √ exp − 2 (yi − η − ui ) (xi − ξ − ui ) + . (5.57) 2σ ( 2πσ)2 The UMP unbiased test for testing H : η = ξ against η > ξ is then given in terms of the differences Wi = Yi − Xi by the rejection region √ 31 nw̄ 1  (wi − w̄)2 > C. n−1 (5.58) (See Problem 5.48.) However, usually one is not willing to trust the assumption ui = ui even after matching, and it again becomes necessary to randomize. Since as a result of the matching the variability of the u’s within each pair is presumably considerably smaller than the overall variation, randomization is carried out only within each pair. For each pair, one of the units is selected with probability 12 to receive the treatment, while the other serves as control. The density of the X’s and Y ’s is then   n   1 1 1  2  2 √ − ξ − u ) + (y − η − u ) (x exp − i i i i 2n ( 2πσ)2n i=1 2σ 2    1  + exp − 2 (xi − ξ − ui )2 + (yi − η − ui )2 . 2σ (5.59) Under the hypothesis η = ξ, and writing zi1 = xi , zi2 = yi , ζi1 = ξ + ui , ζi2 = η + ui (i = 1, . . . , n), this becomes ! " 2 n 1 1  1   2 √ exp − 2 (zij − ζij ) . 2n 2σ i=1 j=1 ( 2πσ)2n (5.60)   Here the outer summation extends over the 2n points ζ  = (ζ11 , . . . , ζn2 ) for which   (ζi1 , ζi2 ) is either (ζi1 , ζi2 ) or (ζi2 , ζi1 ) 184 5. Unbiasedness: Applications to Normal Distributions 5.11 Permutation Tests and Randomization It was shown in the preceding section that randomization provides a basis for testing the hypothesis η = ξ of no treatment effect, without any assumptions concerning the experimental units. In the present section, a specific test will be derived for this problem. When the experimental units are treated as constants, the probability density of the observations is given by (5.55) in the case of complete randomization and by (5.59) in the case of matched pairs. More generally, let the experimental material be divided into c subgroups, let the randomization be applied within each subgroup, and let the observations in the ith subgroup be (Zi1 , . . . , ZiNi ) = (Xi1 , . . . , Ximi ; Yi1 , . . . , Yini ). For any point u = (u11 , . . . , ucNc ), let S(u) denote as before the set of N1 ! . . . Nc ! points obtained from u by permuting the coordinates within each subgroup in all N1 ! . . . Nc ! possible ways. Then the joint density of the Z’s given u is  1 1 √ (5.61) N N1 ! . . . Nc !  ( 2πσ) u ∈S(u) m " ! Ni c i  1    2  2 , × exp − 2 (zij − ξ − uij ) + (zij − η − uij ) 2σ i=1 j=1 j=m +1 i and under the hypothesis of no treatment effect ! " c Ni  1 1 1   2 √ pσ,ζ (z) = exp − 2 (zij − ζij ) . (5.62) N1 ! . . . Nc !  2σ i=1 j=1 ( 2πσ)N ζ ∈S(ζ) It may happen that the coordinates of u or ζ are not distinct. If then some of the points of S(u) or S(ζ) also coincide, each should be counted with its proper multiplicity. More precisely, if the N1 ! . . . Nc ! relevant permutations of N1 + . . . + Nc coordinates are denoted by gk , k = 1, . . . , N1 ! . . . Nc !, then S(ζ) can be taken to be the ordered set of points gk ζ, k = 1, . . . , N1 ! . . . Nc !, and (5.62), for example, becomes  N1 !...Nc !  1 1 1 √ Pσ,ζ (z) = exp − 2 |z − gk ζ|2 N1 ! . . . Nc ! 2σ ( 2πσ)N 2 where |u| stands for c i=1 N k=1 j=1 u2ij . Theorem 5.11.1 A necessary and sufficient condition for a critical function φ to satisfy  φ(z)pσ,ζ (z) dz ≤ α (dz = dz11 . . . dzcNc ) (5.63) for all σ > 0 and all vectors ζ is that  1 N1 ! . . . Nc! φ(z  ) ≤ α z  ∈S(z) The proof will be based on the following lemma. a.e. (5.64) 5.11. Permutation Tests and Randomization 185 Lemma 5.11.1 Let A be a set in N -space with positive Lebesgue measure µ(A). Then for any  > 0 there exist real numbers σ > 0 and ξ1 , . . . , ξN , such that P {(X1 , . . . , XN ) ∈ A} ≥ 1 − , where the X’s are independently normally distributed with means E(Xi ) = ξi and 2 variance σX = σ2 . i Proof. Suppose without loss of generality that µ(A) < ∞. Given any η > 0, there exists a square Q such that µ(Q ∩ Ac ) ≤ ηµ(Q). This follows from the fact that almost every point of A is a density point,9 or from the more elementary fact that a measurable set can be approximated in measure by unions of disjoint squares. Let a be such that   a  1 t2  1/N √ , − dt = 1 − 2 2 2π −a and let η=  2 √ 2π 2a N . If (ξ1 , . . . , ξN ) is the center of Q, and if σ = b/a = (1/2a)[µ(Q)]1/N , where 2b is the length of the side of Q, then    1 1  √ (xi − ξi )2 dx1 . . . dxN exp − 2 2σ ( 2πσ)N Ac ∩Qc    1 1  (xi − ξi )2 dx1 . . . dxN ≤ √ exp − 2 2σ ( 2πσ)N Qc   N  a 1 t2  =1− √ exp − = . dt 2 2 2π −a On the other hand,   1  exp − 2 (xi − ξi )2 dx1 . . . dxN 2σ Ac ∩Q 1  ≤ √ µ(Ac ∩ Q) < , 2 ( 2πσ)N 1 √ ( 2πσ)N  and by adding the two inequalities one obtains the desired result. Proof.[Proof of the theorem] Let φ be any critical function, and let  1 φ(z  ). ψ(z) = N1 ! . . . Nc !  z ∈S(z) If (5.64) does not hold, there exists η > 0 such that φ(z) > α + η on a set A of positive measure. By the Lemma there exists σ > 0 and ζ = (ζ11 , . . . , ζcNc ) 9 See, for example, Billingsley (1995), p.417. 186 5. Unbiasedness: Applications to Normal Distributions such that P {Z ∈ A} > 1 − η when Z11 , . . . , ZcNc are independently normally distributed with common variance σ 2 and means E(Zij ) = ζij . It follows that   φ(z)pσ,ζ (z) dz = ψ(z)pσ,ζ (z) dz (5.65)    1 1  (zij − ζij )2 dz ψ(z) √ exp − 2 ≥ N 2σ ( 2πσ) A > (α + η)(1 − η), which is > α, since α+η < 1. This proves that (5.63) implies (5.64). The converse follows from the first equality in (5.65). Corollary 5.11.1 Let H be the class of densities {pσ,ζ (z) : σ > 0, −∞ < ζij < ∞}. A complete family of tests for H at level of significance α is the class of tests C satisfying  1 φ(z  ) = α a.e. (5.66) N1 ! . . . Nc !  z ∈S(z) Proof. The corollary states that for any given level-α test φ0 there exists an element φ of C which is uniformly at least as powerful as φ0 . By the preceding theorem the average value of φ0 over each set S(z) is ≤ α. On the sets for which this inequality is strict, one can increase φ0 to obtain a critical function φ satisfying (5.66), and such that φ0 (z) ≤ φ(z) for all z. Since against all alternatives the power of φ is at least that of φ0 , this establishes the result. An explicit construction of φ, which shows that it can be chosen to be measurable, is given in Problem 5.51. This corollary shows that the normal randomization model (5.61) leads exactly to the class of tests that was previously found to be relevant when the U ’s constituted a sample but the assumption of normality was not imposed. It therefore follows from Section 5.9 that the most powerful level-α test for testing (5.62) against a simple alternative (5.61) is given by (5.51) with h(z) equal to the probability density (5.61). If η − ξ = ∆, the rejection region of this test reduces to ! " N Ni c i   1     exp zij uij + ∆ (zij − uij ) > C[T (z)], (5.67) σ 2 i=1 j=1  j=m +1 u ∈S(u) i   2 zij are constant on S(z) and therefore functions since both zij and only of T (z). It is seen that this test depends on ∆ and the unit effects uij , so that a UMP test does not exist. Among the alternatives (5.61) a subclass occupies a central position and is of particular interest. This is the class of alternatives specified by the assumption that the unit effects ui constitute a sample from a normal distribution. Although this assumption cannot be expected to hold exactly – in fact, it was just as a safeguard against the possibility of its breakdown that randomization was introduced – it is in many cases reasonable to suppose that it holds at least 5.12. Randomization Model and Confidence Intervals 187 approximately. The resulting subclass of alternatives is given by the probability densities 1 √ (5.68) ( 2πσ)N  " ! mi Ni c  1   . (zij − ui − ξ)2 + (zij − ui − η)2 × exp − 2 2σ i=1 j=1 j=m +1 i These alternatives are suggestive also from a slightly different point of view. The procedure of assigning the experimental units to the treatments at random within each subgroup was seen to be appropriate when the variation of the u’s is small within these groups and is employed when this is believed to be the case. This suggests, at least as an approximation, the assumption of constant uij = ui , which is the limiting case of a normal distribution as the variance tends to zero, and for which the density is also given by (5.68). Since the alternatives (5.68) are the same as the alternatives (5.52) of Section 5.9 with ui − ξ = ξi , ui − η = ξi − ∆, the permutation test (5.53) is seen to be most powerful for testing the hypothesis η = ξ in the normal randomization model (5.61) against the alternatives (5.68) with η − ξ > 0. The test retains this property in the still more general setting in which neither normality nor the sample property of the U ’s is assumed to hold. Let the joint density of the variables be !m  Ni c   i    fi (zij − uij − ξ) fi (zij − uij − η) , (5.69) u ∈S(u) i=1 j=1 j=mi +1 with fi continuous a.e. but otherwise unspecified.10 Under the hypothesis H : η = ξ, this density is symmetric in the variables (zi1 , . . . , ziNi ) of the ith subgroup for each i, so that any permutation test (5.48) has rejection probability α for all distributions of H. By Corollary 5.11.1, these permutation tests therefore constitute a complete class, and the result follows. 5.12 Randomization Model and Confidence Intervals In the preceding section, the unit responses ui were unknown constants (parameters) which were observed with error, the latter represented by the random terms Vi . A limiting case assumes that the variation of the V ’s is so small compared with that of the u’s that these error variables can be taken to be constant, i.e. that Vi = v. The constant v can then be absorbed into the u’s, and can therefore be assumed to be zero. This leads to the following two-sample randomization model : N subjects would give “true” responses u1 , . . . , uN if used as controls. The subjects are assigned at random, n to treatment and m to control. If the responses 10 Actually, all that is needed is that f , . . . , f ∈ F , where F is any family containing c 1 all normal distributions. 188 5. Unbiasedness: Applications to Normal Distributions are denoted by X1 , . . . , Xm and Y1 , . . . , Yn as before, then under the hypothesis H of no treatment effect, the X’s and Y ’s are a random permutation of the u’s. Under this model, in which the random assignment of the subjects to treatment and control constitutes the only random element, the probability of the rejection region (5.54) is the same as under the more elaborate models of the preceding sections. The corresponding limiting model under the alternatives assumes that the treatment has the effect of adding a constant amount ∆ to the unit response, so that the X’s and Y ’s are given by (ui1 , . . . ; uim ; uim+1 + ∆, . . . , uim+n + ∆) for some permutation (i1 , . . . , iN ) of (1, . . . , N ). These models generalize in the obvious way to stratified samples. In particular, for paired comparisons it is assumed under H that the unit effects (ui , ui ) are constants, of which one is assigned at random to treatment and the other to control. Thus the pair (Xi , Yi ) is equal to (ui , ui ) or (ui , ui ) with probability 1 each, and the assignments in the n pairs are independent; the sample space 2 consists of 2n points each of which has probability ( 12 )n . Under the alternative, it is assumed as before that ∆ is added to each treated subject, so that P (Xi = ui , Yi = ui + ∆) = P (Xi = ui , Yi = ui + ∆) = 12 . The distribution generated for the observations by such a randomization model is exactly the conditional distribution given T (z) of the preceding sections. In the two-sample case, for example, this common distribution is specified by the fact that all permutations of (X1 , . . . , Xm ; Y1 − ∆, . . . , Yn − ∆) are equally likely. As a consequence, the power of the test (5.54) in the randomization model is also the conditional power in the two-sample model (5.45). As was pointed out in Section 4.4, the conditional power β(∆ | T (z)) can be interpreted as an unbiased estimate of the unconditional power βF (∆) in the two-sample model. The advantage of β(∆ | T (z)) is that it depends only on ∆, not on the unknown F . Approximations to β(∆ | T (z)) are discussed by J. Robinson (1973), G. Robinson (1982), John and Robinson (1983a), and Gabriel and Hsu (1983). The tests (5.53), which apply to all three models – the sampling model (5.46), the randomization model, and the intermediate model (5.69) – can be inverted in the usual way to produce confidence sets for ∆. We shall now determine these sets explicitly for the paired comparisons and the two-sample case. The derivations will be carried out in the randomization model. However, they apply equally in the other two models, since the tests, and therefore the associated confidence sets, are identical for the three models. Consider first the case of paired observations (xi , yi ), i = 1, . . . , n. The onesided test rejects H : ∆ = 0 in favor of ∆ > 0 when n i=1 yi is among the K largest of the 2n sums obtained by replacing yi by xi for all, some, or none of the values i = 1, . . . , n. (It is assumed here for the sake of simplicity that α = K/2n , so that the test requires no randomization to achieve the exact level α.) Let di = yi − xi = 2yi  − ti , where ti = xi + yi is fixed. Then the test  is equivalent to rejecting when di is one of the K largest of the 2n values ±di , since now an interchange of yi with xi is equivalent to replacing di by −di . Consider  testing H : ∆ = ∆0 against ∆ > ∆0 . The test then accepts when (d i − ∆0 )  is one of the l = 2n − K smallest of the 2n sums ±(di − ∆0 ), since it is now yi − ∆0 that is being interchanged with xi . We shall next invert this statement, replacing ∆0 by ∆, and see that it is equivalent to a lower confidence bound for ∆. 5.12. Randomization Model and Confidence Intervals In the inequality   [±(di − ∆)] , (di − ∆) < 189 (5.70) suppose that on the right side the minus sign attaches to the (di − ∆) with i = i1 , . . . , ir and the plus sign to the remaining terms. Then (5.70) is equivalent to di1 + · · · + dir di1 + · · · + dir − r∆ < 0, or < ∆. r   ±(di − ∆) if and only if at Thus, (di − ∆) is among the l smallest of the least 2n − l of the M = 2n − 1 averages (di1 + · · · + dir )/r are < ∆, i.e. if and only if δ(K) < ∆, where δ(1) < · · · < δ(M ) is the ordered set of averages (di1 + · · · + dir )/r, r = 1, . . . , M . This establishes δ(K) as a lower confidence bound for ∆ at confidence level γ = K/2n . [Among all confidence sets that are unbiased in the model (5.46) with mi = ni = 1 and c = n, these bounds minimize the probability of falling below any value ∆ < ∆ for the normal model (5.52).] By putting successively K = 1, 2, . . . , 2n , it is seen that the M + 1 intervals (−∞, δ(1) ), (δ(1) , δ(2) ), . . . , (δ(M −1) , δ(M ) ), (δM , ∞) (5.71) n each have probability 1/(M + 1) = 1/2 of containing the unknown ∆. The twosided confidence intervals (δ(K) , δ(2n −K) ) with γ = (2n−1 − K)/2n−1 correspond to the two-sided version of the test (5.53) with error probability (1 − γ)/2 in each tail. A suitable subset of the points δ(1) , . . . , δ(M ) constitutes a set of confidence points in the sense of Section 3.5. The inversion procedure for the two-group case is quite analogous. Let (x1 , . . . , xm , y1 , . . . , yn ) denote the m control and n treatment observations, and suppose without loss of generality that m ≤ n. Then the hypothesis ∆ =∆0 is  m+n accepted against ∆ > ∆0 if n (y j=1 j − ∆0 ) is among the l smallest of the n sums obtained by replacing a subset of the (yj − ∆0 )’s with x’s. The inequality  (yj − ∆0 ) < (xi1 + · · · + xir ) + [yj1 + · · · + yjn−r − (n − r)∆], with (i1 , . . . , ir , j1 , . . . , jn−r ) a permutation of (1, . . . , n), is equivalent to yi1 + · · · + yir − r∆0 < xi1 + · · · + xir , or ȳ i1 ,...,ir − x̄i1 ,...,ir < ∆0 . (5.72) Note that the number of such averages with r ≥ 1 (i.e. omitting the empty set of subscripts) is equal to      m  m n m+n = −1=M K K n K=1 (Problem 5.57). Thus, H : ∆ = ∆0 is accepted against ∆ > ∆0 at level α = 1 − l/(M + 1) if and only if at least K of the M differences (5.72) are less than ∆0 , and hence if and only if δ(K) < ∆0 , where δ(1) < · · · < δ(M ) denote the ordered set of differences (5.72). This establishes δ(K) as a lower confidence bound for ∆ with confidence coefficient γ = 1 − α. As in the paired comparisons case, it is seen that the intervals (5.71) each have probability 1/(M + 1) of containing ∆. Thus, two-sided confidence intervals and standard confidence points can be derived as before. For the generalization to stratified samples, see Problem 5.58. 190 5. Unbiasedness: Applications to Normal Distributions Algorithms for computing the order statistics δ(1) , . . . , δ(M ) in the pairedcomparison and two-sample cases are discussed by Tritchler (1984); also see Garthwaite (1996). If M is too large for the computations to be practicable, reduced analyses based on either a fixed or random subset of the set of all M + 1 permutations are discussed, for example, by Gabriel and Hall (1983) and Vadiveloo (1983). [See also Problem 5.60(i).] Different such methods are compared by Forsythe and Hartigan (1970). For some generalizations, and relations to other subsampling plans, see Efron (1982, Chapter 9). 5.13 Testing for Independence in a Bivariate Normal Distribution So far, the methods of the present chapter have been illustrated mainly by the two-sample problem. As a further example, we shall now apply two of the formulations that have been discussed, the normal model of Section 5.3 and the nonparametric one of Section 5.8, to the hypothesis of independence in a bivariate distribution. The probability density of a sample (X1 , Y1 ), . . . , (Xn , Yn ) from a bivariate normal distribution is  1 1 1  exp − (5.73) (xi − ξ)2 2 2(1 − ρ ) σ 2 (2πστ 1 − ρ2 )n  1  2ρ  (xi − ξ)(yi − η) + 2 (yi − η)2 − . στ τ Here (ξ, σ 2 ) and (η, τ 2 ) are the mean and variance of X and Y respectively, and ρ is the correlation coefficient between X and Y . The hypotheses ρ ≤ ρ0 and ρ = ρ0 for arbitrary ρ0 cannot be treated by the methods of the present chapter, and will be taken up in Chapter 6. For the present, we shall consider only the hypothesis ρ = 0 that X and Y are independent, and the corresponding one-sided hypothesis ρ ≤ 0. The family of densities (5.73) is of the exponential form (1) with   2  2   U= Xi Yi , T1 = Xi , T2 = Yi , T3 = Xi , T4 = Yi and θ= ρ , στ (1−ρ2 ) ϑ3 = 1 1−ρ2  ϑ1 = ξ σ2 − ηρ στ −1 , 2σ 2 (1−ρ2 ) , ϑ4 = ϑ2 = η 1 1−ρ2 τ2 −1 , 2τ 2 (1−ρ2 ) − ξρ στ , The hypothesis H : ρ ≤ 0 is equivalent to θ < 0. Since the sample correlation coefficient  (Xi − X̄)(Yi − Ȳ ) R=   (Xi − X̄)2 (Yi − Ȳ )2 is unchanged when the Xi and Yi are replaced by (Xi − ξ)/σ and (Yi − η)/τ , the distribution of R does not depend on ξ, η, σ, or τ , but only on ρ. For θ = 0 it therefore does not depend on ϑ1 , . . . , ϑ4 , and hence by Theorem 5.1.2, R is 5.13. Testing for Independence in a Bivariate Normal Distribution 191 independent of (T1 , . . . , T4 ) when θ = 0. It follows from Theorem 5.1.1 that the UMP unbiased test of H rejects when R ≥ C0 , (5.74) or equivalently when R > K0 . (1 − R2 )/(n − 2) (5.75) The statistic R is linear in U , and its distribution for ρ = 0 is symmetric about 0. The UMP unbiased test of the hypothesis ρ = 0 against the alternative ρ = 0 therefore rejects when |R| > K1 . (1 − R2 )/(n − 2) (5.76) √ √ Since n − 2R/ 1 − R2 has the t-distribution with n − 2 degrees of freedom when ρ = 0 (Problem 5.64), the constants K0 and K1 in the above tests are given by  ∞  ∞ α tn−2 (y) dy = α and tn−2 (y) dy = (5.77) 2 K0 K1 Since the distribution of R depends only on the correlation coefficient ρ, the same is true of the power of these tests. Some large sample properties of the above test will be examined in Problem (11.64). In particular, if (Xi , Yi ) is not bivariate normal, the level of the above test is approximately α in large samples under the hypothesis H1 that Xi and Yi are independent, but not necessarily under the hypothesis H2 that the correlation between Xi and Yi is 0. For the nonparametric model H1 , one can obtain an exact level-α unbiased test of independence in analogy to the permutation test of Section 5.8. For any bivariate distribution of (X, Y ), let Yx denote a random variable whose distribution is the conditional distribution of Y given x. We shall say that there is positive regression dependence between X and Y if for any x < x the variable Yx is stochastically larger than Yx . Generally speaking, larger values of Y will then correspond to larger values of X; this is the intuitive meaning of positive dependence. An example is furnished by any normal bivariate distribution with ρ > 0. (See Problem 5.68.) Regression dependence is a stronger requirement than positive quadrant dependence, which was defined in Problem 4.28. However, both reflect the intuitive meaning that large (small) values of Y will tend to correspond to large (small) values of X. As alternatives to H1 consider positive regression dependence in a general bivariate distribution possessing a density. To see that unbiasedness implies similarity, let F1 , F2 be any two univariate distributions with densities f1 , f2 and consider the one-parameter family of distribution functions F1 (x)F2 (y){1 + ∆[1 − F1 (x)][1 − F2 (y)]}, 0 ≤ ∆ ≤ 1. (5.78) This is positively regression dependent (Problem 5.69), and by letting ∆ → 0 one sees that unbiasedness of φ against these distributions implies that the rejection probability is α when X and Y are independent, and hence that  φ(x1 , . . . , xn ; y1 , . . . , yn )f1 (x1 ) · · · f1 (xn )f2 (y1 ) · · · f2 (yn ) dx dy = α 192 5. Unbiasedness: Applications to Normal Distributions for all probability densities f1 and f2 . By Theorem 5.8.1 this in turn implies 1  φ(xi1 , . . . , xin ; yj1 , . . . , yjn ) = α. (n!)2 Here the summation extends over the (n!)2 points of the set S(x, y), which is obtained from a fixed point (x, y) with x = (x1 , . . . , xn ), y = (y1 , . . . , yn ) by permuting the x-coordinates and the y-coordinates, each among themselves in all possible ways. Among all tests satisfying this condition, the most powerful one against the  normal alternatives (5.73) with ρ > 0 rejects for k largest the  values  of (5.73) in each set S(x, y), where k /(n!)2 = α. Since x2i , yi2 , xi , yi ,  are all constant on S(x, y), the test equivalently rejects for the k largest values of xi yi in each S(x, y).  Of the (n!)2 values that the statistic Xi Yi takes on over S(x, y), only n! are distinct, since the statistic remains unchanged if the X’s and Y ’s are subjected to the same permutation. A simpler form of the test  is therefore obtained, for example by rejecting H1 for the k largest values of x(i) yji , of each set S(x, y), where x(i) < · · · < x(n) and k/n! = α. The test can be shown to be unbiased against all alternatives with positive regression dependence. (See Problem 6.62.) In order to obtain a comparison of the permutation test with the standard normal test based on the sample correlation coefficient R, let T (X, Y ) denote the set of ordered X’s and Y ’s T (X, Y ) = (X(1) , . . . , X(n) ; Y(1) , . . . , Y(n) ). The rejection region of the permutation test can then be written as  Xi Yi > C[T (X, Y )]. or equivalently as R > K[T (X, Y )]. It again turns out that the difference between K[T (X, Y )] and the cutoff point C0 of the corresponding normal test (5.74) tends to zero in an appropriate sense. Such results are developed in Section 15.2; also see Problem 15.13. For large n, the standard normal test (5.74) therefore serves as an approximation for the permutation test. 5.14 Problems Section 5.2 Problem 5.1 Let X1 , . . . , Xn be a sample from N (ξ, σ 2 ). The power of Student’s t-test is an increasing function of ξ/σ in the one-sided case H : ξ ≤ 0, K : ξ > 0, and of |ξ|/σ in the two-sided case H : ξ = 0, K : ξ = 0. [If 1 1  S= (Xi − X̄)2 , n−1 the power in the two-sided case is given by  √ √ √  n(X̄ − ξ) nξ nξ CS CS 1−P − − ≤ − ≤ σ σ σ σ σ 5.14. Problems 193 and the result follows from the fact that it holds conditionally for each fixed value of S/σ.] Problem 5.2 In the situation of the previous problem there exists no test for testing H : ξ = 0 at level α, which for all σ has power ≥ β > α against the alternatives (ξ, σ) with ξ = ξ1 > 0. [Let β(ξ1 , σ) be the power of any level α test of H, and let β(σ) denote the power of the most powerful test for testing ξ = 0 against ξ = ξ1 when σ is known. Then inf σ β(ξ1 , σ) ≤ inf σ β(σ) = α.] Problem 5.3 (i) Let Z and V be independently distributed as N (δ, 1) and χ2 with f degrees of freedom respectively. Then the ratio Z ÷ V /f has the noncentral t-distribution with f degrees of freedom and noncentrality parameter δ, the probability density of which is 11  ∞ 1 1 pδ (t) = y 2 (f −1) (5.79) √ 1 (f −1) 1 22 Γ( 2 f ) πf 0 !  2 " 1 y 1 1 × exp − y exp − − δ dy dy t 2 2 f or equivalently pδ (t) =  1 f δ2 exp − √ 1 2 f + t2 2 2 (f −1) Γ( 12 f ) πf !   1 (f +1)  ∞ 2 1 f f υ exp − υ− × f + t2 2 0 1 2 " δt dv. f + t2 √ √ Another form is obtained by making the substitution w = t y/ f in (5.79). √ (ii) If X1 , . . . , Xn are independently distributed as N (ξ, σ 2 ), then nX̄  (X1 − X̄)2 /(n − 1) has the noncentral t-distribution with n − 1 de÷ √ grees of freedom and noncentrality parameter δ = nξ/σ. In the case δ = 0, show that t-distribution with n − 1 degrees of freedom is given by (5.18). [(i): The first expression is obtained from the joint density of Z and V by transforming to t = z ÷ υ/f and υ.] Problem 5.4 Let X1 , . . . , Xn be a sample from N (ξ, σ 2 ). Denote the power of the one-sided t-test of H : ξ ≤ 0 against the alternative ξ/σ by β(ξ/σ), and by β ∗ (ξ/σ) the power of the test appropriate when σ is known. Determine β(ξ/σ) for n = 5, 10, 15, α = .05, ξ/σ = .07, 0.8, 0.9, 1.0, 1.1, 1.2, and in each case compare it with β ∗ (ξ/σ). Do the same for the two-sided case. Problem 5.5 Let Z1 , . . . , Zn be independently normally distributed with common variance σ 2 and means E(Zi ) = ζi (i = 1, . . . , s), E(Zi ) = 0 (i = s+1, . . . , n). 11 A systematic account of this distribution can be found in in Owen (1985) and Johnson, Kotz and Balakrishnan (1995). 194 5. Unbiasedness: Applications to Normal Distributions There exist UMP unbiased tests for testing ζ1 ≤ ζ10 and ζ1 = ζ10 given by the rejection regions 6 Z1 − ζ10 n  i=s+1 > C0 Zi2 /(n and 6 − s) |Z1 − ζ10 | n  i=s+1 Zi2 /(n > C. − s) When ζ1 = ζ10 , the test statistic has the t-distribution with n − s degrees of freedom. Problem 5.6 Let X1 , . . . , Xn be independently normally distributed with comn mon variance σ 2 and means nζ1 , . . . , ζn , and let Zi = j=1 aij Xj , be an orthogonal transformation (that is, i=1 aij aik = 1 or 0 as j = k or j =  k). The Z’s are normally distributed with common variance σ 2 and means ζi = aij ξj . [The density of the Z’s is obtained from that of the X’s by substituting xi =  bij zj , where (bij ) is the inverse of the matrix (aij ), and multiplying by the Jacobian, which is 1.] Problem 5.7 If X1 , . . . , Xn is a sample from N (ξ, σ 2 ), the UMP unbiased tests of ξ ≤ 0 and ξ = 0 can be obtained from Problems 5.5 and 5.6 by making an √ orthogonal transformation to variables Z1 , . . . , Zn such that Z1 = nX̄. [Then n  Zi2 = n  i=2 Zi2 − Z12 = i=1 n  Xi2 − nX̄ 2 = i=1 n  (Xi − X̄)2 .] i=1 Problem 5.8 Let X1 , X2 , . . . be a sequence of independent variables distributed as N (ξ, σ 2 ), and let Yn = [nXn+1 − (X1 + · · · + Xn )]/ n(n + 1) . Then the variables Y1 , Y2 , . . . are independently distributed as N (0, σ 2 ). Problem 5.9 Let N have the binomial distribution based on 10 trials with success probability p. Given N = n, let X1 , · · · , Xn be i.i.d. normal with mean θ and variance one. The data consists of (N, X1 , · · · , XN ). (i). If p has a known value p0 , show there does not exist a UMP test of θ = 0 versus θ > 0. [In fact, a UMPU test does not exist either.] (ii). If p is unknown (taking values in (0,1)), find a UMPU test of θ = 0 versus θ > 0. Problem 5.10 As in Example 3.9.2, suppose X is multivariate normal with unknown mean ξ = (ξ1 , . . . , ξk )T and known positive definite covariance matrix Σ. Assume a = (a1 , . . . , ak )T is a fixed vector. The problem is to test H: k  i=1 ai ξi = δ vs. K: k  ak ξi = δ . i=1 Find a UMPU level α test. Hint: First consider Σ = Ik , the identity matrix. Problem 5.11 Let Xi = ξ + Ui , and suppose that  2the joint density f of the U ’s is spherically symmetric, that is, a function of Ui only,  2 f (u1 , . . . , un ) = q( ui ) . 5.14. Problems 195 Show that the null distribution of the one-sample t-statistic is independent of q and hence is the same as in the normal case, namely Student’s t with n − 1 degrees of freedom. Hint: Write tn as , n1/2 X̄n / Xj2 , ,  (Xi − X̄n )2 /(n − 1) Xj2 and use the  fact that when ξ = 0, the density of X1 , . . . , Xn is constant over the spheres x2j = c and hence the conditional distribution of the variables ,  2 2 Xj given Xj = c is uniform over the conditioning sphere and hence Xi / independent of q. Note. This model represents one departure from the normaltheory assumption, which does not affect the level of the test. The effect of a much weaker symmetry condition more likely to arise in practice is investigated by Efron (1969). Section 5.3 Problem 5.12 Let X1 , . . . , Xn and Y1 , . . . , Yn be independent samples from N (ξ, σ 2 ) and N (η, τ 2 ) respectively. Determine the sample size necessary to obtain power ≥ β against the alternatives τ /σ > ∆ when α = .05, β = .9, ∆ = 1.5, 2, 3, and the hypothesis being tested is H : τ /σ ≤ 1. Problem 5.13 If m = n, the acceptance region (5.23) can be written as  2 SY2 ∆0 SX 1−C , max , ≤ 2 2 ∆0 SX SY C   2 where SX = (Xi − X̄)2 , SY2 = (Yi − Ȳ )2 and where C is determined by  C α Bn−1,n−1 (w) dw = . 2 0 Problem 5.14 Let X1 , . . . , Xm and Y1 , . . . , Yn be samples from N (ξ, σ 2 ) and N (η, σ 2 ). The UMP unbiased test for testing η − ξ = 0 can be obtained through Problems 5.5 and 5.6 by making an orthogonal transformation from (X1 , . . . Xm , Y 1 , . . . Yn ) to (Z1 , . . . , Zm+n ) such that Z1 = (Ȳ − √  X̄)/ 1/m + (1/n), Z2 = ( Xi + Yi )/ m + n. Problem 5.15 Exponential densities. Let X1 , . . . , Xn , be a sample from a distribution with exponential density a−1 e−(x−b)/a for x ≥ b. (i) For testing a = 1 there exists a UMP unbiased test given by the acceptance region  C1 ≤ 2 [xi − min(x1 , . . . , xn )] ≤ C2 , where the test statistic has a χ2 -distribution with 2n−2 degrees of freedom when α = 1, and C1 , C2 are determined by  C2  C2 χ22n−2 (y) dy = χ22n (y) dy = 1 − α. C1 C1 196 5. Unbiasedness: Applications to Normal Distributions (ii) For testing b = 0 there exists a UMP unbiased test given by the acceptance region n min(x1 , . . . , xn ) 0≤  ≤ C. [xi − min(xi , . . . , xn )] When b = 0, the test statistic has probability density p(u) = n−1 , (1 + u)n u ≥ 0. [These distributions for varying b do not constitute an exponential family, and Theorem 4.4.1 is therefore not directly applicable. For (i), one can restrict attention to the ordered variables X(1) < · · · < X(n) , since these are sufficient for a and b, and transform to new variables Z1 = nX(1) , Zi = (n − i + 1)[X(i) − X(i−1) ] for i = 2, . . . , n, as in Problem 2.15. When a = 1, Z1 is a complete sufficient statistic for b, and the test  is therefore obtained by considering the conditional problem given z1 . Since n of Z1 , the conditional UMP unbiased test i=2 Zi , is independent n has the acceptance region C Z 1 ≤ i ≤ C2 for each z1 , and the result follows. i=2  For (ii), when b = 0, n and the i=1 Zi , is a complete sufficient statistic for a,  n test is therefore obtained by considering the conditional problem given i=1 zi . n The remainder of the argument uses the fact that Z Z is indepen1/ i i=1 n dent of i=1 Zi , when b = 0, and otherwise is similar to that used to prove Theorem 5.1.1.] Problem 5.16 Let X1 , . . . , Xn be a sample from the Pareto distribution P (c, τ ), both parameters unknown. Obtain UMP unbiased tests for the parameters c and τ . [Problems 5.15 and 3.8.] Problem 5.17 Extend the results of the preceding problem to the case, considered in Problem 3.29, that observation is continued only until X(1) , . . . , X(r) have been observed. Problem 5.18 Gamma two-sample problem. Let X1 , . . . Xm ; Y1 , . . . , Yn be independent samples from gamma distributions Γ(g1 , b1 ), Γ(g2 , b2 ) respectively. (i) If g1 , g2 are known, there exists a UMP unbiased test of H : b2 = b1 against one- and two-sided alternatives, which can be based on a beta distribution. [Some applications and generalizations are discussed in Lentner and Buehler (1963).] (ii) If g1 , g2 are unknown, show that a UMP unbiased test of H continues to exist, and describe its general form. (iii) If b2 = b1 = b (unknown), there exists a UMP unbiased test of g2 = g1 against one- and two-sided alternatives; describe its general form. [(i): If Yi (i = 1, 2) are independent Γ(gi , b), then Y1 + Y2 is Γ(g1 + g2 , b) and Y1 /(Y1 + Y2 ) has a beta distribution.] 5.14. Problems 197 Problem 5.19 Inverse Gaussian distribution.12 Let X1 , . . . , Xn be a sample from the inverse Gaussian distribution I(µ, τ ), both parameters unknown. (i) There exists a UMP unbiased test of µ ≤ µ0 against µ > µ0 , which rejects when X̄ > C[ (Xi + 1/Xi )], and a corresponding UMP unbiased test of µ = µ0 against µ0 = µ0 . [The conditional distribution needed to carry out this test is given by Chhikara and Folks (1976).] (ii) There exist UMP unbiased tests of H : τ = τ 0 against both one- and two-sided hypotheses based on the statistic V = (1/Xi − 1/X̄). (iii) When τ = τ0 , the distribution of τ0 V is χ2n−1 . [Tweedie (1957).] Problem 5.20 Let X1 , . . . , Xm and Y1 , . . . , Yn be independent samples from I(µ, σ) and I(ν, τ ) respectively. (i) There exist UMP unbiased tests of τ2 /τ1 against one- and two-sided alternatives. (ii) If τ = σ, there exist UMP unbiased tests of ν/µ against one- and two-sided alternatives. [Chhikara (1975).] Problem 5.21 Suppose X and Y are independent, normally distributed with variance 1, and means ξ and η, respectively. Consider testing the simple null hypothesis ξ = η = 0 against the composite alternative hypothesis ξ > 0, η > 0. Show that a UMPU test does not exist. Section 5.4 Problem 5.22 On the basis of a sample X = (X1 , . . . , Xn ) of fixed size from N (ξ, σ 2 ) there do not exist confidence intervals for ξ with positive confidence coefficient and of bounded length.13 [Consider any family of confidence intervals δ(X) ± L/2 of constant length L. Let ξ1 , . . . ξ2n be such that |ξi − ξj | > L whenever i = j. Then the sets Si {x : |δ(x) − ξi | ≤ L/2} (i = 1, . . . , 2N ) are mutually exclusive. Also, there exists σ0 > 0 such that |Pξi ,σ {X ∈ Si } − Pξ1 ,σ {X ∈ Si }| ≤ 1 2N for σ > σ0 , 12 For additional information concerning inference in inverse Gaussian distributions, see Folks and Chhikara (1978) and Johnson, Kotz and Balakrishnan (1994, volume 1). 13 A similar conclusion holds in the problem of constructing a confidence interval for the ratio of normal means (Fieller’s problem), as discussed in Koschat (1987). For problems where it is impossible to construct confidence intervals with finite expected length, see Gleser and Hwang (1987). 198 5. Unbiasedness: Applications to Normal Distributions as is seen by transforming to new variables Yj = (Xj − ξ1 )/σ and applying Lemmas 5.5.1 and 5.11.1 of the Appendix. Since mini Pξ1 ,σ {X ∈ Si } ≤ 1/(2N ), it follows for σ > σ0 that mini Pξ1 ,σ {X ∈ Si } ≤ 1/N , and hence that   1 L ≤ inf Pξ,σ |δ(X) − ξ| ≤ ξ,σ 2 N The confidence coefficient associated with the intervals δ(X) ± L/2 is therefore zero, and the same must be true a fortiori of any set of confidence intervals of length ≤ L.] Problem 5.23 Stein’s two-stage procedure. (i) If mS 2 /σ 2 has a χ2 = distribution with m degrees of freedom, and if the conditional distribution of Y given S = s is N (0, σ 2 /S 2 ), then Y has Student’s t-distribution with m degrees of freedom. distributed as N (ξ, σ 2 ). Let X̄0 = (ii) Let n0 X1 , X2 , . . .2 beindependently n0 2 i=1 Xi /n0 , S = i=1 (Xi − X̄0 ) /(n0 − 1), and let a1 = · · · = an0 = a, an0 +1 = · · · = an = b and n ≥ n0 be measurable functions of S. Then n  Y = ai (Xi − ξ)  2 S2 n i=1 ai i=1 has Student’s distribution with n0 − 1 degrees of freedom. (iii) Consider a two-stage sampling scheme 1 , in which S 2 is computed from an initial sample of size n0 , and then n − n0 additional observations are taken. The size of the second sample is such that  2   S n = max n0 + 1, +1 c where c is any given constant and where [y] denotes the largest integer ≥ y. There numbers a1 , . . . , an such that a1 = · · · = an0 , an0 +1 =  then exist · · · an , n ai = 1, n a2i = c/S 2 . It follows from (ii) that n i=1 i=1 i=1 ai (Xi − √ ξ)/ c has Student’s t-distribution with n0 − 1 degrees of freedom. (iv) The following sampling scheme 2 , which does not require that the second sample contain at least one observation, is slightly more efficient than 1 , for the applications to be made in Problems 5.24 and 5.25. Let n0 , S 2 , and c be defined as before; let  2   S n = max n0 , +1 c n √ ai = 1/n (i = 1, . . . , n), and X̄ = i=1 ai Xi . Then n(X̄ − ξ)/S has again the t-distribution with n0 − 1 degrees of freedom.  [(ii): Given S = s, the quantities a, b, and n are constants, n i=1 ai (Xi − ξ) = 2 2 n0 a(X̄0 − ξ) is distributed as N (0, n0 a σ ), and the numerator of Y is therefore  2 normally distributed with zero mean and variance σ 2 n a . The result now i i=1 follows from (i).] Problem 5.24 Confidence intervals of fixed length for a normal mean. 5.14. Problems 199 (i) In the two-stage procedure 1 , defined in part (iii) of the preceding problem, let the number c be determined for any given L > 0 and 0 < γ < 1 by  L/2√c t (y) dy = γ, √ n0 −1 −L/2 c where tn0 −1 denotes the density  of the t-distribution with n0 − 1 degrees of freedom. Then the intervals n i=1 ai Xi ± L/2 are confidence intervals for ξ of length L and with confidence coefficient γ. (ii) Let c be defined as in (i), and let the sampling procedure be 2 as defined in part (iv) of Problem 5.23. The intervals X̄ ± L/2 are then confidence intervals of length L for ξ with confidence coefficient ≥ γ, while the expected number of observations required is slightly lower than under 1 . [(i): The probability that the intervals cover ξ equals ⎫ ⎧ n  ⎪ ⎪ ⎪ ⎪ a (X − ξ) i i ⎬ ⎨ L L i=1 √ =γ ≤ √ Pξ,σ − √ ≤ ⎪ 2 c c 2 c⎪ ⎪ ⎪ ⎭ ⎩ (ii): The probability that the intervals cover ξ equals  √ √  √ n|X̄ − ξ| n|X̄ − ξ| nL L Pξ,σ = γ.] ≤ ≤ √ ≥ S 2S S 2 c Problem 5.25 Two-stage t-tests with power independent of σ. (i) For the procedure 1 with any given c, let C be defined by  ∞ tn0 −1 (y) dy = α. C n √ Then the rejection region ( i=1 ai Xi − ξ0 )/ c > C defines a level-α test of H : ξ ≤ ξ0 with strictly increasing power function βc (ξ) depending only on ξ. (ii) Given any alternative ξ1 and any α < β < 1, the number c can be chosen so that βc (ξ1 ) = β. √ (iii) The test with rejection region n(X̄ − ξ0 )/S > C based on 2 and the same c as in (i) is a level-α test of H which is uniformly more powerful than the test given in (i). (iv) Extend parts (i)–(iii) to the problem of testing ξ = ξ0 against ξ = ξ0 . [(i) and (ii): The power of the test is  βc (ξ) = C−(ξ−ξ0 )/ (iii): This follows from the inequality √ t (y) dy. √ n0 −1 c √ n|ξ − ξ0 |/S ≥ |ξ − ξ0 |/ c.] Problem 5.26 Let S(x) be a family of confidence sets for a real-valued parameter θ, and let µ[S(x)] denote its Lebesgue measure. Then for every fixed 200 5. Unbiasedness: Applications to Normal Distributions distribution Q of X (and hence in particular for Q = Pθ0 where θ0 is the true value of θ)  Q{θ ∈ S(X)} dθ EQ {µ[S(X)]} = θ=θ0 provided the necessary measurability conditions hold. [The identity is known as the Ghosh-Pratt identity; see Ghosh (1961) and Pratt (1961a). To prove it, write the expectation on the left side as a double integral, apply Fubini’s theorem, and note that the integral on the right side is unchanged if the point θ = θ0 is added to the region of integration.] Problem 5.27 Use the preceding problem to show that uniformly most accurate confidence sets also uniformly minimize the expected Lebesgue measure (length in the case of intervals) of the confidence sets.14 Section 5.5 Problem 5.28 Let X1 , . . . , Xn be distributed as in Problem 5.15. Then the most accurate unbiased confidence intervals for the scale parameter a are 2  2  [xi − min(x1 , . . . , xn )] ≤ a ≤ [xi − min(x1 , . . . , xn )]. C2 C1 Problem 5.29 Most accurate unbiased confidence intervals exist in the following situations: (i) If X, Y are independent with binomial distributions b(p1 , m) and b(p2 , m), for the parameter p1 q2 /p2 q1 . (ii) In a 2 × 2 table, for the parameter ∆ of Section 4.6. Problem 5.30 Shape parameter of a gamma distribution. Let X1 , . . . , Xn be a sample from the gamma distribution Γ(g, b) defined in Problem 3.34. (i) There exist UMP unbiased tests of H : g ≤ g0 against g > g0 and of H  : g = g0 against g = g0 , and their rejection regions are based on W = (Xi /X̄). (ii) There exist uniformly most accurate confidence intervals for g based on W . [Shorack (1972).] Notes. (1) The null distribution of W is discussed in Bain and Engelhardt (1975), Glaser (1976), and Engelhardt and Bain (1978). (2) For g = 1, Γ(g, b) reduces to an exponential distribution, and (i) becomes the UMP unbiased test for testing that a distribution is exponential against the alternative that it is gamma with g > 1 or with g = 1. 14 For the corresponding result concerning one-sided confidence bounds, see Madansky (1962). 5.14. Problems 201 (3) An alternative treatment of this and some of the following problems is given by Bar-Lev and Reiser (1982). Problem 5.31 Scale parameter of a gamma distribution. Under the assumptions of the preceding problem, there exists (i)  A UMP unbiased test of H : b ≤ b0 against b > b0 which rejects when Xi > C( , Xi ). (ii) Most accurate unbiased confidence intervals for b.  [The conditional distribution of Xi given Xi , which is required for carrying out this test, is discussed by Engelhardt and Bain (1977).] Problem 5.32 In Example 5.5.1, consider a confidence interval for σ 2 of the  2 −1 2 2 2 form I = [d−1 S , c S ], where S = (X i − X̄) and cn < dn are constants. n n n n n i Subject to the level constraint, choose cn and dn to minimize the length of I. Argue that the solution has shorter length that the uniformly most accurate one; however, it is biased and so does not uniformly improve the probability of covering false values. [The solution, given in Tate and Klett (1959), satisfies d χ2n+3 (cn ) = χ2n+3 (dn ) and cnn χ2n−1 (y)dy = 1 − α, where χ2n (y) denotes the Chisquared density with n degrees of freedom. Improvements of this interval which incorporate X̄ into their construction are discussed in Cohen (1972) and Shorrock (1990); also see Goutis and Casella (1991).] Section 5.6 Problem 5.33 (i) Under the assumptions made at the beginning of Section 5.6, the UMP unbiased test of H : ρ = ρ0 is given by (5.44). (ii) Let (ρ, ρ̄) be the associated most accurate unbiased confidence intervals for ρ = aγ + bδ, where ρ = ρ(a, b), ρ̄ = ρ̄(a, b). Then if f1 and f2 are increasing functions, the expected value of f1 (|ρ̄ − ρ|) + f2 (|ρ − ρ|) is an increasing function of a2 /n + b2 . [(i): Make any orthogonal  transformation from y1 , . . . , yn to new  variables z1 , . . √ . , zn , such that z1 = (a2 /n) + b2 , z2 = i [bvi + (a/n)]yi / i (avi − b)yi / a2 + nb2 , and apply Problems 5.5 and 5.6. (ii): If a21 /n + b21 < a22 /n + b22 , the random variable |ρ̄(a2 , b2 ) − ρ| is stochastically larger than |ρ̄(a1 , b1 ) − ρ|, and analogously for ρ.] Section 5.7 Problem 5.34 Verify the posterior distribution of Θ given x in Example 5.7.1. Problem 5.35 If X1 , . . . , Xn , are independent N (θ, 1) and θ has the improper prior π(θ) ≡ 1, determine the posterior distribution of θ given the X’s. Problem 5.36 Verify the posterior distribution of p given x in Example 5.7.2. 202 5. Unbiasedness: Applications to Normal Distributions Problem 5.37 In Example 5.7.3, verify the marginal posterior distribution of ξ given x. Problem 5.38 In Example 5.7.4, show that (i) the posterior density π(σ | x) is of type (c) of Example 5.7.2; (ii) for sufficiently large r, the posterior density of σ r given x is no longer of type (c). Problem 5.39 If X is normal N (θ, 1) and θ has a Cauchy density b/{π[b2 + (θ − µ)2 ]}, determine the possible shapes of the HPD regions for varying µ and b. Problem 5.40 Let θ = (θ1 , . . . , θs ) with θi real-valued, X have density pθ (x), and Θ a prior density π(θ). Then the 100γ% HPD region is the 100γ% credible region R that has minimum volume. [Apply the Neyman–Pearson fundamental lemma to the problem of minimizing the volume of R.] Problem 5.41 Let X1 , . . . , Xm and Y1 , . . . , Yn be independently distributed as N (ξ, σ 2 ) and N (η, σ 2 ) respectively, and let (ξ, η, σ) have the joint improper prior density given by 1 dσ for all − ∞ < ξ, η < ∞, 0 < σ. σ Under these assumptions, extend the results of Examples 5.7.3 and 5.7.4 to inferences concerning (i) η − ξ and (ii) σ. π(ξ, η, σ) dξ dη dσ = dξ dη · Problem 5.42 Let X1 , . . . , Xm and Y1 , . . . , Yn be independently distributed as N (ξ, σ 2 ) and N (η, τ 2 ), respectively and let (ξ, η, σ, τ ) have the joint improper prior density π(ξ, η, σ, τ ) dξ dη dσ dτ = dξ dη(1/σ) dσ(1/τ ) dτ . Extend the result of Example 5.7.4 to inferences concerning τ 2 /σ 2 . Note. The posterior distribution of η − ξ in this case is the so-called Behrens– Fisher distribution. The credible regions for η − ξ obtained from this distribution do not correspond to confidence intervals with fixed coverage probability, and the associated tests of H : η = ξ thus do not have fixed size (which instead depends on τ /σ). From numerical evidence [see Robinson (1976) for a summary of his and earlier results] it appears that the confidence intervals are conservative, that is, the actual coverage probability always exceeds the nominal one. Problem 5.43 Let T1 , . . . , Ts−1 have the multinomial distribution (2.34), and suppose that (p1 , . . . , ps−1 ) has the Dirichlet prior density D(a1 , . . . , as ) with density proportional to p1a1 −1 . . . psas −1 , where ps = 1−(p1 +· · ·+ps−1 ). Determine the posterior distribution of (p1 , . . . , ps−1 ) given the T ’s. Section 5.8 Problem 5.44 Prove Theorem 5.8.1 for arbitrary values of c. 5.14. Problems 203 Section 5.9 Problem 5.45 If c = 1, m = n = 4, α = .1 and the ordered coordinates z(1) , . . . , z(N ) of a point z are 1.97, 2.19, 2.61, 2.79, 2.88, 3.02, 3.28, 3.41, determine the points of S(z) belonging to the rejection region (5.53). Problem 5.46 Confidence intervals for a shift. [Maritz (1979)] (i) Let X1 , . . . , Xm ; Y1 , . . . , Yn be independently distributed according to continuous distributions F (x) and G(y) = F (y − ∆) respectively. Without any further assumptions concerning F , confidence intervals for ∆ can be obtained from permutation tests of the hypotheses H(∆0 ) : ∆ = ∆0 . Specifically, consider  the point (z1 , . . . , zm+n ) = (x1 , . . . , xm , y1 − ∆, . . . , yn − ∆) and the m+n permutations i1 < · · · < im ; im+1 < · · · < m im+n of the integers 1, . . . , m + n. Suppose that the hypothesis H(∆) is accepted for the k of these permutations which lead to the smallest values of % m+n % m %  %  % % zij /n − zij /m% % % % j=m+1 where j=1   m+n k = (1 − α) . m Then the totality of values ∆ for which H(∆) is accepted constitute an interval, and these intervals are confidence intervals for ∆ at confidence level 1 − α. (ii) Let Z1 , . . . , ZN be independently distributed, symmetric about θ, with distribution F (z − θ), where F (z) is continuous and symmetric about 0. Without any further assumptions about F , confidence intervals for θ can be  obtained by considering the 2N points Z1 , . . . , ZN where Zi = ±(Zi − θ0 ), and accepting H(θ 0 ) : θ = θ0 for the k of these points which lead to the smallest values of |Zi |, where k = (1 − α)2N . [(i): A point is in the acceptance region for H(∆) if %  % % (yj − ∆) xi %% % − = |ȳ − x̄ − ∆| % n m %  − k of the quantities |ȳ  − x̄ − γ∆|, where is exceeded by at least m+n n (x1 , . . . , xm , y1 , . . . , yn ) is a permutation of (x1 , . . . , xm , y1 , . . . , yn ), the quantity γ is determined by this permutation, and |γ| ≤ 1. The desired result now follows from the following facts (for an alternative proof, see Section 14): (a) The set of ∆’s for which (ȳ − x̄ − ∆)2 ≤ (ȳ  − x̄ − γ∆)2 is, with probability one, an 2 interval containing ȳ − x̄. (b) The is exceeded m+nset of ∆’s for which (ȳ − x̄ − ∆)  by a particular set of at least m − k of the quantities (ȳ − x̄ − γ∆)2 is the intersection of the corresponding intervals (a) and hence is an interval containing ȳ − x̄. (c) The set of ∆’s of interest is the union of the intervals (b) and, since they have a nonempty intersection, also an interval.] 204 5. Unbiasedness: Applications to Normal Distributions Section 5.10 Problem 5.47 In the matched-pairs experiment for testing the effect of a treatment, suppose that only the differences Zi = Yi − Xi are observable. The Z’s are assumed to be a sample from an unknown continuous distribution, which under the hypothesis of no treatment effect is symmetric with respect to the origin. Under the alternatives it is symmetric with respect to a point ζ > 0. Determine the test which among all unbiased tests maximizes the power against the alternatives that the Z’s are a sample from N (ζ, σ 2 ) with ζ  > 0. n 2 2n [Under the hypothesis, the set of statistics ( n i=1 Zi , . . . , i=1 Zi ) is sufficient; that it is complete is shown as the corresponding result in Theorem 5.8.1. The remainder of the argument follows the lines of Section 11.] Problem 5.48 (i) If X1 , . . . , Xn ; Y1 , . . . , Yn are independent normal variables with common variance σ 2 and means E(Xi ) = ξi , E(Yi ) = ξi + ∆, the UMP unbiased test of ∆ = 0 against ∆ > 0 is given by (5.58). (ii) Determine the most accurate unbiased confidence intervals for ∆. [(i): The structure of the problem becomes clear √ if one makes the orthogonal √ transformation Xi = (Yi − Xi )/ 2, Yi = (Xi + Yi )/ 2.] Problem 5.49 Comparison of two designs. Under the assumptions made at the beginning of Section 12, one has the following comparison of the methods of complete randomization and matched pairs. The unit effects and experimental effects Ui and Vi are independently normally distributed with variances σ12 , σ 2 and means E(Ui ) = µ and E(Vi ) = ξ or η as Vi corresponds to a control or treatment. With complete randomization, the observations are Xi = Ui + Vi (i = 1, . . . , n) for the controls and Yi = Un+i + Vn+i (i = 1, . . . , n) for the treated cases, with E(Xi ) = µ+ξ, E(Yi ) = µ+η. For the matched pairs, if the matching is assumed to be perfect, the X’s are as before, but Yi = Ui + Vm+i . UMP unbiased tests are given by (5.27) for complete randomization and by (5.58) for matched pairs. The distribution of the test statistic under an alternative ∆ = η − ξ is the √ noncentral t-distribution with noncentrality parameter n∆/ 2(σ 2 + σ12 ) and 2n − 2√degrees of freedom in the first case, and with noncentrality parameter √ n∆/ 2σ and n − 1 degrees of freedom in the second. Thus the method of matched pairs has the disadvantage of a smaller number of degrees of freedom and the advantage of a larger noncentrality parameter. For α = .05 and ∆ = 4, compare the power of the two methods as a function of n when σ1 , σ = 2 and when σ1 = 2, σ = 1. Problem 5.50 Continuation. An alternative comparison of the two designs is obtained by considering the expected length of the most accurate unbiased confidence intervals for ∆ = η − ξ in each case. Carry this out for varying n and confidence coefficient 1 − α = .95 when σ1 = 1, σ = 2 and when σ1 = 2, σ = 1. Section 5.11 Problem 5.51 Suppose that a critical function φ0 satisfies (5.64) but not (5.66), and let α < 12 . Then the following construction provides a measurable critical 5.14. Problems 205 function φ satisfying (5.66) and such that φ0 (z) ≤ φ(z) for all z Inductively, sequences of functions φ1 , φ2 , . . . and ψ0 , ψ1 , . . . are defined through the relations ψm (z) =  z  ∈S(z) and φm (z  ) , N1 ! . . . Nc ! m = 0, 1, . . . , ⎧ ⎨ φm−1 (z) + [α − ψm−1 (z)] if both φm−1 (z) and ψm−1 (z) are < α, φm (z) = ⎩ φm−1 (z) otherwise. The function φ(z) = lim φm (z) then satisfies the required conditions. [The functions φm are nondecreasing and between 0 and 1. It is further seen by induction that 0 ≤ α − ψm (z) ≤ (1 − γ)m [α − ψ0 (z)], where γ = 1/(N1 ! . . . Nc !).] Problem 5.52 Consider the problem of testing H : η = ξ in the family of densities (5.61) when it is given that σ > c > 0 and that the point (ζ11 , . . . , ζcNc of (5.62) lies in a bounded region R containing a rectangle, where c and R are known. Then Theorem 5.11.1 is no longer applicable. However, unbiasedness of a test φ of H implies (5.66), and therefore reduces the problem to the class of permutation tests. [Unbiasedness implies (φ(z)pσ,ζ (z) dz = α and hence     1 1  α = ψ(z)pσ,ζ (z) dz = ψ(z) √ (zij − ζij )2 dz exp − 2 2σ ( 2πσ)N for all σ > c and ζ in R. The result follows from completeness of this last family.] Problem 5.53 To generalize Theorem 5.11.1 to other designs, let Z = (Z1 , . . . , ZN ) and let G = {g1 , . . . , gr } be a group of permutations of N coordinates or more generally a group of orthogonal transformations of N -space If  r 1 1 1 2 √ Pσ,ζ (z) = exp − 2 |z − gk ζ| , (5.80) r 2σ ( 2πσ)N 2 where |z| =  k=1 zi2 , then φ(z)pσ,ζ (z) dz ≤ α for all σ > 0 and all ζ implies 1  φ(z  ) ≤ α a.e., (5.81) r  z ∈S(z) where S(z) is the set of points in N -space obtained from z by applying to it all the transformations gk , k = 1, . . . , r. Problem 5.54 Generalization of Corollary 5.11.1. Let H be the class of densities (5.80) with σ > 0 and −∞ < ζi < ∞ (i = 1, . . . , N ). A complete family of tests of H at level of significance α is the class of permutation tests satisfying 1  φ(z  ) = α a.e. (5.82) r  z ∈S(z) 206 5. Unbiasedness: Applications to Normal Distributions Section 5.12 Problem 5.55 If c = 1, m = n = 3, and if the ordered x’s and y’s are respectively 1.97, 2.19, 2.61 and 3.02, 3.28, 3.41, determine the points δ(1) , . . . , δ(19) defined as the ordered values of (5.72). Problem 5.56 If c = 4, mi = ni = 1, and the pairs (xi , yi ) are (1.56,2.01), (1.87,2.22), (2.17,2.73), and (2.31,2.60), determine the points δ(1) , . . . , δ(15) which define the intervals (5.71). Problem 5.57 If m, n are positive integers with m ≤ n, then      m  m n m+n = −1 K K m K=1 Problem 5.58 (i) Generalize the randomization models of Section 14 for paired comparisons (n1 = · · · = nc = 2) and the case of two groups (c = 1) to an arbitrary number c of groups of sizes n1 , . . . , nc . (ii) Generalize the confidence intervals (5.71) and (5.72) to the randomization model of part (i). Problem 5.59 Let Z1 , . . . , Zn be i.i.d. according to a continuous distribution symmetric about θ, and let T(1) < · · · < T(M ) be the ordered set of M = 2n − 1 subsamples; (Zi1 + · · · + Zir )/r, r ≤ 1. If T(0) = −∞, T(M +1) = ∞, then Pθ [T(i) < θ < T(i+1) ] = 1 M +1 for all i = 0, 1, . . . , M. [Hartigan (1969).] Problem 5.60 (i) Given n pairs (x1 , y1 ), . . . , (xn , yn ), let G be the group of 2n permutations of the 2n variables which interchange xi and yi in all, some, or none of the n pairs. Let G0 be any subgroup of G, and let e be the number of elements in G0 . Any element g ∈ G0 (except the identity) is characterized by the numbers i1 , . . . , ir (r ≥ 1) of the pairs in which xi and yi have been switched. Let di = yi − xi , and let δ(1) < · · · < δ(e−1) , denote the ordered values (di1 + · · · + dir )/r corresponding to G0 . Then (5.71) continues to hold with e − 1 in place of M . (ii) State the generalization of Problem 5.59 to the situation of part (i). [Hartigan (1969).] Problem 5.61 The preceding problem establishes a 1 : 1 correspondence between e − 1 permutations T of G0 which are not the identity and e − 1 nonempty subsets {i1 , . . . , ir } of the set {1, . . . , n}. If the permutations T and T  correspond respectively to the subsets R = {i1 , . . . , ir } and R = {j1 , . . . , js }, then the group product T  T corresponds to the subset (R ∩ S̃) ∪ (R̃ ∩ S) = (R ∪ S) − (R ∩ S). [Hartigan (1969).] 5.14. Problems 207 Problem 5.62 Determine for each of the following classes of subsets of {1, . . . , n} whether (together with the empty subset) it forms a group under the group operation of the preceding problem: All subsets {i1 , . . . , ir } with (i) r = 2; (ii) r = even; (iii) r divisible by 3. (iv) Give two other examples of subgroups G0 of G. Note. A class of such subgroups is discussed by Forsythe and Hartigan (1970). Problem 5.63 Generalize Problems 5.60(i) and 5.61 to the case of two groups of sizes m and n (c = 1). Section 5.13 Problem 5.64 (i) If the joint distribution of X and Y is the bivariate normal distribution (5.69), then the conditional distribution of Y given x is the normal distribution with variance τ 2 (1 − ρ2 ) and mean η + (ρτ /σ)(x − ξ). (ii) Let (X1 , Y1 ), . . . , (Xn , Yn ) be a sample from a bivariate normal distribution, let R be the sample correlation√coefficient, √ and suppose that ρ = 0. Then the conditional distribution of n − 2R/ 1 − R2 given x1 , . . . , xn , is Student’s t-distribution with n−2 degrees of freedom provided (xi − x̄)2 > 0. This is therefore also the unconditional distribution of this statistic. (iii) The probability density of R itself is then [(ii): If vi = (x1 − x̄)/ be written as 1 1 1 Γ[ 2 (n − 1)] (1 − r2 ) 2 n−2 . (5.83) p(r) = √ 1 n Γ[ 2 (n − 2)]   2  vi = 0, v1 = 1, the statistic can (xj − x̄)2 so that ,  Yi2 vi Yi .   − nȲ 2 − ( vi Yi )2 /(n − 2) Since its distribution depends only on ρ one can assume η = 0, τ = 1. The desired result follows from Problem 5.6 by making an orthogonal transformation from √ (Y1 , , . . . , Yn ) to (Z1 , . . . , Zn ) such that Z1 = nȲ , Z2 = vi Yi .] Problem 5.65 (i) Let (X1 , Y1 ), . . . , (Xn , Yn ) be a sample from  the bivariate normal distribution (5.69), and let S12 = (Xi − X̄)2 , S22 = (Yi − Ȳ )2 , S12 = (Xi − X̄)(Yi − Ȳ ). There exists a UMP unbiased test for testing the hypothesis τ /σ = ∆. Its acceptance region is |∆2 S12 − S22 | ≤ C, 2 (∆2 S12 + S22 )2 − 4∆2 S12 and the probability density of the test statistic is given by (5.83) when the hypothesis is true. 208 5. Unbiasedness: Applications to Normal Distributions (ii) Under the assumption τ = σ, there exists a UMP unbiased test for testing η = ξ, with acceptance region |Ȳ − X̄|/ S12 + S22 − 2S12 ≤ C. On multiplication by a suitable constant the test statistic has Student’s t-distribution with n − 1 degrees of freedom when η = ξ. [Due to Morgan (1939) and Hsu (1940). (i): The transformation U = ∆X + Y , V = X − (1/∆)Y reduces the problem to that of testing that the correlation coefficient in a bivariate normal distribution is zero. (ii): Transform to new variables Vi = Yi − Xi , Ui = Yi + Xi .] Problem 5.66 (i) Let (X1 , Y1 ), . . . , (Xn , Yn ) be  a sample 2from the bivariate  normal distribution (5.73), and let S12 = (Xi − X̄) , S12 = (Xi −  2 2 X̄)(Yi − Ȳ ), S2 = (Yi − Ȳ ) . Then (S12 , S12 , S22 ) are independently distributed X̄, Ȳ ), andtheir joint n−1  2 of(n−1 2 distribution is the same as that of ( i=1 Xi , i=1 Xi Yi , n−1 i=1 Yi ), where (Xi , Yi ), i = 1, . . . , n − 1, are a sample from the distribution (5.73) with ξ = η = 0. Y1 , . . . , Ym betwo samples  from N (0, 1). Then the (ii) Let X1 , . . . , Xm and  joint density of S12 = Xi2 , S12 = Xi Yi , S22 = Yi2 is   1 1 1 (s21 s22 − s212 ) 2 (m−3) exp − (s21 + s22 ) 4πΓ(m − 1) 2 for s212 ≤ s21 s22 , and zero elsewhere. (iii) The joint density of the statistics (S12 , S12 , S22 ) of part (i) is   1 (s21 s22 − s212 ) 2 (n−4) 1 2ρs12 s21 s22 exp − − +  n−1 2(1 − ρ2 ) σ 2 στ τ2 4πΓ(n − 2) στ 1 − ρ2 (5.84) for s212 ≤ s21 s22 and zero elsewhere. [(i): Make an orthogonal transformation from X1 , . . . , Xn to X1 , . . . , Xn such that √ Xn = nX̄, and apply the same orthogonal transformation also to Y1 , . . . , Yn . Then Yn = √ nȲ , n−1  Xi Yi = i=1 n−1  i=1 Xi 2 = n  (Xi − X̄)(Yi − Ȳ ), i=1 n  (Xi − X̄)2 , n−1  i=1 i=1 Yi 2 = n  (Yi − Ȳ )2 . i=1 The pairs of variables (X1 , Y1 ), . . . , (Xn , Yn ) are independent, each with a bivariate normal distribution with the same variances and correlation as those of (X, Y ) and with means E(Xi ) − E(Yi ) = 0 for i = 1, . . . , n − 1.  2 (ii): Consider first the joint distribution of S xi Yi and S22 = Yi given 12 =  2 x1 . . . , xm . Letting Z1 = S12 / xi and making an orthogonal transformation m 2 from Y1 , . . . , Ym to Z1 , . . . , Zm so that S22 = i=1 Zi , the variables Z1 and  m 2 2 2 2 i=2 Zi = S2 − Z1 are independently distributed as N (0, 1) and χm−1 respectively. From this the joint conditional density of S12 = s1 Z1 and S22 is obtained by a simple transformation of variables. Since the conditional distribution depends on the x’s only through s21 , the joint density of S12 , S12 , S22 is found by multiplying 5.14. Problems 209 the above conditional density by the marginal one of S12 , which is χ2m . The proof is completed through use of the identity √ # $   πΓ(m − 1) Γ 12 (m − 1) Γ 21 m = . 2m−2   , Ym ) isa sample a bivariate normal (iii): If (X  , Y  ) = (X1 , Y1 ; . . . ; Xm  from  distribution with ξ = η = 0, then T = ( Xi 2 , Xi Yi , Yi 2 ) is sufficient for θ(σ, ρ, τ ), and the density of T is obtained from that given in part (ii) for θ0 = (1, 0, 1) through the identity [Problem 3.39 (i)] pTθ (t) = pTθ0 (t) pX θ  ,Y  (x , y  )  ,Y  pX (x , y  ) θ0 . The result now follows from part (i) with m = n − 1.] Problem 5.67 If (X1 , Y1 ), . . . , (Xn , Yn ) is a sample from a bivariate normal distribution, the probability density of the sample correlation coefficient R is15 pρ (r) 1 1 2n−3 (1 − ρ2 ) 2 (n−1) (1 − r2 ) 2 (n−4) π(n − 3)! ∞ $ (2ρr)k #  × Γ2 12 (n + k − 1) k! = (5.85) k=0 or alternatively pρ (r) = 1 1 n−2 (1 − ρ2 ) 2 (n−1) (1 − r2 ) 2 (n−4) π  1 tn−2 1 √ × dt. n−1 (1 − ρrt) 1 − t2 0 (5.86) Another form is obtained by making the transformation t = (1 − v)/(1 − ρrv) in the integral on the right-hand side of (5.86). The integral then becomes  1 $−1/2 (1 − v)n−2 # 1 √ dv. (5.87) 1 − 12 v(1 + ρr) 1 (2n−3) 2v (1 − ρr) 2 0 Expanding the last factor in powers of v, the density becomes 1 3 n − 2 Γ(n − 1) 2 1 (n−1) 2 √ (1 − r2 ) 2 (n−4) (1 − ρr)−n+ 2 1 (1 − ρ ) 2π Γ(n − 2 )  1 + ρr ×F 21 ; 12 ; n − 12 ; , 2 (5.88) where F (a, b, c, x) = ∞  Γ(a + j) Γ(b + j) Γ(c) xj Γ(a) Γ(b) Γ(c + j) j! j=0 (5.89) is a hypergeometric function. 15 The distribution of R is reviewed by Johnson and Kotz (1970, Vol. 2, Section 32) and Patel and Read (1982). 210 5. Unbiasedness: Applications to Normal Distributions [To obtain the first expression make a transformation from (S12 , S22 , S12 ) with density (5.84) to (S12 , S22 , R) and expand the factor exp{ρs12 /(1 − ρ2 )στ } = exp{ρrs1 s2 /(1 − ρ2 )στ } into a power series. The resulting series can be integrated term by term with respect to s21 and s22 . The equivalence with the second expression is seen by expanding the factor (1 − ρrt)−(n−1) under the integral in (5.86) and integrating term by term.] Problem 5.68 If X and Y have a bivariate normal distribution with correlation coefficient ρ > 0, they are positively regression-dependent. [The conditional distribution of Y given x is normal with mean η + ρτ σ −1 (x − ξ) and variance τ 2 (l − ρ2 ). Through addition to such a variable of the positive quantity ρτ σ −1 (x −x) it is transformed into one with the conditional distribution of Y given x > x.] Problem 5.69 functions. (i) The functions (5.78) are bivariate cumulative distributions (ii) A pair of random variables with distribution (5.78) is positively regressiondependent. [The distributions (5.78) were introduced by Morgenstem (1956).] Problem 5.70 If X, Y are positively regression dependent, they are positively quadrant dependent. [Positive regression dependence implies that P [Y ≤ y | X ≤ x] ≥ P [Y ≤ y | X ≤ x ] for all x < x and y, (5.90) and (5.90) implies positive quadrant dependence.] 5.15 Notes The optimal properties of the one- and two-sample normal-theory tests were obtained by Neyman and Pearson (1933) as some of the principal applications of their general theory. Theorem 5.1.2 is due to Basu (1955), and its uses are reviewed in Boos and Hughes-Oliver (1998). For converse aspects of this theorem see Basu (1958), Koehn and Thomas (1975), Bahadur (1979), Lehmann (1980) and Basu (1982). An interesting application is discussed in Boos and Hughes-Oliver (1998). In some exponential family regression models where UMPU tests do not exist, classes of admissible, unbiased tests are obtained in Cohen, Kemperman and Sackrowitz (1994). The roots of the randomization model of Section 5.10 can be traced to Neyman (1923); see Speed (1990) and Fienberg and Tanur (1996). Permutation tests, as alternatives to the standard tests having fixed critical levels, were initiated by Fisher (1935a) and further developed, among others, by Pitman (1937, 1938a), Lehmann and Stein (1949), Hoeffding (1952), and Box and Andersen (1955). Some aspects of these tests are reviewed in Bell and Sen (1984) and Good (1994). Applications to various experimental designs are given in Welch (1990). Optimality of permutation tests in a multivariate nonparametric two-sample setting are 5.15. Notes 211 studied in Runger and Eaton (1992). Explicit confidence intervals based on subsampling were given by Hartigan (1969). The theory of unbiased confidence sets and its relation to that of unbiased tests is due to Neyman (1937a). 6 Invariance 6.1 Symmetry and Invariance Many statistical problems exhibit symmetries, which provide natural restrictions to impose on the statistical procedures that are to be employed. Suppose, for example, that X1 , . . . , Xn are independently distributed with probability densities pθ1 (x1 ), . . . , pθn (xn ). For testing the hypothesis H : θ1 = · · · = θn against the alternative that the θ’s are not all equal, the test should be symmetric in x1 , . . . , xn , since otherwise the acceptance or rejection of the hypothesis would depend on the (presumably quite irrelevant) numbering of these variables. As another example consider a circular target with center O, on which are marked the impacts of a number of shots. Suppose that the points of impact are independent observations on a bivariate normal distribution centered on O. In testing this distribution for circular symmetry with respect to O, it seems reasonable to require that the test itself exhibit such symmetry. For if it lacks this feature, a two-dimensional (for example, Cartesian) coordinate system is required to describe the test, and acceptance or rejection will depend on the choice of this system, which under the assumptions made is quite arbitrary and has no bearing on the problem. The mathematical expression of symmetry is invariance under a suitable group of transformations. In the first of the two examples above the group is that of all permutations of the variables x1 , . . . , xn since a function of n variables is symmetric if and only if it remains invariant under all permutations of these variables. In the second example, circular symmetry with respect to the center O is equivalent to invariance under all rotations about O. In general, let X be distributed according to a probability distribution Pθ , θ ∈ Ω, and let g be a transformation of the sample space X . All such transformations 6.1. Symmetry and Invariance 213 considered in connection with invariance will be assumed to be 1 : 1 transformations of X onto itself. Denote by gX the random variable that takes on the value gx when X = x, and suppose that when the distribution of X is Pθ , θ ∈ Ω, the distribution of gX is Pθ with θ also in Ω. The element θ of Ω which is associated with θ in this manner will be denoted by ḡθ, so that Pθ {gX ∈ A} = Pḡθ {X ∈ A}. (6.1) Here the subscript θ on the left member indicates the distribution of X, not that of gX. Equation (6.1) can also be written as Pθ (g −1 A) = Pḡθ (A) and hence as Pḡθ (gA) = Pθ (A). (6.2) The parameter set Ω remains invariant under g (or is preserved by g) if ḡθ ∈ Ω for all θ ∈ Ω, and if in addition for any θ ∈ Ω there exists θ ∈ Ω such that ḡθ = θ . These two conditions can be expressed by the equation ḡΩ = Ω. (6.3) The transformation ḡ of Ω onto itself defined in this way is 1 : 1 provided the distributions Pθ corresponding to different values of θ are distinct. To see this let ḡθ1 = ḡθ2 . Then Pḡθ1 (gA) = Pḡθ2 (gA) and therefore Pθ1 (A) = Pθ2 (A) for all A, so that θ1 = θ2 . Lemma 6.1.1 Let g, g  be two transformations preserving Ω. Then the transformations g  g and g −1 defined by (g  g)x = g  (gx) and g(g −1 x) = x for all x∈X also preserve Ω and satisfy g  g = g  · ḡ and (g −1 ) = (ḡ)−1 . (6.4)  Proof. If the distribution of X is Pθ then that of gX is Pḡθ and that of g gX = g  (gX) is therefore Pḡ ḡθ . This establishes the first equation of (6.4); the proof of the second one is analogous. We shall say that the problem of testing H : θ ∈ ΩH against K : θ ∈ ΩK remains invariant under a transformation g if ḡ preserves both ΩH and ΩK , so that the equation ḡΩH = ΩH (6.5) holds in addition to (6.3). Let C be a class of transformations satisfying these two conditions, and let G be the smallest class of transformations containing C such that g, g  ∈ G implies that g  g and g −1 belong to G. Then G is a group of transformations, all of which by Lemma 6.1.1 preserve both Ω and ΩH . Any class C of transformations leaving the problem invariant can therefore be extended to a group G. It follows further from Lemma 6.1.1 that the class of induced transformations ḡ form a group Ḡ. The two equations (6.4) express the fact that Ḡ is a homomorphism of G. In the presence of symmetries in both sample and parameter space represented by the groups G and Ḡ, it is natural to restrict attention to tests φ which are also symmetric, that is, which satisfy φ(gx) = φ(x) for all x∈X and g ∈ G. (6.6) 214 6. Invariance A test φ satisfying (6.6) is said to be invariant under G. The restriction to invariant tests is a particular case of the principle of invariance formulated in Section 1.5. As was indicated there and in the examples above, a transformation g can be interpreted as a change of coordinates. From this point of view, a test is invariant if it is independent of the particular coordinate system in which the data are expressed.1 A transformation g, in order to leave a problem invariant, must in particular preserve the class A of measurable sets over which the distributions Pθ are defined. This means that any set A ∈ A is transformed into a set of A and is the image of such a set, so that gA and g −1 A both belong to A. Any transformation satisfying this condition is said to be bimeasurable. Since a group with each element g also contains g −1 its elements are automatically bimeasurable if all of them are measurable. If g  and g are bimeasurable, so are g  g and g −1 . The transformations of the group G above generated by a class C are therefore all bimeasurable provided this is the case for the transformations of C. 6.2 Maximal Invariants If a problem is invariant under a group of transformations, the principle of invariance restricts attention to invariant tests. In order to obtain the best of these, it is convenient first to characterize the totality of invariant tests. Let two points x1 , x2 be considered equivalent under G, x1 ∼ x2 ( mod G), if there exists a transformation g ∈ G for which x2 = gx1 . This is a true equivalence relation, since G is a group and the sets of equivalent points, the orbits of G, therefore constitute a partition of the sample space. (Cf. Appendix, Section A.1.) A point x traces out an orbit as all transformations g of G are applied to it; this means that the orbit containing x consists of the totality of points gx with g ∈ G. It follows from the definition of invariance that a function is invariant if and only if it is constant on each orbit. A function M is said to be maximal invariant if it is invariant and if M (x1 ) = M (x2 ) implies x2 = gx1 for some g ∈ G, (6.7) that is, if it is constant on the orbits but for each orbit takes on a different value. All maximal invariants are equivalent in the sense that their sets of constancy coincide. Theorem 6.2.1 Let M (x) be a maximal invariant with respect to G. Then, a necessary and sufficient condition for φ to be invariant is that it depends on x only through M (x); that is, that there exists a function h for which φ(x) = h[M (x)] for all x. 1 The relationship between this concept of invariance under reparametrization and that considered in differential geometry is discussed in Barndorff-Nielson, Cox and Reid (1986). 6.2. Maximal Invariants 215 Proof. If φ(x) = h[M (x)] for all x, then φ(gx) = h[M (gx)] = h[M (x)] = φ(x) so that φ is invariant. On the other hand, if φ is invariant and if M (x1 ) = M (x2 ), then x2 = gx1 for some g and therefore φ(x2 ) = φ(x1 ). Example 6.2.1 (i) Let x = (x1 , . . . , xn ), and let G be the group of translations gx = (x1 + c, . . . , xn + c), −∞ < c < ∞. Then the set of differences y = (x1 − xn , . . . , xn−1 − xn ) is invariant under G. To see that it is maximal invariant suppose that xi −xn = xi −xn for i = 1, . . . , n−1. Putting xn −xn = c, one has xi = xi +c for all i, as was to be shown. The function y is of course only one representation of the maximal invariant. Others are for example (x1 −x2 , x2 −x3 , . . . , xn−1 −xn ) or the redundant (x1 − x̄, . . . , xn − x̄). In the particular case that n = 1, there are no invariants. The whole space is a single orbit, so that for any two points there exists a transformation of G taking one into the other. In such a case the transformation group G is said to be transitive. The only invariant functions are then the constant functions φ(x) ≡ c. (ii) if G is the group of transformations gx = (cx1 , . . . , cxn ), c = 0, a special role is played by any zero coordinates. However, in statistical applications the set of points for which none of the coordinates is zero typically has probability 1; attention can then be restricted to this part of the sample space, and the set of ratios x1 /xn , . . . , xn−1 /xn is a maximal invariant. Without this restriction, two points x, x are equivalent with respect to the maximal invariant partition if among their coordinates there are the same number of zeros (if any), if these occur at the same places, and if for any two nonzero coordinates xi , xj the ratios xj /xi and xj /xi are equal. (iii) Let x = (x1 , . . . , xn ), and  let G be the group of all orthogonal transformations x = Γx of n-space. Then x2i is maximal invariant, that is, two points ∗ x and x can be transformed into each other by an orthogonal transformation if and only if they have the same distance from the origin. The proof of this is immediate if one restricts attention to the plane containing the points x, x∗ and the origin. Example 6.2.2 (i) Let x = (x1 , . . . , xn ), and let G be the set of n! permutations of the coordinates of x. Then the set of ordered coordinates (order statistics) x(1) ≤ · · · ≤ x(n) is maximal invariant. A permutation of the xi obviously does not change the set of values of the coordinates and therefore not the x(i) . On the other hand, two points with the same set of ordered coordinates can be obtained from each other through a permutation of coordinates. (ii) Let G be the totality of transformations xi = f (xi ), i = 1, . . . , n, such that f is continuous and strictly increasing, and suppose that attention can be restricted to the points that have n distinct coordinates. If the xi are considered as n points on the real line, any such transformation preserves their order. Conversely, if x1 , . . . , xn and x1 , . . . , xn are two sets of points in the same order, say xi1 < · · · < xin and xi1 < · · · < xin , there exists a transformation f satisfying the required conditions and such that xi = f (xi ) for all i. It can be defined for example as f (x) = x + (xi1 − xi1 ) for x ≤ xi1 , f (x) = x + (xin − xin ) for x ≥ xin , and to be linear between xik and xik+1 for k = 1, . . . , n − 1. A formal expression for 216 6. Invariance the maximal invariant in this case is the set of ranks (r1 , . . . , rn ) of (x1 , . . . , xn ). Here the rank ri of xi is defined through Xi = X(ri ) so that ri is the number of x’s ≤ xi . In particular, ri = 1 if xi is the smallest x, ri = 2 if it is the second smallest, and so on. Example 6.2.3 Let x be an n × s matrix (s ≤ n) of rank s, and let G be the group of linear transformations gx = xB, where B is any nonsingular s×s matrix. Then a maximal invariant under G is the matrix t(x) = x(xT x)−1 xT , where xT denotes the transpose of x. Here (xT x)−1 is meaningful because the s × s matrix xT x is nonsingular; see Problem 6.3. That t(x) is invariant is clear, since t(gx) = xB(B T xT xB)−1 B T xT = x(xT x)−1 xT = t(x). To see that t(x) is maximal invariant, suppose that x1 (xT1 x1 )−1 xT1 = x2 (xT2 x2 )−1 x2 . Since (xTi xi )−1 is positive definite, there exist nonsingular matrices Ci such that (xTi xi )−1 = Ci CiT and hence (x1 C1 )(x1 C1 )T = (x2 C2 )(x2 C2 )T . This implies the existence of an orthogonal matrix Q such that x2 C2 = x1 C1 Q and thus x2 = x1 B with B = C1 QC2−1 , as was to be shown. In the special case s = n, we have t(x) = I, so that there are no nontrivial invariants. This corresponds to the fact that in this case G is transitive, since any two nonsingular n× n matrices x1 and x2 satisfy x2 = x1 B with B = x−1 1 x2 . This result can be made more intuitive through a geometric interpretation. Consider the s-dimensional subspace S of Rn spanned by the s columns of x. Then P = x(xT x)−1 xT has the property that for any y in Rn , the vector P y is the projection of y onto S. (This will be proved in Section 7.2.) The invariance of P expresses the fact that the projection of y onto S is independent of the choice of vectors spanning S. To see that it is maximal invariant, suppose that the projection of every y onto the spaces S1 and S2 spanned by two different sets of s vectors is the same. Then S1 = S2 , so that the two sets of vectors span the same space. There then exists a nonsingular transformation taking one of these sets into the other. A somewhat more systematic way of determining maximal invariants is obtained by selecting, by means of a specified rule, a unique point M (x) on each orbit. Then clearly M (X) is maximal invariant. To illustrate this method, consider once more two of the earlier examples. Example 6.2.1(i) (continued). The orbit containing the point (a1 , . . . , an ) under the group of translations is the set (a1 + c, . . . , an + c), −∞ < c < ∞}, which is a line in En . (a) As representative point M (x) on this line, take its intersection with the hyperplane xn = 0. Since then an + c = 0, this point corresponds to the value c = −an and thus has coordinates (a1 − an , . . . , an−1 − an , 0). This leads to the maximal invariant (x1 − xn , . . . , xn−1 − xn ). 6.2. Maximal Invariants 217 (b)  An alternative point on the line is its intersection with the hyperplane xi = 0. Then c = −ā, and M (a) = (a1 − ā, . . . , an − ā). (c) The point need not be specified by an intersection property. It can for instance be taken as the point  on the line that is closest to the origin. Since the value of c minimizing (ai + c)2 is c = −ā, this leads to the same point as (b). Example 6.2.1(iii) (continued). The orbit containing the point (a1 , . . . , an ) under the group of orthogonal transformations is the hypersphere containing (a1 , . . . , an ) and with center at the origin. As representative point on this sphere, take its north pole, i.e. thepoint with a1 = · · · = an−1 = 0. The coordinates  of this point are (0, . . . , 0, a2i ) and hence lead to the maximal invariant x2i . (Note that in this example, the determination of the orbit is essentially equivalent to the determination of the maximal invariant.) Frequently, it is convenient to obtain a maximal invariant in a number of steps, each corresponding to a subgroup of G. To illustrate the process and a difficulty that may arise in its application, let x = (x1 , . . . , xn ), suppose that the coordinates are distinct, and consider the group of transformations gx = (ax1 + b, . . . , axn + b), a = 0, −∞ < b < ∞. xi Applying first the subgroup of translations = xi + b, a maximal invariant is y = (y1 , . . . , yn−1 ) with yi = xi − xn . Another subgroup consists of the scale changes xi = axi . This induces a corresponding change of scale in the y’s: yi = ayi , and a maximal invariant with respect to this group acting on the y-space is z = (z1 , . . . , zn−2 ) with zi = yi /yn−1 . Expressing this in terms of the x’s, we get zi = (xi − xn )/(xn−1 − xn ), which is maximal invariant with respect to G. Suppose now the process is carried out in the reverse order. Application first of the subgroup xi = axi yields as maximal invariant u = (u1 , . . . , un−1 ) with ui = xi /xn . However, the translations xi = xi + b do not induce transformations in u-space, since (xi + b)/(xn + b) is not a function of xi /xn . Quite generally, let a transformation group G be generated by two subgroups D and E in the sense that it is the smallest group containing D and E. Then G consists of the totality of products em dm . . . e1 d1 for m = 1, 2, . . . , with di ∈ D, ei ∈ E (i = 1, . . . , m).2 The following theorem shows that whenever the process of determining a maximal invariant in steps can be carried out at all, it leads to a maximal invariant with respect to G. Theorem 6.2.2 Let G be a group of transformations, and let D and E be two subgroups generating G. Suppose that y = s(x) is maximal invariant with respect to D, and that for any e ∈ E s(xi ) = s(x2 ) implies s(ex1 ) = s(ex2 ). ∗ (6.8) ∗ If z = t(y) is maximal invariant under the group E of transformations e defined by e∗ y = s(ex) 2 See Section A.1 of the Appendix. when y = s(x), 218 6. Invariance then z = t[s(x)] is maximal invariant with respect to G. Proof. To show that t[s(x)] is invariant, let x = gx, g = em dm · · · e1 d1 . Then t[s(x )] = t[s(em dm · · · e1 d1 x)] = t[e∗m s(dm · · · e1 d1 x)] = t[s(em−1 dm−1 · · · e1 d1 x)], and the last expression can be reduced by induction to t[s(x)]. To see that t[s(x)] is in fact maximal invariant, suppose that t[s(x )] = t[s(x)]. Setting y  = s(x ), y = s(x), one has t(y  ) = t(y), and since t(y) is maximal invariant with respect to E ∗ , there exists e∗ such that y  = e∗ y. Then s(x ) = e∗ s(x) = s(ex), and by the maximal invariance of s(x) with respect to D there exists d ∈ D such that x = dex. Since de is an element of G this completes the proof. Techniques for obtaining the distribution of maximal invariants are discussed by Andersson (1982), Eaton (1983, 1989), Farrell (1985), Wijsman (1990) and Anderson (2003). 6.3 Most Powerful Invariant Tests In the presence of symmetries, one may wish to restrict attention to invariant tests, and it then becomes of interest to determine the most powerful invariant test. The following is a simple example. Example 6.3.1 Let X1 , . . . , Xn be i.i.d. on (0, 1) and consider testing the hypothesis H0 that the the common distribution of the X’s is uniform on (0, 1) against the two alternatives H1 : p1 (x1 , . . . , xn ) = f (x1 ) · · · f (xn ) and p2 (x1 , . . . , xn ) = f (1 − x1 ) · · · f (1 − xn ) , where f is a fixed (known) density. (i) This problem remains invariant under the 2 element group G consisting of the transformations g : xi = 1 − xi , i = 1, . . . , n and the identity transformation xi = xi for i = 1, . . . , n. (ii) The induced transformation ḡ is the space of alternatives takes p1 into p2 and p2 into p1 . (iii) A test φ(x1 , . . . , xn ) remains invariant under G if and only if φ(x1 , . . . , xn ) = φ(1 − x1 , . . . , 1 − xn ) . (iv) There exists a UMP invariant test (i.e. an invariant test which is simultaneously most powerful against both p1 and p2 ), and it rejects H0 when the average p̄(x1 , . . . , xn ) = is sufficiently large. 1 [p1 (x1 , . . . , xn ) + p2 (x1 , . . . , xn )] 2 6.3. Most Powerful Invariant Tests 219 We leave the proof of (i)-(iii) to Problem 6.5. To prove (iv), note that any invariant test satisfies Ep1 [φ(X1 , . . . , Xn )] = Ep2 [φ(X1 , . . . , Xn )] = Ep̄ [φ(X1 , . . . , Xn )] . Therefore, maximizing the power against p1 or p2 is equivalent to maximizing the power under p̄, and the result follows from the Neyman-Pearson Lemma. This example is a special case of the following result. Theorem 6.3.1 Suppose the problem of testing Ω0 against Ω1 remains invariant under a finite group G = {g1 , . . . , gN } and that Ḡ is transitive over Ω0 and over Ω1 . Then there exists a UMP invariant test of Ω0 against Ω1 , and it rejects Ω0 when N pḡi θ1 (x)/N (6.9) i=1 N i=1 pḡi θ0 (x)/N is sufficiently large, where θ0 and θ1 are any elements of Ω0 and Ω1 , respectively. The proof is exactly analogous to that of the preceding example; see Problem 6.6. The results of the previous section provide an alternative approach to the determination of most powerful invariant tests. By Theorem 6.2.1, the class of all invariant functions can be obtained as the totality of functions of a maximal invariant M (x). Therefore, in particular the class of all invariant tests is the totality of tests depending only on the maximal invariant statistic M . The latter statement, while correct for all the usual situations, actually requires certain qualifications regarding the class of measurable sets in M -space. These conditions will be discussed at the end of the section; they are satisfied in the examples below. Example 6.3.2 Let X = (X1 , . . . , Xn ), and suppose that the density of X is fi (x1 − θ, . . . , xn − θ) under Hi (i = 0, 1), where θ ranges from −∞ to ∞. The problem of testing H0 against H1 is invariant under the group G of transformations gx = (x1 + c, . . . , xn + c), −∞ < c < ∞. which in the parameter space induces the transformations ḡθ = θ + c. By Example 6.2.1, a maximal invariant under G is Y = (X1 −Xn , . . . , Xn−1 −Xn ). The distribution of Y is independent of 0 and under Hi has the density  ∞ fi (y1 + z, . . . , yn−1 + z, z) dz. −∞ When referred to Y , the problem of testing H0 against H1 therefore becomes one of testing a simple hypothesis against a simple alternative. The most powerful test is then independent of θ, and therefore UMP among all invariant tests. Its rejection region by the Neyman–Pearson lemma is ∞ ∞ ∞ ∞ f1 (y1 + z, . . . , yn−1 + z, z) dz = f0 (y1 + z, . . . , yn−1 + z, z) dz ∞ −∞ ∞ −∞ f1 (x1 + u, . . . , xn + u) du f0 (x1 + u, . . . , xn + u) du > C. (6.10) 220 6. Invariance A general theory of separate families of hypotheses (in which the family K of alternatives does not adjoin the hypothesis H but, as above, is separated from it) was initiated by Cox (1961, 1962). A bibliography of the subject is given in Pereira (1977); see also Loh (1985), Pace and Salvan (1990) and Rukhin (1993). Example 6.3.2 illustrates the fact, also utilized in Theorem 6.3.1, that if the group Ḡ is transitive over both Ω0 and Ω1 , then the problem reduces to one of testing a simple hypothesis against a simple alternative, and a UMP invariant test is then obtained by the Neyman-Pearson Lemma. Note also the close similarity between Theorem 6.3.1 and Example 6.3.2 shown by a comparison of (6.9) and the right side of (6.10), where the summation in (6.9) is replaced by integration with respect to Lebesgue measure. Before applying invariance, it is frequently convenient first to reduce the data to a sufficient statistic T . If there exists a test φ0 (T ) that is UMP among all invariant tests depending only on T , one would like to be able to conclude that φ0 (T ) is also UMP among all invariant tests based on the original X. Unfortunately, this does not follow, since it is not clear that for any invariant test based on X there exists an equivalent test based on T , which is also invariant. Sufficient conditions for φ0 (T ) to have this property are provided by Hall, Wijsman, and Ghosh (1965) and Hooper (1982a), and a simple version of such a result (applicable to Examples 6.3.3 and 6.3.4 below) will be given by Theorem 6.5.3 in Section 6.5. For a review and clarification of this and later work on invariance and sufficiency see Berk, Nogales, and Oyola (1996), Nogales and Oyola (1996) and Nogales, Oyola and Pérez (2000). Example 6.3.3 If X1 , . . . , Xn is a sample from N (ξ, σ 2 ), the hypothesis H : σ ≥ σ0 remains invariant under the transformations Xi = Xi + c, −∞ < c < ∞. In terms of the sufficient statistics Y = X̄, S 2 = Σ(Xi − X̄)2 these transformations become Y  = Y + c, (S 2 ) = S 2 , and a maximal invariant is S 2 . The class of invariant tests is therefore the class of tests depending on S 2 . It follows from Theorem 3.4.1 that there exists a UMP invariant test, with rejection region Σ(Xi − X̄)2 ≤ C. This coincides with the UMP unbiased test (6.11). Example 6.3.4 If X1 , . . . , Xm and Y1 , . . . , Yn are samples from N (ξ, σ 2 ) and N (η, τ 2 ), a set of sufficient statistics is T1 = X̄, T2 = Ȳ , T3 = Σ(Xi − X̄)2 , and T4 = Σ(Yj − Ȳ )2 . The problem of testing H : τ 2 /σ 2 ≤ ∆0 remains invariant under the transformations T1 = T1 + c1 , T2 = T2 + c2 , T3 = T3 , T4 = T4 , −∞ < c1 , c2 < ∞, and also under a common change of scale of all four variables. A maximal invariant with respect to the first group is (T3 , T4 ). In the space of this maximal invariant, the group of scale changes induces the transformations T3 = cT3 , T4 = cT4 , 0 < c, which has as maximal invariant the ratio T4 /T3 . The statistic Z = [T42 /(n − 1)] ÷ [T32 /(m − 1)] on division by ∆ = τ 2 /σ 2 has an F -distribution with density given by (5.21), so that the density of Z is 1 c(∆)z 2 (n−3)  1 (m+n−2) , 2 n−1 z ∆+ m−1 z > 0. For varying ∆, these densities constitute a family with monotone likelihood ratio, so that among all tests of H based on Z, and therefore among all invariant tests, 6.3. Most Powerful Invariant Tests 221 there exists a UMP one given by the rejection region Z > C. This coincides with the UMP unbiased test (5.20). Example 6.3.5 In the method of paired comparisons for testing whether a treatment has a beneficial effect, the experimental material consists of n pairs of subjects. From each pair, a subject is selected at random for treatment while the other serves as control. Let Xi be 1 or 0 as for the ith pair the experiment turns out in favor of the treated subject or the control, and let pi = P {Xi = 1}. The hypothesis of no effect, H : pi = 12 for i = 1, . . . , n, is to be tested against the alternatives that pi > 12 for all i. The problem remains invariant under all permutations of the n variables X1 , . . . , Xn , and a maximal invariant under this group is the total number of successes X = X1 + · · · + Xn . The distribution of X is  pi pi 1 ··· k , P {X = k} = q1 · · · qn qi1 qik  where qi = 1 − pi and where the summation extends over all nk choices of subscripts i1 < · · · < ik . The most powerful invariant test against an alternative (p1 , . . . , pn ) rejects H when pi 1  pi1 · · ·  k > C. f (k) = n  qi1 qik k To see that f is an increasing function of k, note that ai = pi /qi > 1, and that   aj ai1 · · · aik = (k + 1) ai1 · · · aik+1 j and  ai1 · · · aik = (n − k)  ai1 · · · aik1 . j Here, in both equations, the second summation on the left-hand side extends over all subscripts i1 < · · · < ik of which none is equal to j, and the summation on the right-hand side extends over all subscripts i1 < · · · < ik+1 and i1 < · · · < ik respectively without restriction. Then  1  1 n f (k + 1) =  n aj ai1 · · · aik ai1 · · · aik+1 = (n − k) k j k+1 1  > n ai1 · · · aik = f (k), k as was to be shown. Regardless of the alternative chosen, the test therefore rejects when k > C, and hence is UMP invariant. If the ith comparison is considered plus or minus as Xi is 1 or 0, this is seen to be another example of the sign test. (Cf. Example 3.8.1 and Section 4.9.) Sufficient statistics provide a simplification of a problem by reducing the sample space; this process involves no change in the parameter space. Invariance, on the other hand, by reducing the data to a maximal invariant statistic M , whose distribution may depend only on a function of the parameter, typically also shrinks the parameter space. The details are given in the following theorem. 222 6. Invariance Theorem 6.3.2 If M (x) is invariant under G, and if υ(θ) maximal invariant under the induced group Ḡ, then the distribution of M (X) depends only on v(θ). Proof. Let υ(θ1 ) = υ(θ2 ). Then θ2 = ḡθ1 , and hence Pθ2 {M (X) ∈ B} = Pḡθ1 {M (X) ∈ B} = Pθ1 {M (gX) ∈ B} = Pθ1 {M (X) ∈ B}. This result can be paraphrased by saying that the principle of invariance identifies all parameter points that are equivalent with respect to Ḡ. In application, for instance in Examples 6.3.3 and 6.3.4, the maximal invariants M (x) and δ = v(θ) under G and Ḡ are frequently real-valued, and the family of probability densities pδ (m) of M has monotone likelihood ratio. For testing the hypothesis H : δ ≤ δ0 there exists then a UMP test among those depending only on M , and hence a UMP invariant test. Its rejection region is M ≥ C, where  ∞ Pδ0 (m) dm = α. (6.11) C Consider this problem now as a two-decision problem with decisions d0 and d1 of accepting or rejecting H, and a loss function L(θ, di ) = Li (θ). Suppose that Li (θ) depends only on the parameter δ, Li (θ) = Li (δ) say, and satisfies L1 (δ) − L0 (δ) > <0 as δ < > δ0 . (6.12) It then follows from Theorem 3.4.2 that the family of rejection regions M ≥ C(α), as α varies from 0 to 1, forms a complete family of decision procedures among those depending only on M , and hence a complete family of invariant procedures. As before, the choice of a particular significance level α can be considered as a convenient way of specifying a test from this family. At the beginning of the section it was stated that the class of invariant tests coincides with the class of tests based on a maximal invariant statistic M = M (X). However, a statistic is not completely specified by a function, but requires also specification of a class B of measurable sets. If in the present case B is the class of all sets B for which M −1 (B) ∈ A, the desired statement is correct. For let φ(x) = ψ[M (x)] and φ by A-measurable, and let C be a Borel set on the line. Then φ−1 (C) = M −1 [ψ −1 (C)] ∈ A and hence ψ −1 (C) ∈ B, so that ψ is B-measurable and φ(x) = ψ[M (x)] is a test based on the statistic M . In most applications, M (x) is a measurable function taking on values in a Euclidean space and it is convenient to take B as the class of Borel sets. If φ(x) = ψ[M (x)] is then an arbitrary measurable function depending only on M (x), it is not clear that ψ(m) is necessarily B-measurable. This measurability can be concluded if X is also Euclidean with A the class of Borel sets, and if the range of M is a Borel set. We shall prove it here only under the additional assumption (which in applications is usually obvious, and which will not be verified explicitly in each case) that there exists a vector-valued Borel-measurable function Y (x) such that [M (x), Y (x)] maps X onto a Borel subset of the product space M × Y, that this mapping is 1 : 1, and that the inverse mapping is also Borel-measurable. Given any measurable function φ of x, there exists then a measurable function φ of (m, y) such that φ(x) ≡ φ [M (x), Y (x)]. If φ depends only on M (x), then φ depends only on m, so that φ (m, y) = ψ(m) say, and ψ is a measurable 6.4. Sample Inspection by Variables 223 function of m.3 In Example 6.2.1(i) for instance, where x = (x1 , . . . xn ) and M (x) = (x1 − xn , . . . , xn−1 − xn ), the function Y (x) can be taken as Y (x) = xn . 6.4 Sample Inspection by Variables A sample is drawn from a lot of some manufactured product in order to decide whether the lot is of acceptable quality. In the simplest case, each sample item is classified directly as satisfactory or defective (inspection by attributes), and the decision is based on the total number of defectives. More generally, the quality of an item is characterized by a variable Y (inspection by variables), and an item is considered satisfactory if Y exceeds a given constant u. The probability of a defective is then p = P {Y ≤ u} and the problem becomes that of testing the hypothesis H : p ≥ p0 . As was seen in Example 3.8.1, no use can be made of the actual value of Y unless something is known concerning the distribution of Y . In the absence of such information, the decision will be based, as before, simply on the number of defectives in the sample. We shall consider the problem now under the assumption that the measurements Y1 , . . . , Yn constitute a sample from N (η, σ 2 ). Then    u u − η 1 1 √ p= , exp − 2 (y − η)2 dy = Φ 2σ σ 2πσ −∞ where  y Φ(y) = −∞  1 √ exp − 12 t2 dt 2π denotes the cumulative distribution function of a standard normal distribution, and the hypothesis H becomes (u − η)/σ ≥ Φ−1 (p0 ). In terms of the variables X1 = Yi − u, which have mean ξ = η − u and variance σ 2 , this reduces to H: ξ ≤ θ0 σ with θ0 = −Φ−1 (p0 ). This hypothesis, which was considered in Section 5.2, for θ0 = 0, occurs also in other contexts. It is appropriate when one is interested in the mean ξ of a normal distribution, expressed in σ units rather than on a fixed scale. For  testing H, attention can be restricted to the pair of variables X̄ and S= (Xi − X̄)2 , since they form a set of sufficient statistics for (ξ, σ), which satisfy the conditions of Theorem 6.5.3 of the next section. These variables are independent, the distribution of X̄ being N (ξ, σ 2 /n) and that of S/σ being χn−1 . Multiplication of X̄ and S by a common constant c > 0 transforms the parameters into ξ  = cξ, σ  = cσ, so that ξ/σ and hence the problem of testing H remain 3 The last statement follows, for example, from Theorem 18.1 of Billingsley (1995). 224 6. Invariance invariant. A maximal invariant under these transformations is x̄/s or √ nx̄ t= √ , s/ n − 1 the distribution of which depends only on the maximal invariant in the parameter space θ = ξ/σ (cf. Section 5.2). Thus, the invariant tests are those depending only on t, and it remains to find the most powerful test of H : θ ≤ θ0 within this class. The probability density of t is (Problem 5.3) ! 2 " 1  ∞  1 w 1 pδ (t) = C exp − −δ w 2 (n−2) exp − 12 w dw, t 2 n−1 0 √ where δ = nθ is the noncentrality parameter, and this will now be shown to constitute a family with monotone likelihood ratio. To see that the ratio  , 2  1 ∞ 1 w exp − − δ t w 2 (n−2) exp(− 12 w) dw 1 2 n−1 0  r(t) = , 2  1 ∞ 1 w exp − − δ t w 2 (n−2) exp(− 12 w) dw 0 2 n−1 0 is an increasing function of t for δ0 < δ1 , suppose first that t < 0 and let υ = −t w/(n − 1) . The ratio then becomes proportional to ∞ 0   (n−1)υ 2 dv f (υ) exp −(δ1 −δ0 )υ− 2t2   ∞ (n−1)υ 2 dv 2 0 f (υ) exp − 2t = exp[−(δ1 − δ0 )υ]gt2 (υ) dv where f (υ) = exp(−δ0 υ)υ n−1 exp(−υ 2 /2) and # $ 2 f (υ) exp − (n−1)υ 2 2t # $ . gt2 (υ) = ∞ (n−1)z 2 f (z) exp − 2t2 dz 0 Since the family of probability densities gt2 (υ) is a family with monotone likelihood ratio, the integral of exp[−(δ1 − δ0 )υ] with respect to this density is a decreasing function of t2 (Problem 3.39), and hence an increasing function of t for t < 0. Similarly one finds that r(t) is an increasing function of t for t > 0 by making the transformation v = t w/(n − 1). By continuity it is then an increasing function of t for all t. There exists therefore a UMP invariant test of H : ξ/σ ≤ θ0 , which rejects when t > C, where C is determined by (6.11). In terms of the original variables Yi the rejection region of the UMP invariant test of H : p ≥ p0 becomes √ n(ȳ − u) > C. (6.13)  (yi − ȳ)2 /(n − 1) If the problem is considered as a two-decision problem with losses L0 (p) and L1 (p) for accepting or rejecting p ≥ p0 , which depend only on p and satisfy the 6.5. Almost Invariance 225 condition corresponding to (6.12), the class of tests (6.13) constitutes a complete family of invariant procedures as C varies from −∞ to ∞. Consider next the comparison of two products on the basis of samples X1 , . . . , Xm ; Y1 , . . . , Yn from N (ξ, σ 2 ) and N (η, σ 2 ). If  u − η u−ξ p=Φ , , π=Φ σ σ one wishes to test the hypothesis p ≤ π, which is equivalent to H : η ≤ ξ.   (Xi − X̄)2 + (Yj − Ȳ )2 are a set of sufficient The statistics X̄, Ȳ , and S = statistics for ξ, η, σ. The problem remains invariant under the addition of an arbitrary common constant to X̄ and Ȳ , which leaves Ȳ − X̄ and S as maximal invariants. It is also invariant under multiplication of X̄, Ȳ , and S, and hence of Ȳ − X̄ and S, by a common positive constant, which reduces the data to the maximal invariant (Ȳ − X̄)/S. Since , 1 (ȳ − x̄)/ m + n1 √ t= s/ m + n − 2 √ has a noncentral t-distribution with noncentrality parameter δ = mn(η − ξ)/ √ m + nσ, the UMP invariant test of H : η − ξ ≤ 0 rejects when t > C. This coincides with the UMP unbiased test (5.27). Analogously, the corresponding two-sided test (5.30), with rejection region |t| ≥ C, is UMP invariant for testing the hypothesis p = π against the alternatives p = π (Problem 6.18). 6.5 Almost Invariance Let G be a group of transformations leaving a family P = {Pθ , θ ∈ ⊗} of distributions of X invariant. A test φ is said to be equivalent to an invariant test if there exists an invariant test φ such that φ(x) = ψ(x) for all x except possibly on a P-null set N ; φ is said to be almost invariant with respect to G if φ(gx) = φ(x) for all x ∈ X − Ng , g∈G (6.14) where the exceptional null set Ng is permitted to depend on g. This concept is required for investigating the relationship of invariance to unbiasedness and to certain other desirable properties. In this connection it is important to know whether a UMP invariant test is also UMP among almost invariant tests. This turns out to be the case under assumptions which are made precise in Theorem 6.5.1 below and which are satisfied in all the usual applications. If φ is equivalent to an invariant test, then φ(gx) = φ(x) for all x ∈ / N ∪ g −1 N . −1 Since Pθ (g N ) = Pḡθ (N ) = 0, it follows that φ is then almost invariant. The following theorem gives conditions under which conversely any almost invariant test is equivalent to an invariant one. Theorem 6.5.1 Let G be a group of transformations of X , and let A and B be σ-fields of subsets of X and G such that for any set A ∈ A the set of pairs (x, g) 226 6. Invariance for which gx ∈ A is measurable A × B. Suppose further that there exists a σ-finite measure ν over G such that ν(B) = 0 implies ν(Bg) = 0 for all g ∈ G. Then any measurable function that is almost invariant under G (where “almost” refers to some σ-finite measure µ) is equivalent to an invariant function. Proof. Because of the measurability assumptions, the function φ(gx) considered as a function of the two variables x and g is measurable A × B. It follows that φ(gx) − φ(x) is measurable A × B, and so therefore is the set S of points (x, g) with φ(gx) = φ(x). If φ is almost invariant, any section of S with fixed g is a µ-null set. By Fubini’s theorem (Theorem 2.2.4), there exists therefore a µ-null set N such that for all x ∈ X − N φ(gx) = φ(x) a.e. ν. Without loss of generality suppose that ν(G) = 1, and let A be the set of points x for which  φ(g  x) dν(g  ) = φ(gx) a.e. ν. If % % % % f (x, g) = %% φ(g  x) dν(g  ) − φ(gx)%% then A is the set of points x for which  f (x, g) dν(g) = 0. Since this integral is a measurable function of x, it follows that A is measurable. Let  φ(gx)dν(g) if x ∈ A, ψ(x) = 0 if x ∈ / A. Then ψ is measurable and ψ(x) = φ(x) for x ∈ / N , since φ(gx) = φ(x) a.e. ν implies that φ(g  x) dν(g  ) = φ(x) and that x ∈ A. To show that ψ is invariant it is enough to prove that the set A is invariant. For any point x ∈ A, the function φ(gx) is constant except on a null subset Nx of G. Then φ(ghx) has the same constant value for all g ∈ / Nx h−1 , which by assumption is again a ν-null set; and hence hx ∈ A, which completes the proof. Additional results concerning the relation of invariance and almost invariance are given by Berk and Bickel (1968) and Berk (1970). In particular, the basic idea of the following example is due to Berk (1970). Example 6.5.1 (Counterexample) Let Z, Y1 , . . . , Yn be independently distributed as N (θ, 1), and consider the 1 : 1 transformations yi = yi (i = 1, . . . , n) and z  = z except for a finite number of points a1 , . . . , ak for which ai = aji , for some permutation (j1 , . . . , jk ) of (1, . . . , k). If the group G is generated by taking for (a1 , . . . , ak ), k = 1, 2, . . . , all finite sets and for (j1 , . . . , jk ) all permutations of (1, . . . , k), then (z, y1 , . . . , yn ) is almost invariant It is however not equivalent to an invariant function, since (y1 , . . . , yn ) is maximal invariant. 6.5. Almost Invariance 227 Corollary 6.5.1 Suppose that the problem of testing H : θ ∈ ω against K : θ ∈ Ω − ω remains invariant under G and that the assumptions of Theorem 6.5.1 hold. Then if φ0 is UMP invariant, it is also UMP within the class of almost invariant tests. Proof. If φ is almost invariant, it is equivalent to an invariant test ψ by Theorem 6.5.1. The tests φ and ψ have the same power function, and hence φ0 is uniformly at least as powerful as φ. In applications, P is usually a dominated family, and µ any σ-finite measure equivalent to P (which exists by Theorem A.4.2 of the Appendix). If φ is almost invariant with respect to P, it is then almost invariant with respect to µ and hence equivalent to an invariant test. Typically, the sample space X is an ndimensional Euclidean space, A is the class of Borel sets, and the elements of G are transformations of the form y = f (x, τ ), where τ ranges over a set of positive measure in an m-dimensional space and f is a Borel-measurable vector-valued function of m + n variables. If B is taken as the class of Hotel sets in m-space the measurability conditions of the theorem are satisfied. The requirement that for all g ∈ G and B ∈ B ν(B) = 0 implies ν(Bg) = 0 (6.15) g ∈ G, (6.16) is satisfied in particular when ν(Bg) = ν(B) for all B ∈ B. The existence of such a right invariant measure is guaranteed for a large class of groups by the theory of Haar measure. (See, for example, Eaton (1989).) Alternatively, it is usually not difficult to check the condition (6.15) directly. Example 6.5.2 Let G be the group of all nonsingular linear transformations of n-space. Relative to a fixed coordinate system the elements of G can be represented by nonsingular n × n matrices A = (aij ), A = (aij ), . . . with the matrix product serving as the group product of two such elements. The σ-field B can be taken to be the class of Borel sets in the space of the n2 elements of the matrices, and the measure ν can be taken as Lebesgue measure over B. Consider now a set S of matrices with ν(S) = 0, and the set S ∗ of matrices A A with A ∈ S and A fixed. If a = max |aij |, C  = A A, and C  = A A, the inequalities |aij − aij | ≤  for all i, j imply |cij − cij | ≤ na. Since a set has ν-measure zero if and only if it can be covered by a union of rectangles whose total measure does not exceed any given  > 0, it follows that ν(S ∗ ) = 0, as was to be proved. In the preceding chapters, tests were compared purely in terms of their power functions (possibly weighted according to the seriousness of the losses involved). Since the restriction to invariant tests is a departure from this point of view, it is of interest to consider the implications of applying invariance to the power functions rather than to the tests themselves. Any test that is invariant or almost invariant under a group G has a power function which is invariant under the group Ḡ induced by G in the parameter space. To see that the converse is in general not true, let X1 , X2 , X3 be independently, normally distributed with mean ξ and variance σ 2 , and consider the hypothesis 228 6. Invariance σ ≥ σ0 . The test with rejection region |X2 − X1 | > k when X̄ < 0, |X3 − X2 | > k when X̄ ≥ 0 is not invariant under the group G of transformations Xi = Xi + c, but its power function is invariant under the associated group Ḡ. The two properties, almost invariance of a test φ and invariance of its power function, become equivalent if before the application of invariance considerations the problem is reduced to a sufficient statistic whose distributions constitute a boundedly complete family. Lemma 6.5.1 Let the family P T = {PθT , θ ∈ Ω} of distributions of T be boundedly complete, and let the problem of testing H : θ ∈ ΩH remain invariant under a group G of transformations of T . Then a necessary and sufficient condition for the power function of a test ψ(t) to be invariant under the induced group Ḡ over Ω is that ψ(t) is almost invariant under G. Proof. For all θ ∈ Ω we have Eḡθ ψ(T ) = Eθ ψ(gT ). If ψ is almost invariant, Eθ ψ(T ) = Eθ ψ(gT ) and hence Eḡθ ψ(T ) = Eθ ψ(T ), so that the power function of ψ is invariant. Conversely, if Eθ ψ(T ) = Eḡθ ψ(T ), then Eθ ψ(T ) = Eθ ψ(gT ), and by the bounded completeness of P T , we have ψ(gt) = ψ(t) a.e. P T . As a consequence, it is seen that UMP almost invariant tests also possess the following optimum property. Theorem 6.5.2 Under the assumptions of Lemma 6.5.1, let v(θ) be maximal invariant with respect to Ḡ, and suppose that among the tests of H based on the sufficient statistic T there exists a UMP almost invariant one, say ψ0 (t). Then ψ0 (t) is UMP in the class of all tests based on the original observations X, whose power function depends only on v(θ). Proof. Let φ(x) be any such test, and let ψ(t) = E[φ(X)|t]. The power function of ψ(t), being identical with that of φ(x), depends then only on v(θ), and hence is invariant under Ḡ. It follows from Lemma 6.5.1 that ψ(t) is almost invariant under G, and ψ0 (t) is uniformly at least as powerful as ψ(t) and therefore as φ(x). Example 6.5.3 For the hypothesis τ 2 ≤ σ 2 concerning the variances of two normal distributions, the statistics (X̄, Ȳ , Sx2 , SY2 ) constitute a complete set of sufficient statistics. It was shown in Example 6.3.4 that there exists a UMP invariant test with respect to a suitable group G, which has rejection region 2 SY2 /SX > C0 . Since in the present case almost invariance of a test with respect to G implies that it is equivalent to an invariant one (Problem 6.21), Theorem 6.5.2 is applicable with v(θ) = ∆ = τ 2 /σ 2 , and the test is therefore UMP among all tests whose power function depends only on ∆. Theorem 6.5.1 makes it possible to establish a simple condition under which reduction to sufficiency before the application of invariance is legitimate. 6.6. Unbiasedness and Invariance 229 Theorem 6.5.3 Let X be distributed according to Pθ , θ ∈ Ω, and let T be sufficient for θ. Suppose G leaves invariant the problem of testing H : θ ∈ ΩH , and that T satisfies T (x1 ) = T (x2 ) implies T (gx1 ) = T (gx2 ) for all g ∈ G, so that G induces a group G̃ of transformations of T -space through g̃T (x) = T (gx). (i) If ϕ(x) is any invariant test of H, there exists an almost invariant test ψ based on T , which has the same power function as ϕ. (ii) If in addition the assumptions of Theorem 6.5.1 are satisfied, the test ψ of (i) can be taken to be invariant. (iii) If there exists a test ψ0 (T ) which is UMP among all G̃-invariant tests based on T , then under the assumptions of (ii), ψ0 , is also UMP among all G-invariant tests based on X. This theorem justifies the derivation of the UMP invariant tests of Examples 6.3.3 and 6.3.4. Proof. (i): Let ψ(t) = E[ϕ(X)|t]. Then ψ has the same power function as ϕ. To complete the proof, it suffices to show that ψ(t) is almost invariant, i.e. that ψ(g̃t) = ψ(t) (a.e. P T ). It follows from (1) that Eθ [ϕ(gX)|g̃t] = Eḡθ [ϕ(X)|t] (a.e. Pθ ). Since T is sufficient, both sides of this equation are independent of θ. Furthermore ϕ(gx) = ϕ(x) for all x and g, and this completes the proof. Part (ii) follows immediately from (i) and Theorem 6.5.1, and part (iii) from (ii). 6.6 Unbiasedness and Invariance The principles of unbiasedness and invariance complement each other in that each is successful in cases where the other is not. For example, there exist UMP unbiased tests for the comparison of two binomial or Poisson distributions, problems to which invariance considerations are not applicable. UMP unbiased tests also exist for testing the hypothesis σ = σ0 against σ = σ0 in a normal distribution, while invariance does not reduce this problem sufficiently far. Conversely, there exist UMP invariant tests of hypotheses specifying the values of more than one parameter (to be considered in Chapter 7) but for which the class of unbiased tests has no UMP member. There are also hypotheses, for example the one-sided hypothesis ξ/σ ≤ θ0 in a univariate normal distribution or ρ ≤ ρ0 in a bivariate one (Problem 6.19) with θ0 , ρ0 = 0, where a UMP invariant test exists but the existence of a UMP unbiased test does not follow by the methods of Chapter 5 and is an open question. On the other hand, to some problems both principles have been applied successfully. These include Student’s hypotheses ξ ≤ ξ0 and ξ = ξ0 concerning the mean 230 6. Invariance of a normal distribution, and the corresponding two sample problems η − ξ ≤ ∆0 and η − ξ = ∆0 when the variances of the two samples are assumed equal. Other examples are the one-sided hypotheses σ 2 ≥ σ02 and τ 2 /σ 2 ≥ ∆0 concerning the variances of one or two normal distributions. The hypothesis of independence ρ = 0 in a bivariate normal distribution is still another case in point (Problem 6.19). In all these examples the two optimum procedures coincide. We shall now show that this is not accidental but is the case whenever the UMP invariant test is UMP also among all almost invariant tests and the UMP unbiased test is unique. In this sense, the principles of unbiasedness and of almost invariance are consistent. Theorem 6.6.1 Suppose that for a given testing problem there exists a UMP unbiased test φ∗ which is unique (up to sets of measure zero), and that there also exists a UMP almost invariant test with respect to some group G. Then the latter is also unique (up to sets of measure zero), and the two tests coincide a.e. Proof. If U (α) is the class of unbiased level-α tests, and if g ∈ G, then φ ∈ U (α) if and only if φg ∈ U (α).4 Denoting the power function of the test φ by βφ (θ), we thus have βφ∗ g (θ) = βφ∗ (ḡθ) = sup βφ (ḡθ) = sup βφg (θ) φ∈U (α) = sup φg∈U (α) φ∈U (α) βφg (θ) = βφ∗ (θ). ∗ It follows that φ and φ∗ g have the same power function, and, because of the uniqueness assumption, that φ∗ is almost invariant. Therefore, if φ is UMP almost invariant, we have βφ (θ) ≥ βφ∗ (θ) for all θ. On the other hand, φ is unbiased, as is seen by comparing it with the invariant test φ(x) ≡ α, and hence βφ (θ) ≤ βφ∗ (θ) for all θ. Since φ and φ∗ therefore have the same power function, they are equal a.e. because of the uniqueness of φ∗ , as was to be proved. This theorem provides an alternative derivation for some of the tests of Chapter 5. In Theorem 4.4.1, the existence of UMP unbiased tests was established for oneand two-sided hypotheses concerning the parameter θ of the exponential family (4.10). For this family, the statistics (U, T ) are sufficient and complete, and in terms of these statistics the UMP unbiased test is therefore unique. Convenient explicit expressions for some of these tests, which were derived in Chapter 5, can instead be obtained by noting that when a UMP almost invariant test exists, the same test by Theorem 6.6.1 must also be UMP unbiased. This proves for example that the tests of Examples 6.3.3 and 6.3.4 are UMP unbiased. The principles of unbiasedness and invariance can be used to supplement each other in cases where neither principle alone leads to a solution but where they do so when applied in conjunction. As an example consider a sample X1 , . . . , Xn from N (ξ, σ 2 ) and the problem of testing H : ξ/σ = θ0 = 0 against the two-sided alternatives that ξ/σ = θ0 . Here sufficiency and invariance reduce the problem  √ to the consideration of t = nx̄/ (xi − x̄)2 /(n − 1). The distribution of this √ statistic is the noncentral t-distribution with noncentrality parameter δ = nξ/σ and n − 1 degrees of freedom. For varying δ, the family of these distributions can 4 φg denotes the critical function which assigns to x the value φ(gx). 6.6. Unbiasedness and Invariance 231 be shown to be STP∞ . [Karlin (1968, pp. 118–119; see Problem 3.50] and hence in particular STP3 . It follows by Problem 6.42 that among all tests of H based on t, there exists a UMP unbiased one with acceptance region C1 ≤ t ≤ C2 , where C1 , C2 are determined by the conditions % % ∂Pδ {C1 ≤ t ≤ C2 } %% Pδ0 {C1 ≤ t ≤ C2 } = 1 − α and = 0. % ∂δ % δ=δ0 In terms of the original observations, this test then has the property of being UMP among all tests that are unbiased and invariant. Whether it is also UMP unbiased without the restriction to invariant tests is an open problem. An analogous example occurs in the testing of the hypotheses H : ρ = ρ0 and H  : ρ1 ≤ ρ ≤ ρ2 against two-sided alternatives on the basis of a sample from a bivariate normal distribution with correlation coefficient ρ. (The testing of ρ ≤ ρ0 against ρ > ρ0 is treated in Problem 6.19.) The distribution of the sample correlation coefficient has not only monotone likelihood ratio as shown in Problem 6.19, but is in fact STP∞ . [Karlin (1968, Section 3.4)]. Hence there exist tests of both H and H  which are UMP among all tests that are both invariant and unbiased. Another case in which the combination of invariance and unbiasedness appears to offer a promising approach is the Behrens–Fisher problem. Let X1 , . . . , Xm and Y1 , . . . , Yn be samples from normal distributions N (ξ, σ 2 ) and N (η, τ 2 ) respectively. The problem is that of testing H : η ≤ ξ (or η = ξ) without assuming equality of the variances σ 2 and τ 2 . Aset of sufficient statistics 2 2 for (ξ, η, σ, τ ) is then (X̄, Ȳ , SX , SY2 ), where SX = (Xi − X̄)2 /(m − 1) and 2 2 SY = (Yj − Ȳ ) /(n − 1). Adding the same constant to X̄ and Ȳ reduces the 2 problem to Ȳ − X̄, SX , SY2 , and multiplication of all variables by a common 2 2 positive constant to (Ȳ − X̄)/ SX + SY2 and SY2 /SX . One would expect any reasonable invariant rejection region to be of the form  SY2 Ȳ − X̄ ≥ g (6.17) 2 2 SX SX + SY2 for some suitable function g. If this test is also to be unbiased, the probability of (6.17) must equal α when η = ξ for all values of τ /σ. It has been shown by Linnik and others that only pathological functions g with this property can exist. [This work is reviewed by Pfanzagl (1974).] However, approximate solutions are available which provide tests that are satisfactory for all practical purposes. These are the Welch approximate t-solution described in Section 11.3, and the Welch–Aspin test. Both are discussed, and evaluated, in Scheffé (1970) and Wang (1971); see also Chernoff (1949), Wallace (1958), Davenport and Webster (1975) and Robinson (1982). The Behrens-Fisher problem will be revisited in Examples 13.5.4 and 15.6.3 and Section 15.2. The property of a test φ1 being UMP invariant is relative to a particular group G1 , and does not exclude the possibility that there might exist another test φ2 which is UMP invariant with respect to a different group G2 . Simple instances can be obtained from Examples 6.5.1 and 6.6.11. Example 6.6.8 (continued) If G1 is the group G of Example 6.5.1, a UMP invariant test of H : θ ≤ θ0 against θ > θ0 rejects when Y1 + · · · + Yn > C. 232 6. Invariance Let G2 be the group obtained by interchanging the role of Z and Y1 . Then a UMP invariant test with respect to G2 rejects when Z + Y2 + · · · + Yn > C. Analogous UMP invariant tests are obtained by interchanging the role of Z and any one of the other Y ’s and further examples by applying the transformations of G in Example 6.5.1 to more than one variable. In particular, if it is applied independently to all n + 1 variables, only the constants remain invariant, and the test φ ≡ α is UMP invariant. Example 6.6.11 For another example (due to Charles Stein), let (X11 , X12 ) and (X21 , X22 ) be independent and have bivariate normal distributions with zero means and covariance matrices   σ12 ρσ1 σ2 ∆ρσ1 σ2 ∆σ12 and . ρσ1 σ2 σ22 ∆ρσ1 σ2 ∆σ22 Suppose that these matrices are nonsingular, or equivalently that |ρ| = 1, but that all σ1 , σ2 , ρ, and ∆ are otherwise unknown. The problem of testing ∆ = 1 against ∆ > 1 remains invariant under the group G1 of all nonsingular transformations  Xi1 = bXi1 ,  Xi2 = a1 Xi1 + a2 Xi2 (a2 , b > 0). Since the probability is 0 that X11 X22 = X12 X21 , the 2 × 2 matrix (Xij ) is nonsingular with probability 1, and the sample space can therefore be restricted to be the set of all nonsingular such matrices. A maximal invariant under the subgroup corresponding to b = 1 is the pair (X11 , X21 ). The argument of Example 6.3.4 then shows that there exists a UMP invariant test under G1 which rejects 2 2 when X21 X11 > C. By interchanging 1 and 2 in the second subscript of the X’s one sees that under 2 2 the corresponding group G2 the UMP invariant test rejects when X22 X12 > C. A third group leaving the problem invariant is the smallest group containing both G1 and G2 , namely the group G of all common nonsingular transformations  = ai1 Xi1 + a12 Xi2 Xi1 ,  Xi2 = a21 Xi1 + a22 Xi2 (i = 1, 2).  ), there exists Given any two nonsingular sample points Z = (Xij ) and Z  = (Xij  a nonsingular linear transformation A such that Z = AZ. There are therefore no invariants under G, and the only invariant size-α test is φ ≡ α. It follows vacuously that this is UMP invariant under G. 6.7 Admissibility Any UMP unbiased test has the important property of admissibility (Problem 4.1), in the sense that there cannot exist another test which is uniformly at least as powerful and against some alternatives actually more powerful than the given one. The corresponding property does not necessarily hold for UMP invariant tests, as is shown by the following example. Example 6.7.11 (continued) Under the assumptions of Example 6.6.11 it was seen that the UMP invariant test under G is the test ϕ ≡ α which has power 6.7. Admissibility 233 β(∆) ≡ α. On the other hand, X11 and X21 are independently distributed as N (0, σ12 ) and N (0, ∆σ12 ). On the basis of these observations there exists a UMP 2 2 test for testing ∆ = 1 against ∆ > 1 with rejection region X21 /X11 > C (Problem 3.62). The power function of this test is strictly increasing in ∆ and hence > α for all ∆ > 1. Admissibility of optimum invariant tests therefore cannot be taken for granted but must be established separately for each case. We shall distinguish two slightly different concepts of admissibility. A test ϕ0 will be called α-admissible for testing H : θ ∈ ΩH against a class of alternatives θ ∈ Ω if for any other level-α test ϕ Eθ ϕ(X) ≥ Eθ ϕ0 (X) for all θ ∈ Ω (6.18) implies Eθ ϕ(X) = Eθ ϕ0 (X) for all θ ∈ Ω . This definition takes no account of the relationship of Eθ ϕ(X) and Eθ ϕ0 (X) for θ ∈ ΩH beyond the requirement that both tests are of level α. For some unexpected, and possibly undesirable consequences of α-admissibility, see Perlman and Wu (1999). A concept closer to the decision-theoretic notion of admissibility discussed in Section 1.8, defines ϕ0 to be d-admissible for testing H against Ω if (6.18) and Eθ ϕ(X) ≤ Eθ ϕ0 (X) for all θ ∈ ΩH (6.19) jointly imply Eθ ϕ(X) = Eθ ϕ0 (X) for all θ ∈ ΩH ∪ Ω (see Problem 6.32). Any level-α test ϕ0 that is α-admissible is also d-admissible provided no other test ϕ exists with Eθ ϕ(X) = Eθ ϕ0 (X) for all θ ∈ Ω but Eθ ϕ(X) = Eθ ϕ0 (X) for some θ ∈ ΩH . That the converse does not hold is shown by the following example. Example 6.7.12 Let X be normally distributed with mean ξ and known variance σ 2 . For testing H : ξ ≤ −1 or ≥ 1 against Ω : ξ = 0, there exists a level-α test ϕ0 , which rejects when C1 ≤ X ≤ C2 and accepts otherwise, such that (Problem 6.33) Eξ ϕ0 (X) ≤ Eξ=−1 ϕ0 (X) = α for ξ ≤ −1 and Eξ ϕ0 (X) ≤ Eξ=+1 ϕ0 (X) = α < α for ξ ≥ +1. A slight modification of the proof of Theorem 3.7.1 shows that ϕ0 is the unique test maximizing the power at ξ = 0 subject to Eξ ϕ(X) ≤ α for ξ ≤ −1 and Eξ ϕ(X) ≤ α for ξ ≥ 1, and hence that ϕ0 is d-admissible. On the other hand, the test ϕ with rejection region |X| ≤ C, where Eξ=−1 ϕ(X) = Eξ=1 ϕ(X) = α, is the unique test maximizing the power at ξ = 0 subject to Eξ ϕ(X) ≤ α for ξ ≤ −1 or ≥ 1, and hence is more powerful against Ω than ϕ0 , so that ϕ0 is not α-admissible. A test that is admissible under either definition against Ω is also admissible against any Ω containing Ω and hence in particular against the class of all alternatives ΩK = Ω − ΩH . The terms α- and d-admissible without qualification 234 6. Invariance will be reserved for admissibility against ΩK . Unless a UMP test exists, any αadmissible test will be admissible against some Ω ⊂ ΩK and inadmissible against others. Both the strength of an admissibility result and the method of proof will depend on the set Ω . Consider in particular the admissibility of a UMP unbiased test mentioned at the beginning of the section. This does not rule out the existence of a test with greater power for all alternatives of practical importance and smaller power only for alternatives so close to H that the value of the power there is immaterial. In the present section, we shall discuss two methods for proving admissibility against various classes of alternatives. Theorem 6.7.1 Let X be distributed according to an exponential family with density  s   pθ (x) = C(θ) exp θj Tj (x) j=1 with respect to a σ-finite measure µ over a Euclidean sample space (X , A), and let Ω be the natural parameter space of this family. Let ΩH and Ω be disjoint nonempty subsets of Ω, and suppose that ϕ0 is a test of H : θ ∈ ΩH based on T = (T1 , . . . , Ts ) with acceptance region A0which is a closed convex subset of Rs possessing the following property: If A0 ∩ { ai ti > c} is empty for some c, there exists a point θ∗ ∈ Ω and a sequence λn → ∞ such that θ∗ + λn a ∈ Ω [where λn is a scalar and a = (a1 , . . . , as )]. Then if A is any other acceptance region for H satisfying Pθ (X ∈ A) ≤ Pθ (X ∈ AO ) for all θ ∈ Ω , A is contained in A0 , except for a subset of measure 0, i.e. µ(A ∩ Ã0 ) = 0. Proof. Suppose to the contrary that µ(A ∩ Ã0 ) > 0. Then it follows from the closure and convexity of A0 , that there exist a ∈ Rs and a real number c such that -  . ai ti > c is empty (6.20) A0 ∩ t : and . -  A∩ t: ai ti > c has positive µ-measure, (6.21) that is, the set A protrudes in some direction from the convex set A0 . We shall show that this fact and the exponential nature of the densities imply that Pθ (A) > Pθ (A0 ) for some θ ∈ Ω , (6.22) which provides the required contradiction. Let ϕ0 and ϕ denote the indicators of Ã0 and à respectively, so that (6.22) is equivalent to  for some θ ∈ Ω . [ϕ0 (t) − ϕ(t)] dPθ (t) > 0 If θ = θ∗ + λn a ∈ Ω , the left side becomes   C(θ∗ + λn a) cλn [ϕ0 (t) − ϕ(t)]eλn ( ai ti −c) dPθ∗ (t). e ∗ C(θ ) 6.7. Admissibility 235 − + − denote the contributions over the Let this integral be In+ + I n , where In and In  regions of integration {t : ai ti > c} and {t : ai ti ≤ c} respectively. Since In− + is bounded, it is enough to show that  In → ∞ as n → ∞. By (6.20), ϕ0 (t) = 1 and hence ϕ0 (t) − ϕ(t) ≥ 0 when ai ti > c, and by (6.21) .  ai ti > c > 0. µ ϕ0 (t) − ϕ(t) > 0 and This shows that In+ → ∞ as λn → ∞ and therefore completes the proof. Corollary 6.7.1 Under the assumptions of Theorem 6.7.1, the test with acceptance region A0 is d-admissible. If its size is α and there exists a finite point θ0 in the closure Ω̄H of ΩH for which Eθ0 ϕ0 (X) = α, then ϕ0 is also α-admissible. Proof. (i) Suppose ϕ satisfies (6.18). Then by Theorem 6.7.1, ϕ0 (x) ≤ ϕ(x) (a.e. µ). If ϕ0 (x) < ϕ(x) on a set of positive measure, then Eθ ϕ0 (X) < Eθ ϕ(X) for all θ and hence (6.19) cannot hold. (ii) By the argument of part (i), (6.18) implies α = Eθ0 ϕ0 (X) < Eθ0 ϕ(X), and hence by the continuity of Eθ ϕ(X) there exists a point θ ∈ ΩH for which α < Eθ ϕ(X). Thus ϕ is not a level-α test. Theorem 6.7.1 and the corollary easily extend to the case where the competitors ϕ of ϕ0 are permitted to be randomized but the assumption that ϕ0 is nonrandomized is essential. Thus, the main applications of these results are to the case that µ is absolutely continuous with respect to Lebesgue measure. The boundary of A0 will then typically have measure zero, so that the closure requirement for A0 can be dropped. Example 6.7.13 (Normal mean) If X1 , . . . , Xn is a sample from the normal 2 distribution family of distributions is exponential with T1 = X̄,  2 N (ξ, σ ), the T2 = Xi , θ1 = nξ/σ 2 , θ2 = −1/2σ 2 . Consider first the one-sided problem 1 H : θ1 ≤ √ 0, K : θ1 > 0 with α < 2 . Then the acceptance region of the t-test is A : T1 / T2 ≤ C (C > 0), which is convex [Problem 6.34(i)]. The alternatives θ ∈ Ω ⊂ K will satisfy the conditions of Theorem 6.7.1 √ if for any half plane a1 t1 + a2 t2 > c that does not intersect the set t1 ≤ C t2 there exists a ray (θ1∗ + λa1 , θ2∗ + λa2 ) in the direction of the vector (a1 , a2 ) for which (θ1∗ + λa1 , θ2∗ + λa2 ) ∈ Ω for all sufficiently large λ. In the present case, this condition must hold for all a1 > 0 > a2 . Examples of sets Ω satisfying this requirement (and against which the t-test is therefore admissible) are Ω1 : θ1 > k1 or ξ > k1 σ2 and θ1 ξ > k2 or > k2 . Ω2 : √ σ −θ2 On the other hand, the condition is not satisfied for Ω : ξ > k (Problem 6.34). Analogously, the acceptance region A : T12 ≤ CT2 of the two-sided t-test for testing H : θ1 = 0 against θ1 = 0 is convex, and the test is admissible against Ω1 : |ξ/σ 2 | > k1 and Ω2 : |ξ/σ| > k2 . 236 6. Invariance In decision theory, a quite general method for proving admissibility consists in exhibiting a procedure as a unique Bayes solution. In the present case, this is justified by the following result, which is closely related to Theorem 3.8.1. Theorem 6.7.2 Suppose the set {x : fθ (x) > 0} is independent of θ, and let a σ-field be defined over the parameter space Ω, containing both ΩH and ΩK and such that the densities fθ (x) (with respect to µ) of X are jointly measurable in θ and x. Let Λ0 and Λ1 be probability distributions over this σ-field with Λ0 (ΩH ) = Λ1 (ΩK ) = 1, and let  hi (x) = fθ (x) dΛi (θ). Suppose ϕ0 is a nonrandomized test of H against K defined by  h1 (x) > 1 ϕ0 (x) = if < k, h0 (x) 0 and that µ{x : h1 (x)/h0 (x) = k} = 0. (i) Then ϕ0 is d-admissible for testing H against K. (ii) Let supΩH Eθ ϕ0 (X) = α and ω = {θ : Eθ ϕ0 (X) = α}. If ω ⊂ ΩH and Λ0 (ω) = 1, then ϕ0 is also α-admissible. (iii) If Λ1 assigns probability 1 to Ω ⊂ ΩK , the conclusions of (i) and (ii) apply with Ω in place of ΩK . Proof. (i): Suppose ϕ is any other test, satisfying (6.18) and (6.19) with Ω = ΩK . Then also   Eθ ϕ(X) dΛ0 (θ) ≤ Eθ ϕ0 (X) dΛ0 (θ) and   Eθ ϕ(X) dΛ1 (θ) ≥ Eθ ϕ0 (X) dΛ1 (θ). By the argument of Theorem 3.8.1, these inequalities are equivalent to   ϕ(x)h0 (x) dµ(x) ≤ ϕ0 (x)h0 (x) dµ(x) and   ϕ(x)h1 (x) dµ(x) ≥ ϕ0 (x)h1 (x) dµ(x), and the hi (x) (i = 0, 1) are probability densities with respect to µ. This contradicts the uniqueness of the most powerful test of h0 against h1 at level ϕ(x)h0 (x) dµ(x). (ii): By assumption, Eθ ϕ0 (x) dΛ0 (θ) = α, so that ϕ0 is a level-α test of h0 . If ϕ is any other level-α test of H satisfying (6.18) with Ω = ΩK , it is also a level-α test of h0 and the argument of part (i) can be applied as before. (iii): This follows immediately from the proofs of (i) and (ii). Example 6.7.13 (continued) In the two-sided normal problem of Example 6.7.13 with H : ξ = 0, K : ξ = 0 consider the class Ωa,b of alternatives (ξ, σ) 6.7. Admissibility 237 satisfying σ2 = 1 , a + η2 ξ= bη , a + η2 −∞ < η < ∞ (6.23) for some fixed a, b > 0, and the subset ω, of ΩH of points (0, σ 2 ) with σ 2 < 1/a. Let Λ0 , Λ1 be distributions over ω and Ωa,b defined by the densities [Problem 6.35(i)] λ0 (η) = C0 (a + η 2 )n/2 and 2 2 λ1 (η) = 2 C1 e(n/2)b η /(a+η ) . (a + η 2 )n/2 Straightforward calculation then shows [Problem 6.35(ii)] that the densities h0 and h1 of Theorem 6.7.2 become  2 xi C0 e−(a/2) h0 (x) =  2 xi and   2 xi + C1 exp − a2 h1 (x) =  2 xi   b2 ( xi )2  2 2 xi ,  so that the Bayes test ϕ0 of Theorem 6.7.2 rejects when x̄2 / x2i > k and hence reduces to the two-sided t-test. The condition of part (ii) of the theorem is clearly satisfied so that the t-test is both d- and α-admissible against Ωa,b . When dealing with invariant tests, it is of particular interest to consider admissibility against invariant classes of alternatives. In the case of the two-sided test ϕ0 , this means sets Ω depending only on |ξ/σ|. It was seen in Example 6.7.13 that ϕ0 is admissible against Ω : |ξ/σ| ≥ B for any B, that is, against distant alternatives, and it follows from the test being UMP unbiased or from Example 6.7.13 (continued) that ϕ0 , is admissible against Ω : |ξ/σ| ≤ A for any A > 0, that is, against alternatives close to H. This leaves open the question whether ϕ0 is admissible against sets Ω : 0 < A < |ξ/σ| < B < ∞, which include neither nearby nor distant alternatives. It was in fact shown by Lehmann and Stein (1953) that ϕ0 is admissible for testing H against |ξ|/σ = δ for any δ > 0 and hence that it is admissible against any invariant Ω . It was also shown there that the one-sided t-test of H : ξ = 0 is admissible against ξ/σ = δ  for any δ  > 0. These results will not be proved here. The proof is based on assigning to log σ the uniform density on (−N, N ) and letting N → ∞, thereby approximating the “improper” prior distribution which assigns to log a the uniform distribution on (−∞, ∞), that is, Lebesgue measure. That the one-sided t-test ϕ1 of H : ξ < 0 is not admissible against all Ω is shown by Brown and Sackrowitz (1984), who exhibit a test ϕ satisfying Eξ,σ ϕ(X) < Eξ,σ ϕ1 (X) for all ξ < 0, 0 < σ < ∞ and Eξ,σ ϕ(X) > Eξ,σ ϕ1 (X) for all 0 < ξ1 < ξ < ξ2 < ∞, 0 < σ < ∞. 238 6. Invariance Example 6.7.14 (Normal variance) For testing the variance σ 2 of a normal distribution on the basis of a sample X1 , . . . , Xn from N (ξ, σ 2 ), the Bayes approach of Theorem 6.7.2 easily proves α-admissibility of the standard test against any location invariant set of alternatives Ω , that is, any set Ω depending only on σ 2 . Consider first the one-sided hypothesis H : σ ≤ σ0 and the alternatives Ω : σ = σ1 for any  σ1 > σ0 . Admissibility of the UMP invariant (and unbiased) rejection region (Xi − X̄)2 > C follows immediately from Section 3.9, where it was shown that this test is Bayes for a pair of prior distributions (Λ0 , Λ1 ): namely, Λ1 assigning probability 1 to any point (ξ1 , σ1 ), and Λ0 putting σ = σ0 and assigning to ξ the normal distribution N (ξ1 , (σ12 − σ02 )/n). Admissibility of  (Xi − X̄)2 ≤ C when the hypothesis is H : σ ≥ σ0 and Ω = {(ξ, σ) : σ = σ1 }, σ1 < σ0 , is seen by interchanging Λ0 and Λ1 , σ0 and σ1 . A similar approach proves α-admissibility of any size-α rejection region  (Xi − X̄)2 ≤ C1 or ≥ C2 (6.24) for testing H : σ = σ0 against Ω : {σ = σ1 } ∪ {σ = σ2 } (σ1 < σ0 < σ2 ). On ΩH , where the only variable is ξ, the distribution Λ0 for ξ can be taken as the normal distribution with an arbitrary mean ξ1 and variance (σ22 − σ02 )/n. On Ω , let the conditional distribution of ξ given σ = σ2 assign probability 1 to the value ξ1 , and let the conditional distribution of ξ given σ = σ1 be N (ξ1 , (σ22 − σ12 )/n). Finally, let Λ1 assign probabilities p and 1 − p to σ = σ1 and σ = σ2 , respectively. Then the rejection region satisfies (6.24), and any constants C1 and C2 for which the test has size a can be attained by proper choice of p [Problem 6.36(i)]. The results of Examples 6.7.13 and 6.7.14 can be used as the basis for proving admissibility results in many other situations involving normal distributions. The main new difficulty tends to be the presence of additional (nuisance) means. These can often be eliminated by use of the following lemma. Lemma 6.7.1 For any given σ 2 and M 2 > σ 2 there exists a distribution Λσ such that  2 2 1 √ I(z) = e−(1/2σ )(z−ζ) dΛσ (ζ) 2πσ is the normal density with mean zero and variance M 2 . Proof. Let θ = ζ/σ, and let θ be normally distributed with zero mean and variance τ 2 . Then it is seen [Problem 6.36(ii)] that   1 1 2 √ exp − 2 z . I(z) = √ 2σ (1 + τ 2 ) 2πσ 1 + τ 2 The result now follows by letting τ 2 = (M 2 /σ 2 ) − 1, so that σ 2 (1 + τ 2 ) = M 2 . Example 6.7.15 Let X1 , . . . , Xm ; Y1 , . . . , Yn be samples from N (ξ, σ 2 ) and N (η, τ 2 ) respectively, and consider the problem of testing H : τ /σ = 1 against τ /σ = ∆ > 1. (i) Suppose first that ξ = η = 0. If Λ0 and Λ1 assign probability 1 to the points (σ0 , τ0 = σ0 ) and (σ1 , τ1 = ∆σ1 ) respectively, the ratio h1 /h0 of Theorem 6.8. Rank Tests 6.7.2 is proportional to    1 1 1  2 exp − yj − − 2 ∆2 σ12 σ02 1 1 − 2 σ02 σ1  x2i 239  , and for suitable choice of critical value and σ1 < σ0 , the rejection region of the Bayes test reduces to  2 y ∆2 σ12 − σ02  j2 > . xi σ02 − σ12 The values σ02 and σ12 can then be chosen to e this test any preassigned  size α. 2 2 (ii) If ξ and η are unknown, then X̄, Ȳ , SX = (Xi − X̄)2 , SY2 =  (Yj − Ȳ ) m−1 2 2 2 2 are sufficient statistics, and S and S can be represented as S = X Y X i=1 Ui , n−1 2 2 SY = j=1 Vj , with the Ui , Vj independent normal with means 0 and variances σ 2 and τ 2 respectively. To σ and τ assign the distributions Λ0 and Λ1 of part (i) and conditionally, given σ and τ , let ξ and η be independently distributed according to Λ0σ , Λ0τ , over ΩH and Λ1σ , Λ1τ over ΩK , with these four conditional distributions determined from Lemma 6.7.1 in such a way that  √  √ m −(m/2σ02 )(x̄−ξ)2 m −(m/2σ12 )(x̄−ξ)2 √ √ e dΛ0σ0 (ξ) = e dΛ0σ1 (ξ), 2πσ0 2πσ1 and analogously for η. This is possible by choosing the constant M 2 of Lemma 6.7.1 greater than both σ02 and σ12 . With this choice of priors, the contribution from x̄ and ȳ to the ratio h1 /h0 of Theorem 6.7.2 disappears, so h1 /h0  2  that 2 reduces to the expression for this ratio in part (i), with x and y replaced i j   by (xi − x̄)2 and (yj − ȳ)2 respectively. This approach applies quite generally in normal problems with nuisance means, provided the prior distribution of the variances σ 2 , τ 2 , . . . assigns probability 1 to a bounded set, so that M 2 can be chosen to exceed all possible values of these variances. Admissibility questions have been considered not only for tests but also for confidence sets. These will not be treated here (but see Example 8.5.4); convenient entries to the literature are Cohen and Strawderman (1973) and Joshi (1982). For additional results, see Hooper (1982b) and Arnold (1984). 6.8 Rank Tests One of the basic problems of statistics is the two-sample problem of testing the equality of two distributions. A typical example is the comparison of a treatment with a control, where the hypothesis of no treatment effect is tested against the alternatives of a beneficial effect. This was considered in Chapter 5 under the assumption of normality, and the appropriate test was seen to be based on Student’s t. It was also shown that when approximate normality is suspected but the assumption cannot be trusted, one is led to replacing the t-test by its permutation analogue, which in turn can be approximated by the original t-test. 240 6. Invariance We shall consider the same problem below without, at least for the moment, making any assumptions concerning even the approximate form of the underlying distributions, assuming only that they are continuous. The observations then consist of samples X1 , . . . , Xm and Y1 , . . . , Yn from two distributions with continuous cumulative distribution functions F and G, and the problem becomes that of testing the hypothesis H1 : G = F. If the treatment effect is assumed to be additive, the alternatives are G(y) = F (y − ∆). We shall here consider the more general possibility that the size of the effect may depend on the value of y (so that ∆ becomes a nonnegative function of y) and therefore test H1 against the one-sided alternatives that the Y ’s are stochastically larger than the X’s, K1 : G(z) ≤ F (z) for all z, and G = F. An alternative experiment that can be performed to test the effect of a treatment consists of the comparison of N pairs of subjects, which have been matched so as to eliminate as far as possible any differences not due to the treatment. One member of each pair is chosen at random to receive the treatment while the other serves as control. If the normality assumption of Section 5.10 is dropped and the pairs of subjects can be considered to constitute a sample, the observations (X1 , Y1 ), . . . , (XN , YN ) are a sample from a continuous bivariate distribution F . The hypothesis of no effect is then equivalent to the assumption that F is symmetric with respect to the line y = x: H2 : F (x, y) = F (y, x). Another basic problem, which occurs in many different contexts, concerns the dependence or independence of two variables. In particular, if (X1 , Y1 ), . . . , (XN , YN ) is a sample from a bivariate distribution F , one will be interested in the hypothesis H3 : F (x, y) = G1 (x)G2 (y) that X and Y are independent, which was considered for normal distributions in Section 5.13. The alternatives of interest may, for example, be that X and Y are positively dependent. An alternative formulation results when x, instead of being random, can be selected for the experiment. If the chosen values are x1 < · · · < xN and Fi denotes the distribution of Y given xi , the Y ’s are independently distributed with continuous cumulative distribution functions F1 , . . . , FN . The hypothesis of independence of Y from x becomes H 4 : F 1 = · · · = FN , while under the alternatives of positive regression dependence the variables Yi are stochastically increasing with i. In these and other similar problems, invariance reduces the data so completely that the actual values of the observations are discarded and only certain order relations between different groups of variables are retained. It is nevertheless possible on this basis to test the various hypotheses in question, and the resulting tests frequently are nearly as powerful as the standard normal tests. We shall now carry out this reduction for the four problems above. 6.8. Rank Tests 241 The two-sample problem of testing H1 against K1 remains invariant under the group G of all transformations xi = ρ(xi ), yj = ρ(yj ) (i = 1, . . . , m, j = 1, . . . , n) such that ρ is continuous and strictly increasing. This follows from the fact that these transformations preserve both the continuity of a distribution and the property of two variables being either identically distributed or one being stochastically larger than the other. As was seen (with a different notation) in Example 6.2.3, a maximal invariant under G is the set of ranks  (R ; S  ) = (R1 , . . . , Rm ; S1 , . . . , Sn ) of X1 , . . . , Xm ; Y1 , . . . , Yn in the combined sample. Since the distribution of  (R1 , . . . , Rm ; S1 , . . . , Sn ) is symmetric in the first m and in the last n variables for all distributions F and G, a set of sufficient statistics for (R , S  ) is the set of the X-ranks and that of the Y -ranks without regard to the subscripts of the X’s and Y ’s This can be represented by the ordered X-ranks and Y -ranks R1 < · · · < Rm and S1 < · · · < Sn , and therefore by one of these sets alone since each of them determines the other. Any invariant test is thus a rank test, that is, it depends only on the ranks of the observations, for example on (S1 , . . . , Sn ). That almost invariant tests are equivalent to invariant ones in the present context was shown first by Bell (1964). A streamlined and generalized version of his approach is given by Berk and Bickel (1968) and Berk (1970), who also show that the conclusion of Theorem 6.5.3 remains valid in this case. To obtain a similar reduction for H2 , it is convenient first to make the transformation Zi = Yi − Xi , Wi = Xi + Yi . The pairs of variables (Zi , Wi ) are then again a sample from a continuous bivariate distribution. Under the hypothesis this distribution is symmetric with respect to the w-axis, while under the alternatives the distribution is shifted in the direction of the positive z-axis The problem is unchanged if all the w’s are subjected to the same transformation wi = λ(wi ), where λ is 1 : 1 and has at most a finite number of discontinuities, and (Z1 , . . . , ZN ) constitutes a maximal invariant under this group. [Cf. Problem 6.2(ii).] The Z’s are a sample from a continuous univariate distribution D, for which the hypothesis of symmetry with respect to the origin, H2 : D(z) + D(−z) = 1 for all z, is to be tested against the alternatives that the distribution is shifted toward positive z-values This problem is invariant under the group G of all transformations zi = ρ(zi ) (i = 1, . . . , N ) such that ρ is continuous, odd, and strictly increasing. If zi1 , . . . , zim < 0 < zj1 , . . . , zjn , where i1 < · · · < im and j1 < · · · < jn , let s1 , . . . , sn denote the  ranks of zji , . . . , zjn , among the absolute values |z1 |, . . . , |zN |, and r1 , . . . , rm the ranks of |zi1 |, . . . , |zim | among |z1 |, . . . , |zN |. The transformations ρ preserve the sign of each observation, and hence in particular also the numbers m and n. Since ρ is a continuous, strictly increasing function of |z|, it leaves the order of 242 6. Invariance the absolute values invariant and therefore the ranks ri and sj . To see that the  latter are maximal invariant, let (z1 , . . . , zN ) and (z1 , . . . , zN ) be two sets of points     with m = m, n = n, and the same ri and sj . There exists a continuous, strictly increasing function on the positive real axis such that |zi | = ρ(|zi |) and ρ(0) = 0. If ρ is defined for negative z by ρ(−z) = −ρ(z), it belongs to G and zi = ρ(zi ) for all i, as was to be proved. As in the preceding problem, sufficiency permits the further reduction to the ordered ranks r1 < · · · < rm and s1 < · · · < sn . This retains the information for the rank of each absolute value whether it belongs to a positive or negative observation, but not with which positive or negative observation it is associated. The situation is very similar for the hypotheses H3 and H4 . The problem of testing for independence in a bivariate distribution against the alternatives of positive dependence is unchanged if the Xi and Yi are subjected to transformations Xi = ρ(Xi ), Yi = λ(Yi ) such that ρ and λ are continuous and  strictly increasing. This leaves as maximal invariant the ranks (R1 , . . . , RN ) of   (X1 , . . . , XN ) among the X’s and the ranks (S1 , . . . , SN ) of (Y1 , . . . , YN ) among   the Y ’s. The distribution of (R1 , S1 ), . . . , (RN , SN ) is symmetric in these N pairs for all distributions of (X, Y ). It follows that a sufficient statistic is (S1 , . . . , SN )   where (1, S1 ), . . . , (N, SN ) is a permutation of (R1 , S1 ), . . . , (RN , SN ) and where therefore Si is the rank of the variable Y associated with the ith smallest X. The hypothesis H4 that Y1 , . . . , Yn constitutes a sample is to be tested against the alternatives K4 that the Yi are stochastically increasing with i. This problem is invariant under the group of transformations yi = ρ(yi ) where ρ is continuous and strictly increasing. A maximal invariant under this group is the set of ranks S1 , . . . , SN of Y1 , . . . , YN . Some invariant tests of the hypotheses H1 and H2 will be considered in the next two sections. Corresponding results concerning H3 and H4 are given in Problems 6.60–6.62. 6.9 The Two-Sample Problem The problem of testing the two-sample hypothesis H : G = F against the onesided alternatives K that the Y ’s are stochastically larger than the X’s is reduced by the principle of invariance to the consideration of tests based on the ranks S1 < · · · < Sn of the Y ’s. The specification of the Si is equivalent to specifying for each of the N = m + n positions within the combined sample (the smallest, the next smallest, etc.) whether it is occupied by an x or a y. Since for  any set of observations n of the N positions are occupied by y’s and since the N possible n assignments of n positions to the y’s are all equally likely when G = F , the joint distribution of the Si under H is 5  N P {S1 = s1 , . . . , Sn = sn } = 1 (6.25) n for each set 1 ≤ s1 < s2 < · · · < sn ≤ N . Any rank test of H of size 5  N α=k n 6.9. The Two-Sample Problem 243 therefore has a rejection region consisting of exactly k points (s1 , . . . , sn ). For testing H against K there exists no UMP rank test, and hence no UMP invariant test. This follows for example from a consideration of two of the standard tests for this problem, since each is most powerful among all rank tests against some alternative. The two tests in question have rejection regions of the form h(s1 ) + · · · + h(sn ) > C. (6.26) One, the Wilcoxon two-sample test, is obtained from (6.26) by letting h(s) = s, so that it rejects H when the sum of the y-ranks is too large. We shall show below that for sufficiently small ∆, this is most powerful against the alternatives that F is the logistic distribution F (x) = 1/(1 + e−x ), and that G(y) = F (y − ∆). The other test, the normal-scores test, has the rejection region (6.26) with h(s) = E(W(s) ), where W(1) < · · · < W(N ) , is an ordered sample of size N from a standard normal distribution.5 This is most powerful against the alternatives that F and G are normal distributions with common variance and means ξ and η = ξ + ∆, when ∆ is sufficiently small. To prove that these tests have the stated properties it is necessary to know the distribution of (S1 , . . . , Sn ) under the alternatives. If F and G have densities f and g such that f is positive whenever g is, the joint distribution of the Si is given by  3  g(V(s1 ) ) g(V(sn ) ) N ··· P {S1 = s1 , . . . , Sn = sn } = E , (6.27) f (V(s1 ) ) f (V(sn ) ) n where V(1) < · · · < V(N ) is an ordered sample of size N from the distribution F . (See Problem 6.42.) Consider in particular the translation (or shift) alternatives g(y) = f (y − ∆), and the problem of maximizing the power for small values of ∆. Suppose that f is differentiable and that the probability (6.27), which is now a function of ∆, can be differentiated with respect to ∆ under the expectation sign. The derivative of (6.27) at ∆ = 0 is then %   3  n  % f (V(si ) ) ∂ N =− E P∆ {S1 = s1 , . . . , Sn = Sn }%% . ∂∆ f (V ) n (s ) i ∆=0 i=1 Since under the hypothesis the probability of any ranking is given by (6.25), it follows from the Neyman–Pearson lemma in the extended form of Theorem 3.6.1, that the derivative of the power function at ∆ = 0 is maximized by the rejection region    n  f (V(si ) ) E > C. (6.28) − f (V(si ) ) i=1 The same test maximizes the power itself for sufficiently small ∆. To see this let s denote a general rank point (s1 , . . . , sn ), and denote by s(j) the rank point 5 Tables of the expected order statistics from a normal distribution are given in Biometrika Tables for Statisticians, Vol. 2, Cambridge U. P., 1972, Table 9. For additional references, see David (1981, Appendix, Section 3.2). 244 6. Invariance giving the jth largest value to the left-hand side of (6.28). If 5  N α=k , n the power of the test is then β(∆) = k  (j) P∆ (s )= j=1 k  ! j=1 1 N n % % ∂ (j) % +∆ P∆ (s )% % ∂∆ " + ··· . ∆=0 Since there is only a finite number of points s, there exists for each j a number ∆j > 0 such that the point s(j) also gives the jth largest value to P∆ (s) for all ∆ < ∆j . If ∆ is less than the smallest of the numbers   N j = 1, . . . , ∆j , , n the test also maximizes β(∆). If f (x) is the normal density N (ξ, σ 2 ), then − f  (x) d x−ξ , =− log f (x) = f (x) dx σ2 and the left-hand side of (6.28) becomes  V(s ) − ξ 1 i E = E(W(si ) ) 2 σ σ where W(1) < · · · < W(N ) is an ordered sample from N (0, 1). The test that maximizes the power against these alternatives (for sufficiently small ∆) is therefore the normal-scores test. In the case of the logistic distribution, F (x) = 1 , 1 + e−x f (x) = e−x , (1 + e−x )2 and hence − f  (x) = 2F (x) − 1. f (x)  The locally most powerful rank test therefore rejects when E[F (V(xi ) )] > C. If V has the distribution F , then U = F (V ) is uniformly distributed  over (0, 1) (Problem 3.22). The rejection region can therefore be written as E(U(si ) ) > C, where U(1) < · · · < U(N ) is an ordered sample of size N from the uniform distribution U (0, 1). Since E(U(si ) ) = si /(N + 1), the test is seen to be the Wilcoxon test. Both the normal-scores test and the Wilcoxon test are unbiased against the one-sided alternatives K. In fact, let φ be the critical function of any test determined by (6.26) with h nondecreasing. Then φ is nondecreasing in the y’s and the probability of rejection is α for all F = G. By Lemma 5.9.1 the test is therefore unbiased against all alternatives of K. It follows from the unbiasedness properties of these tests that the most powerful invariant tests in the two cases considered are also most powerful against their respective alternatives among all tests that are invariant and unbiased. The 6.9. The Two-Sample Problem 245 nonexistence of a UMP test is thus not relieved by restricting the tests to be unbiased as well as invariant. Nor does the application of the unbiasedness principle alone lead to a solution, as was seen in the discussion of permutation tests in Section 5.9. With the failure of these two principles, both singly and in conjunction, the problem is left not only without a solution but even without a formulation. A possible formulation (stringency) will be discussed in Chapter 8. However, the determination of a most stringent test for the two-sample hypothesis is an open problem. For testing H : G = F against the two-sided alternatives that the Y ’s are either stochastically smaller or larger than the X’s two-sided versions of the rank tests of this section can be used. In particular, suppose that h is increasing and that h(s)+h(N +1−s) is independent of s, as is the case for the Wilcoxon and normalscores statistics. Then under H, the statistic Σh(sj ) is symmetrically distributed about nΣN i=1 h(i)/N = µ, and (6.26) suggests the rejection region % % n m % % % %  1 %  % % % h(sj ) − µ% = h(sj ) − n h(ri )% > C. % %m % N % j=1 i=1 The theory here is still less satisfactory than in the one-sided case. These tests need not even be unbiased [Sugiura (1965)], and it is not known whether they are admissible within the class of all rank tests. On the other hand, the relative asymptotic efficiencies are the same as in the one-sided case. The two-sample hypothesis G = F can also be tested against the general alternatives G = F . This problem arises in deciding whether two products, two sets of data, or the like can be pooled when nothing is known about the underlying distributions. Since the alternatives are now unrestricted, the problem remains invariant under all transformations xi = f (xi ), yj = f (yj ), i = 1, . . . , m, j = 1, . . . , n, such that f has only a finite number of discontinuities. There are no invariants under this group, so that the only invariant test is φ(x, y) ≡ α. This is however not admissible, since there do exist tests of H that are strictly unbiased against all alternatives G = F (Problem 6.54). One of the tests most commonly employed for this problem is the Smirnov test. Let the empirical distribution functions of the two samples be defined by Sx1 ,...,xm (z) = a , m Sy1 ,...,yn (z) = b , n where a and b are the numbers of x’s and y’s less or equal to z respectively. Then H is rejected according to this test when sup |Sx1 ,...,xm (z) − Sy1 ,...,yn (z)| > C. z Accounts of the theory of this and related tests are given, for example, in Durbin (1973), Serfling (1980), Gibbons and Chakraborti (1992) and Hájek, Sidák, and Sen (1999). Two-sample rank tests are distribution-free for testing H : G = F but not for the nonparametric: Behrens-Fisher situation of testing H : η = ξ when the X’s and Y ’s are samples from F ((x − ξ)/σ) and F ((y − η)/τ ) with σ, τ unknown. A detailed study of the effect of the difference in scales on the levels of the Wilcoxon and normal-scores tests is provided by Pratt (1964). 246 6. Invariance 6.10 The Hypothesis of Symmetry When the method of paired comparisons is used to test the hypothesis of no treatment effect, the problem was seen in Section 6.8 to reduce through invariance to that of testing the hypothesis H2 : D(z) + D(−z) = 1 for all z, which states that the distribution D of the differences Zi = Yi −Xi (i = 1, . . . , N ) is symmetric with respect to the origin. The distribution D can be specified by the triple (ρ, F, G) where ρ = P {Z ≤ 0}, F (z) = P {|Z| ≤ z | Z > 0}, G(z) = P {Z ≤ z | Z > 0}, and the hypothesis of symmetry with respect to the origin then becomes H : p = 12 , G = F. Invariance and sufficiency were shown to reduce the data to the ranks S1 < · · · < Sn of the positive Z’s among the absolute values |Z1 |, . . . , |ZN |. The probability of S1 = s1 , . . . , Sn = sn is the probability of this event given that there are n positive observations multiplied by the probability that the number of positive observations is n. Hence P {S1 = s1 , . . . , Sn = sn }   N = (1 − ρ)n ρN −n PF,G {S1 = s1 , . . . , Sn = sn | n} n where the second factor is given by (6.27). Under H, this becomes P {S1 = s1 , . . . , Sn = sn } = for each of the 1 2N   N = 2N n n=0 N  n-tuples (s1 , . . . , sn ) satisfying 1 ≤ s1 < · · · < sn ≤ N . Any rank test of size α = k/2N therefore has a rejection region containing exactly k such points (s1 , . . . , sn ). The alternatives K of a beneficial treatment effect are characterized by the fact that the variable Z being sampled is stochastically larger than some random variable which is symmetrically distributed about 0. It is again suggestive to use rejection regions of the form h(s1 ) + · · · + h(sn ) > C, where however n is no longer a constant as it was in the two-sample problem, but depends on the observations. Two particular cases are the Wilcoxon one-sample test, which is obtained by putting h(s) = s, and the analogue of the normal-scores test with h(s) = E(W(s) ) where W(1) < · · · < W(N ) are the ordered values of |V1 |, . . . , |VN |, the V ’s being a sample from N (0, 1). The W ’s are therefore an ordered sample 2 of size N from a distribution with density 2/πe−w /2 for w ≥ 0. As in the two-sample problem, it can be shown that each of these tests is most powerful (among all invariant tests) against certain alternatives, and that they 6.10. The Hypothesis of Symmetry 247 are both unbiased against the class K. Their asymptotic efficiencies relative to the t-test for testing that the mean of Z is zero have the same values 3/π and 1 as the corresponding two-sample tests, when the distribution of Z is normal. In certain applications, for example when the various comparisons are made under different experimental conditions or by different methods, it may be unrealistic to assume that the variables Z1 , . . . , ZN have a common distribution. Suppose instead that the Zi are still independently distributed but with arbitrary continuous distributions Di . The hypothesis to be tested is that each of these distributions is symmetric with respect to the origin. This problem remains invariant under all transformations zi = fi (zi ) i = 1, . . . , N , such that each fi is continuous, odd, and strictly increasing. A maximal invariant is then the number n of positive observations, and it follows from Example 6.5.1 that there exists a UMP invariant test, the sign test, which rejects when n is too large. This test reflects the fact that the magnitude of the observations or of their absolute values can be explained entirely in terms of the spread of the distributions Di , so that only the signs of the Z’s are relevant. Frequently, it seems reasonable to assume that the Z’s are identically distributed, but the assumption cannot be trusted. One would then prefer to use the information provided by the ranks si but require a test which controls the probability of false rejection even when the assumption fails. As is shown by the following lemma, this requirement is in fact satisfied for every (symmetric) rank test. Actually, the lemma will not require even the independence of the Z’s; it will show that any symmetric rank test continues to correspond to the stated level of significance provided only the treatment is assigned at random within each pair. Lemma 6.10.1 Let φ(z1 , . . . , zN ) be symmetric in its N variables and such that ED φ(Z1 , . . . , ZN ) = α (6.29) when the Z’s are a sample from any continuous distribution D which is symmetric with respect to the origin. Then Eφ(Z1 , . . . , ZN ) = α (6.30) if the joint distribution of the Z’s is unchanged under the 2N transformations  Z1 = ±Z1 , . . . , ZN = ±ZN . Proof. The condition (6.29) implies  (j1 ,...,jN )  φ(±zj , . . . , ±zj ) 1 N =α 2N · N ! a.e., (6.31) where the outer summation extends over all N ! permutations (j1 , . . . , jN ) and the inner one over all 2N possible choices of the signs + and −. This is proved exactly as was Theorem 5.8.1. If in addition φ is symmetric, (6.31) implies  φ(±z1 , . . . , ±zN ) = α. 2N (6.32) Suppose that the distribution of the Z’s is invariant under the 2N transformations in question. Then the conditional probability of any sign combination of 248 6. Invariance Z1 , . . . , ZN given |Z1 |, . . . , |ZN | is 1/2N . Hence (6.32) is equivalent to E[φ(Z1 , . . . , ZN ) | |Z1 |, . . . , |ZN |] = α a.e., (6.33) and this implies (6.30) which was to be proved. The tests discussed above can be used to test symmetry about any known value θ0 by applying them to the variables Zi − θ0 . The more difficult problem of testing for symmetry about an unknown point θ will not be considered here. Tests of this hypothesis are discussed, among others, by Antille, Kersting, and Zucchini (1982), Bhattacharya, Gastwirth, and Wright (1982), Boos (1982), and Koziol (1983). As will be seen in Section 11.3.1, the one-sample t-test is not robust against dependence. Unfortunately, this is also true-although to a somewhat lesser extent—of the sign and one-sample Wilcoxon tests [Gastwirth and Rubin (1971)]. 6.11 Equivariant Confidence Sets Confidence sets for a parameter θ in the presence of nuisance parameters ϑ were discussed in Chapter 5 (Sections 5.4 and 5.5) under the assumption that θ is realvalued. The correspondence between acceptance regions A(θ0 ) of the hypotheses H(θ0 ) : θ = θ0 and confidence sets S(x) for θ given by (5.33) and (5.34) is, however, independent of this assumption; it is valid regardless of whether θ is realvalued, vector-valued, or possibly a label for a completely unknown distribution function (in the latter case, confidence intervals become confidence bands for the distribution function). This correspondence, which can be summarized by the relationship θ ∈ S(x) if and only if x ∈ A(θ), (6.34) was the basis for deriving uniformly most accurate and uniformly most accurate unbiased confidence sets. In the present section, it will be used to obtain uniformly most accurate equivariant confidence sets. We begin by defining equivariance for confidence sets. Let G be a group of transformations of the variable X preserving the family of distributions {Pθ,ϑ , (θ, ϑ) ∈ Ω} and let Ḡ be the induced group of transformations of Ω. If ḡ(θ, ϑ) = (θ , ϑ ), we shall suppose that θ depends only on ḡ and θ and not on ϑ, so that ḡ induces a transformation in the space of θ. In order to keep the notation from becoming unnecessarily complex, it will then be convenient to write also θ = ḡθ. For each transformation g ∈ G, denote by g ∗ the transformation acting on sets S in θ-space and defined by g ∗ S = {ḡθ : θ ∈ S}, (6.35) ∗ so that g S is the set obtained by applying the transformation ḡ to each point θ of S. The invariance argument of Section 1.5, then suggests restricting consideration to confidence sets satisfying g ∗ S(x) = S(gx) for all x ∈ X, g ∈ G. (6.36) We shall say that such confidence sets are equivariant under G. This terminology is preferable to the older term invariance which creates the impression that the 6.11. Equivariant Confidence Sets 249 confidence sets remain unchanged under the transformation X  = gX. If the transformation g is interpreted as a change of coordinates, (6.36) means that the confidence statement does not depend on the coordinate system used to express the data. The statement that the transformed parameter ḡθ lies in S(gx) is equivalent to stating that θ ∈ g ∗−1 S(gx), which is equivalent to the original statement θ ∈ S(x) provided (6.36) holds. Example 6.11.1 Let X, Y be independently normally distributed with means ξ, η and unit variance, and let G be the group of all rigid motions of the plane, which is generated by all translations and orthogonal transformations. Here ḡ = g for all g ∈ G. An example of an equivariant class of confidence sets is given by   S(x, y) = (ξ, η) : (x − ξ)2 + (y − η)2 ≤ C , √ the class of circles with radius C and center (x, y). The set g ∗ S(x, y) is the set of all points g(ξ, η) with (ξ, η) ∈ S(x, y) and hence is obtained√by subjecting S(x, y) to the rigid motion g. The result is the circle with radius C and center g(x, y), and (6.36) is therefore satisfied. In accordance with the definitions given in Chapters 3 and 5, a class of confidence sets for θ will be said to be uniformly most accurate equivariant at confidence level 1 − α if among all equivariant classes of sets S(x) at that level it minimizes the probability Pθ,ϑ {θ ∈ S(X)} for all θ = θ. In order to derive confidence sets with this property from families of UMP invariant tests, we shall now investigate the relationship between equivariance of confidence sets and invariance of the associated tests. Suppose that for each θ0 there exists a group of transformations Gθ0 which leaves invariant the problem of testing H(θ0 ) : θ = θ0 , and denote by G the group of transformations generated by the totality of groups Gθ . Lemma 6.11.1 (i) Let S(x) be any class of confidence sets that is equivariant under G, and let A(θ) = {x : θ ∈ S(x)}; then the acceptance region A(θ) is invariant under Gθ for each θ. (ii) If in addition, for each θ0 the acceptance region A(θ0 ) is UMP invariant for testing H(θ0 ) at level α, the class of confidence sets S(x) is uniformly most accurate among all equivariant confidence sets at confidence level 1 − α. Proof. (i): Consider any fixed θ, and let g ∈ Gθ . Then gA(θ) = {gx : θ ∈ S(x)} = {x : θ ∈ S(g −1 x)} = {x : θ ∈ g ∗−1 S(x)} = {x : ḡθ ∈ S(x)} = {x : θ ∈ S(x)} = A(θ). Here the third equality holds because S(x) is equivariant, and the fifth one because g ∈ Gθ and therefore ḡθ = θ. (ii): If S  (x) is any other equivariant class of confidence sets at the prescribed level, the associated acceptance regions A (θ) by (i) define invariant tests of the hypotheses H(θ). It follows that these tests are uniformly at most as powerful as those with acceptance regions A(θ) and hence that Pθ,ϑ {θ ∈ S(X)} ≤ Pθ,ϑ {θ ∈ S  (X)} for all θ = θ, 250 6. Invariance as was to be proved. It is an immediate consequence of the lemma that if UMP invariant acceptance regions A(θ) have been found for each hypothesis H(θ) (invariant with respect to Gθ ), and if the confidence sets S(x) = {θ : x ∈ A(θ)} are equivariant under G, then they are uniformly most accurate equivariant. Example 6.11.2 Under the assumptions of Example 6.11.1, the problem of testing ξ = ξ0 , η = η0 is invariant under the group Gξ0 ,η0 of orthogonal transformations about the point (ξ0 , η0 ): X  − ξ0  Y − η0 = a11 (X − ξ0 ) + a12 (Y − η0 ), = a21 (X − ξ0 ) + a22 (Y − η0 ), where the matrix (aij ) is orthogonal. There exists under this group a UMP invariant test, which has the acceptance region (Problem 7.8) (X − ξ0 )2 + (Y − η0 )2 ≤ C. Let G0 be the smallest group containing the groups Gξ,η , for all ξ, η. Since this is a subgroup of the group G of Example 6.11.1 (the two groups actually coincide, but this is immaterial for the argument), the confidence sets (X − ξ)2 + (Y − η)2 ≤ C are equivariant under G0 and hence uniformly most accurate equivariant. Example 6.11.3 Let X1 , . . . , Xn be independently normally distributed with mean ξ and variance σ 2 . Confidence intervals for ξ are based on the hypotheses H(ξ0 ) : ξ = ξ0 , which are invariant under the groups Gξ0 of transformations Xi = a(Xi − ξ0 ) + ξ0 (a = 0). The UMP invariant test of H(ξ0 ) has acceptance region (n − 1)n|X̄ − ξ0 | ≤ C,  (Xi − X̄)2 and the associated confidence intervals are , C (Xi − X̄)2 ≤ ξ ≤ X̄ + X̄ − n(n − 1) C n(n − 1) , (Xi − X̄)2 . (6.37) The group G in the present case consists of all transformations g : Xi = aXi + b (a = 0), which on ξ induces the transformation ḡ : ξ  = aξ + b. Application of the associated transformation g ∗ to the interval (6.37) takes it into the set of points aξ + b for which ξ satisfies (6.37), that is, into the interval with end points , , |a|C |a|C (Xi − X̄)2 , (Xi − X̄)2 aX̄ + b − aX̄ + b + n(n − 1) n(n − 1) Since this coincides with the interval obtained by replacing Xi in (6.37) with aXi + b, the confidence intervals (6.37) are equivariant under G0 and hence uniformly most accurate equivariant. Example 6.11.4 In the two-sample problem of Section 6.9, assume the shift model in which the X’s and Y ’s have densities f (x) and g(y) = f (y − ∆) respectively, and consider the problem of obtaining confidence intervals for the shift parameter ∆ which are distribution-free in the sense that the coverage probability is independent of the true f . The hypothesis H(∆0 ) : ∆ = ∆0 can be 6.12. Average Smallest Equivariant Confidence Sets 251 tested, for example, by means of the Wilcoxon test applied to the observations Xi , Yj −∆0 , and confidence sets for ∆ can then be obtained by the usual inversion process. The resulting confidence intervals are of the form D(k) < ∆ < D(mn+1−k) where D(1) < · · · < D(mn) are the mn ordered differences Yj − Xi . [For details see Problem 6.52 and for fuller accounts nonparametric books such as Randles and Wolfe (1979), Gibbons and Chakraborti (1992) and Lehmann (1998).] By their construction, these intervals have coverage probability 1 − α, which is independent of f . However, the invariance considerations of Sections 6.8 and 6.9 do not apply. The hypothesis H(∆0 ) is invariant under the transformations Xi = ρ(Xi ), Yj = ρ(Yj − ∆0 ) + ∆0 with ρ continuous and strictly increasing, but the shift model, and hence the problem under consideration, is not invariant under these transformations. 6.12 Average Smallest Equivariant Confidence Sets In the examples considered so far, the invariance and equivariance properties of the confidence sets corresponded to invariant properties of the associated tests. In the following examples this is no longer the case. Example 6.12.1 Let X1 , . . . , Xn , be a sample from N (ξ, σ 2 ), and consider the problem of estimating σ 2 . The model is invariant under translations Xi = Xi + a, and sufficiency and  2 invariance reduce the data to S = (Xi − X̄)2 . The problem of estimating σ 2 by confidence sets also remains invariant under scale changes Xi = bXi , S  = bS, σ  = bσ (0 < b), although these do not leave the corresponding problem of testing the hypothesis σ = σ0 invariant. (Instead, they leave invariant the family of these testing problems, in the sense that they transform one such hypothesis into another.) The totality of equivariant confidence sets based on S is given by σ2 ∈ A, S2 where A is any fixed set on the line satisfying  1 Pσ=1 ∈ A = 1 − α. S2 (6.38) (6.39) That any set σ 2 ∈ S 2 · A is equivariant is obvious. Conversely, suppose that σ 2 ∈ C(S 2 ) is an equivariant family of confidence sets for σ 2 . Then C(S 2 ) must satisfy b2 C(S 2 ) = C(b2 S 2 ) and hence σ 2 ∈ C(S 2 ) if and only if σ2 1 ∈ 2 C(S 2 ) = C(1), S2 S which establishes (6.38) with A = C(1). Among the confidence sets (6.38) with A satisfying (6.39) there does not exist one that uniformly minimizes the probability of covering false values (Problem 6.73). Consider instead the problem of determining the confidence sets that are physically smallest in the sense of having minimum Lebesgue measure. This requires minimizing A dv subject to (6.39). It follows from the Neyman-Pearson 252 6. Invariance lemma that the minimizing A∗ is A∗ = {v : p(v) > C}, (6.40) 2 where p(v) is the density of V = 1/S when σ = 1, and where C is determined by (6.39). Since p(v) is unimodal (Problem 6.74), these smallest confidence sets are intervals, aS 2 < σ 2 < bS 2 . Values of a and b are tabled by Tate and Klett (1959), who also table the corresponding (different) values a , b for the uniformly most accurate unbiased confidence intervals a S 2 < σ 2 < b S 2 (given in Example 5.5.1). Instead of minimizing the Lebesgue measure A dv of the confidence sets A, one may prefer to minimize the scale-invariant measure  1 dv. (6.41) A v To an interval (a, b), (6.41) assigns, in place of its length b − a, its logarithmic length log b − log a = log(b/a). The optimum solution A∗∗ with respect to this new measure is again obtained by applying the Neyman Pearson lemma, and is given by A∗∗ = {v : vp(v) > C}, (6.42) which coincides with the uniformly most accurate unbiased confidence sets [Problem 6.75(i)]. One advantage of minimizing (6.41) instead of Lebesgue measure is that it then does not matter whether one estimates σ or σ 2 (or σ r for some other power of r), since under (6.41), if (a, b) is the best interval for σ, then (ar , br ) is the best interval for σ r [Problem 6.75(ii)]. Example 6.12.2 Let Xi (i = 1, . . . , r) be independently normally distributed as N (ξ, 1). A slight generalization of Example 6.11.2 shows that uniformly most accurate equivariant confidence sets for (ξ1 , . . . , ξr ) exist with respect to the group G of all rigid transformations and are given by  (6.43) (Xi − ξi )2 ≤ C. Suppose that the context of the problem does not possess the symmetry which would justify invoking invariance with respect to G, but does allow the weaker assumption of invariance under the group G0 of translations Xi = Xi + ai . The totality of equivariant confidence sets with respect to G0 is given by (X1 − ξ1 , . . . , Xr − ξr ) ∈ A, (6.44) where A is any fixed set in r-space satisfying Pξ1 =···=ξr =0 ((X1 , . . . , Xr ) ∈ A) = 1 − α. (6.45) Since uniformly most accurate equivariant confidence sets do not exist (Problem 6.73), let us consider instead the problem of determining the confidence sets of smallest Lebesgue measure. (This measure is invariant under G0 .) This is given by (6.40) with v = (v1 , . . . , vr ) and p(v) the density of (X1 , . . . , Xr ) when ξ1 = · · · = ξr = 0, and hence coincides with (6.43). Quite surprisingly, the confidence sets (6.43) are inadmissible if and only if r ≥ 3. A further discussion of this fact and references are deferred to Example 8.5.4. 6.12. Average Smallest Equivariant Confidence Sets 253 Example 6.12.3 In the preceding example, suppose that the Xi are distributed as N (ξi , σ 2 ) with σ 2 unknown, and that a variable S 2 is available for estimating σ 2 . Of S 2 assume that it is independent of the X’s and that S 2 /σ 2 has a χ2 -distribution with f degrees of freedom. The estimation of (ξ1 , . . . , ξr ) by confidence sets on the basis of X’s and S 2 remains invariant under the group G0 of transformations Xi = bXi + ai , S  = bS, ξi = bξi + ai , σ  = bσ, and the most general equivariant confidence set is of the form  X1 − ξ1 Xr − ξr ,..., ∈ A, S S where A is any fixed set in r-space satisfying    X1 Xr Pξ1 =···=ξr =0 ,..., ∈ A = 1 − α. S S (6.46) (6.47) The confidence sets (6.46) can be written as (ξ1 , . . . , ξr ) ∈ (X1 , . . . , Xr ) − SA, (6.48) where −SA is the set obtained by multiplying each point of A by the scalar −S. To see (6.48), suppose that C(X1 , . . . , Xr ; S) is an equivariant confidence set for (ξ1 , . . . , ξr ). Then the r-dimensional set C must satisfy C(bX1 + a1 , . . . , bXr + ar ; bS) = b[C(X1 , . . . , Xr ; S)] + (a1 , . . . , ar ) for all a1 , . . . , ar and all b > 0. It follows that (ξ1 , . . . , ξr ) ∈ C if and only if  (X1 , . . . , Xr ) − C(X1 , . . . , Xr ; S) Xr − ξr X1 − ξ1 ,..., = C(0, . . . , 0; 1) ∈ S S S = A. The equivariant confidence sets of smallest volume are obtained by choosing for A the set A∗ given by (6.40) with v = (v1 , . . . , vr ) and p(v) the joint density of (X 1 /S, . . . , Xr /S) when ξ1 = · · · = ξr = 0. This density is a decreasing function of vi2 (Problem 6.76), and the smallest equivariant confidence sets are therefore given by  (Xi − ξi )2 ≤ CS 2 . (6.49) [Under the larger group G generated by all rigid transformations of (X1 , . . . , Xr ) together with the scale changes Xi = bXi , S  = bS, the same sets have the stronger property of being uniformly most accurate equivariant; see Problem 6.77.] Examples 6.12.1–6.12.3 have the common feature that the equivariant confidence sets S(X) for θ = (θ1 , . . . , θr ) are characterized by an r-valued pivotal quantity, that is, a function h(X, θ) = (h1 (X, θ), . . . , hr (X, θ)) of the observations X and parameters θ being estimated that has a fixed distribution, and such that the most general equivariant confidence sets are of the form h(X, θ) ∈ A (6.50) 254 6. Invariance for some fixed set A.6 When the functions hi are linear in θ, the confidence sets C(X) obtained by solving (6.50) for θ are linear transforms of A (with random coefficients), so that the volume or invariant measure of C(X) is minimized by minimizing  ρ(v1 , . . . , vr ) dv1 . . . dvr (6.51) A for the appropriate ρ. The problem thus reduces to that of minimizing (6.51) subject to  Pθ0 {h(X, θ0 ) ∈ A} = p(v1 , . . . , vr ) dv1 . . . dvr = 1 − α, (6.52) A where p(v1 , . . . , vr ) is the density of the pivotal quantity h(X, θ). The minimizing A is given by   p(v1 , . . . , vr ) A∗ = v : >C , (6.53) ρ(v1 , . . . , vr ) with C determined by (6.52). The following is one more illustration of this approach. Example 6.12.4 Let X1 , . . . , Xm and Y1 , . . . , Yn be samples from N (ξ, σ 2 ) and N (η, τ 2 ) respectively, and consider the problem of estimating ∆ = τ 2 /σ 2 . Suffi  ciency and invariance under translations   Xi = Xi + a1 , Yj = Yj + a2 reduce the 2 data to SX = (Xi , −X̄)2 and SY2 = (Yj − Ȳ )2 . The problem of estimating ∆ also remains invariant under the scale changes Xi = b1 Xi , Yj = b2 Yj , 0 < b1 , b2 < ∞, which induce the transformations  = b1 SX , SX SY = b2 SY , σ  = b1 σ, τ  = b2 τ. (6.54) The totality of equivariant confidence sets for ∆ is given by ∆/V ∈ A, where 2 and A is any fixed set on the line satisfying V = SY2 /SX  1 ∈ A = 1 − α. (6.55) P∆=1 V To see this, suppose that C(SX , SY ) are any equivariant confidence sets for ∆. Then C must satisfy C(b1 SX , b2 SY ) = b22 C(SX , SY ), b21 (6.56) and hence ∆ ∈ C(SX , SY ) if and only if the pivotal quantity V /∆ satisfies ∆ S2 S2 ∆ C(SX , SY ) = C(1, 1) = A. = X2 ∈ X V SY SY2 As in Example 6.12.1, one may now wish to choose A so as to minimize either its Lebesgue measure A dv or the invariant measure A (1/v) dv. The resulting 6 More general results concerning the relationship of equivariant confidence sets and pivotal quantities are given in Problems 6.69–6.72. 6.13. Confidence Bands for a Distribution Function 255 confidence sets are of the form p(v) > C and vp(v) > C (6.57) respectively. In both cases, they are intervals V /b < ∆ < V /a [Problem 6.78(i)]. The values of a and b minimizing Lebesgue measure are tabled by Levy and Narula (1974); those for the invariant measure coincide with the uniformly most accurate unbiased intervals [Problem 6.78(ii)]. 6.13 Confidence Bands for a Distribution Function Suppose that X = (X1 , . . . , Xn ) is a sample from an unknown continuous cumulative distribution function F , and that lower and upper bounds LX and MX are to be determined such that with preassigned probability 1 − α the inequalities LX (u) ≤ F (u) ≤ MX (u) for all u hold for all continuous cumulative distribution functions F . This problem is invariant under the group G of transformations Xi = g(Xi ), i = 1, . . . , n, where g is any continuous strictly increasing function. The induced transformation in the parameter space is ḡF = F (g −1 ). If S(x) is the set of continuous cumulative distribution functions S(x) = {F : Lx (u) ≤ F (u) ≤ Mx (u) for all u}, then g ∗ S(x) = {ḡF : Lx (u) ≤ F (u) ≤ Mx (u) for all u} = {F : Lx [g −1 (u)] ≤ F (u) ≤ Mx [g −1 (u)] for all u}. For an equivariant procedure, this must coincide with the set   S(gx) = F : Lg(x1 ),...,g(xn ) (u) ≤ F (u) ≤ Mg(x1 ),...,g(xn ) (u) for all u . The condition of equivariance is therefore Lg(x1 ),...,g(xn ) [g(u)] = Lx (u), Mg(x1 ),...,g(xn ) [g(u)] = Mx (u) for all x and u. To characterize the totality of equivariant procedures, consider the empirical distribution function (EDF) Tx given by Tx (u) = i n for x(i) ≤ u < x(i+1) , i = 0, . . . , n, where x(1) < · · · < x(n) is the ordered sample and where x(0) = −∞, x(n+1) = ∞. Then a necessary and sufficient condition for L and M to satisfy the above equivariance condition is the existence of numbers a0 , . . . , an ; a0 , . . . , an such that Lx (u) = ai , Mx (u) = ai for x(i) < u < x(i+1) . 256 6. Invariance That this condition is sufficient is immediate. To see that it is also necessary, let u, u be any two points satisfying x(i) < u < u < x(i+1) . Given any y1 , . . . , yn and v with y(i) < v < y(i+1) , there exist g, g  ∈ G such that g(y(i) ) = g  (y(i) ) = x(i) , g  (v) = u . g(v) = u, If Lx , Mx are equivariant, it then follows that Lx (u ) = Ly (v) and Lx (u) = Ly (v), and hence that Lx (u ) = Lx (u) and similarly Mx (u ) = Mx (u), as was to be proved. This characterization shows Lx and Mx to be step functions whose discontinuity points are restricted to those of Tx . Since any two continuous strictly increasing cumulative distribution functions can be transformed into one another through a transformation ḡ, it follows that all these distributions have the same probability of being covered by an equivariant confidence band. (See Problem 6.84.) Suppose now that F is continuous but no longer strictly increasing. If I is any interval of constancy of F , there are no observations in I, so that I is also an interval of constancy of the sample cumulative distribution function. It follows that the probability of the confidence band covering F is not affected by the presence of I and hence is the same for all continuous cumulative distribution functions F . For any numbers ai , ai let ∆i , ∆i be determined by ai = i − ∆i , n ai = i − ∆i n Then it was seen above that any numbers ∆0 , . . . , ∆n ; ∆0 , . . . , ∆n define a confidence band for F , which is equivariant and hence has constant probability of covering the true F . From these confidence bands a test can be obtained of the hypothesis of goodness of fit F = F0 that the unknown F equals a hypothetical distribution F0 . The hypothesis is accepted if F0 ties entirely within the band, that is, if −∆i < F0 (u) − Tx (u) < ∆i for all x(i) < u < x(i+1) and all i = 1, . . . , n. Within this class of tests there exists no UMP member, and the most common choice of the ∆’s is ∆i = ∆i = ∆ for all i. The acceptance region of the resulting Kolmogorov-Smirnov test can be written as sup −∞ 0. Then a maximal invariant under G is (sgn xn , x1 /xn , . . . , xn−1 /xn ) where sgn x is 1 or −1 as x is positive or negative. (ii) Let X be the space of points x = (x1 , . . . , xn ) for which all coordinates are distinct, and let G be the group of all transformations xi = f (xi ), i = 1, . . . , n, such that f is a 1 : 1 transformation of the real line onto itself with at most a finite number of discontinuities. Then G is transitive over X. [(ii): Let x = (x1 , . . . , xn ) and x = (x1 , . . . , xn ) be any two points of X . Let I1 , . . . , In be a set of mutually exclusive open intervals which (together with their end points) cover the real line and such that xj ∈ Ij . Let I1 , . . . , In be a corresponding set of intervals for x1 , . . . , xn . Then there exists a transformation f which maps each Ij continuously onto Ij , maps xj into xj , and maps the set of n − 1 end points of I1 , . . . , In onto the set of end points of I1 , . . . , In .] Problem 6.3 Suppose M is any m × p matrix. Show that M T M is positive semidefinite. Also, show the rank of M T M equals the rank of M , so that in particular M T M is nonsingular if and only if m ≥ p and M is of rank p. Problem 6.4 (i) A sufficient condition for (6.8) to hold is that D is a normal subgroup of G. (ii) If G is the group of transformations x = ax + b, a = 0, −∞ < b < ∞, then the subgroup of translations x = x + b is normal but the subgroup x = ax is not. [The defining property of a normal subgroup is that given d ∈ D, g ∈ G, there exists d ∈ D such that gd = d g. The equality s(x1 ) = s(x2 ) implies x2 = dx1 for some d ∈ D, and hence ex2 = edx1 = d ex1 . The result (i) now follows, since s is invariant under D.] Section 6.3 Problem 6.5 Prove statements (i)-(iii) of Example 6.3.1. 258 6. Invariance Problem 6.6 Prove Theorem 6.3.1 (i) by analogy with Example 6.3.1, and (ii) by the method of Example 6.3.2. [Hint: A maximal invariant under G is the set {g1 x, . . . , gN x}. Problem 6.7 Consider the situation of Example 6.3.1 with n = 1, and suppose that f is strictly increasing on (0, 1). (i) The likelihood ratio test rejects if X < α/2 or X > 1 − α/2. (ii) The MP invariant test agrees with the likelihood ratio test when f is convex. (iii) When f is concave, the MP invariant test rejects when 1 α 1 α − C −∞ 0  (ii) Let X = (X1 , . . . , Xn ) have probability density f (x1 − kj=1 w1j βj , . . . , xn − k j=1 wnj βj ) where k < n, the w’s are given constants, the matrix (wij ) is of rank k, the β’s are unknown, and we wish to test f = f0 against f = f1 . The problem remains invariant under the transformations xi = xi + Σkj=1 wij γj , −∞ < γ1 , . . . , γk < ∞, and the most powerful invariant test is given by the rejection region     · · · f1 (x1 − w1j βj , . . . , xn − wnj βj )dβ1 , . . . , dβk     > C. · · · f0 (x1 − w1j βj , . . . , xn − wnj βj )dβ1 , . . . , dβk [A maximal invariant is given by y =  n n   x1 − a1r xr , x2 − r=n−k+1 r=n−k+1 for suitably chosen constants air .] a2r xr , . . . , xn−k − n  r=n−k+1  an−k,r xr 6.14. Problems 259 Problem 6.10 Let X1 , . . . , Xm ; Y1 , . . . , Yn be samples from exponential distributions with densities for σ −1 e−(x−ξ)/σ , for x ≥ ξ, and τ −1 e−(y−n)/τ for y ≥ η. (i) For testing τ /σ ≤ ∆ against τ /σ > ∆, there exists a UMP invariant test with respect to the group G : Xi = aXi + b, Yj = aYj + c, a > 0, −∞ < b, c < ∞, and its rejection region is  [y − min(y1 , . . . , yn )]  j > C. [xi − min(x1 , . . . , xm )] (ii) This test is also UMP unbiased. (iii) Extend these results to the case that only the r smallest X’s and the s smallest Y ’s are observed. [(ii): See Problem 5.15.] Problem 6.11 If X1 , . . . , Xn and Y1 , . . . , Yn are samples from N (ξ, σ 2 ) and N (η, τ 2 ) respectively, the problem of testing τ 2 = σ 2 against the two-sided alternatives τ 2 = σ 2 remains invariant under the group G generated by the transformations Xi = aXi + b, Yi = aYi + c, (a = 0), and Xi = Yi , Yi = Xi . There exists a UMP invariant test under G with rejection region    (Yi − Ȳ )2 (Xi = X̄)2 W = max  ,  ≥ k. (Xi = X̄) (Yi − Ȳ )2 [The ratio of the probability densities of W for τ 2 /σ 2 = ∆ and τ 2 /σ 2 = 1 is proportional to [(1 + w)/(∆ + w)]n−1 + [(1 + w)/(1 + ∆w)]n−1 for w ≥ 1. The derivative of this expression is ≥ 0 for all ∆.] Problem 6.12 Let X1 , . . . , Xn be a sample from a distribution with density x  1  x1  n f ...f , n τ τ τ where f (x) is either zero for x < 0 or symmetric about zero. The most powerful scale-invariant test for testing H : f = f0 against K : f = f1 rejects when ∞ 0 ∞ 0 v n−1 f1 (vx1 ) . . . f1 (vxn ) dv > C. v n−1 f0 (vx1 ) . . . f0 (vxn ) dv √ 2 Problem 6.13 Normal vs. double exponential. For f0 (x) = e−x /2 / 2π, −|x| f1 (x) e /2, the test of the preceding problem reduces to rejecting when  =  x2i / |xi | < C. (Hogg, 1972.) Note. The corresponding test when both location and scale are unknown is obtained in Uthoff (1973). Testing normality against Cauchy alternatives is discussed by Franck (1981). Problem 6.14 Uniform vs. triangular. 260 6. Invariance (i) For f0 (x) = 1 (0 < x < 1), f1 (x) = 2x (0 < x < 1), the test of Problem 6.12 reduces to rejecting when T = x(n) /x̄ < C. (ii) Under f0 , the statistic 2n log T is distributed as χ22n . (Quesenberry and Starbuck, 1976.) Problem 6.15 Show that the test of Problem 6.9(i) reduces to (i) [x(n) − x(1) ]/S < c for normal vs. uniform; (ii) [x̄ − x(1) ]/S < c for normal vs. exponential; (iii) [x̄ − x(1) ]/[x(n) − x(1) ] < c for uniform vs. exponential. (Uthoff, 1970.) Note. When testing for normality, one is typically not interested in distinguishing the normal from some other given shape but would like to know more generally whether the data are or are not consonant with a normal distribution. This is a special case of the problem of testing for goodness of fit, which is briefly discussed at the end of Section 6.13 and forms the topic of Chapter 14; also, see the many references in the notes to Chapter 14. Problem 6.16 Let X1 , . . . , Xn be independent and normally distributed. Suppose Xi has mean µi and variance σ 2 (which is the same for all i). Consider testing the null hypothesis that µi = 0 for all i. Using invariance considerations, find a UMP invariant test with respect to a suitable group of transformations in each of the following cases: (i). σ 2 is known and equal to one. (ii). σ 2 is unknown. Section 6.4 Problem 6.17 (i) When testing H : p ≤ p0 against K : p > p0 by means of the test corresponding to (6.13), determine the sample size required to obtain power β against p = p1 , α = .05, β = .9 for the cases p0 = .1, p1 = .15, .20, .25; p0 = .05, p1 = .10, .15, .20, .25; p0 = .01, p1 = .02, .05, .10, .15, .20. (ii) Compare this with the sample size required if the inspection is by attributes and the test is based on the total number of defectives. Problem 6.18 Two-sided t-test. (i) Let X1 , . . . , Xn be a sample from N (ξ, σ 2 ). For testing ξ = 0 against ξ = 0, there exists a UMP invariant test with respect to the group Xi = cXi , c = 0, given by the two-sided t-test (5.17). (ii) Let X1 , . . . , Xm , and Y1 , . . . , Yn be samples from N (ξ, σ 2 ) and N (η, σ 2 ) respectively. For testing η = ξ against η = ξ there exists a UMP invariant test with respect to the group Xi = aXi + b, Yj = aYj + b, a = 0, given by the two-sided t-test (5.30). 6.14. Problems 261 [(i): Sufficiency and invariance reduce the problem to |t|, which in the notation of Section 4 has the probability density pδ(t) + pδ (−t) for t > 0. The ratio of ∞ this density for δ = δ1 to its value for δ = 0 is proportional to 0 (eδ1 v + −δ1 v 2 )gt2 (v) dv, which is an increasing function of t and hence of |t|.] e Problem 6.19 Testing a correlation coefficient. Let (X1 , Y1 ), . . . , (Xn , Yn ) be a sample from a bivariate normal distribution. (i) For testing ρ ≤ ρ0 against ρ > ρ0 there exists a UMP invariant test with respect to the group of all transformations Xi = aXi + b, Yi = cY1 + d for which a, c > 0. This test rejects when the sample correlation coefficient R is too large. (ii) The problem of testing ρ = 0 against ρ = 0 remains invariant in addition under the transformation Yi = −Yi , Xi = Xi . With respect to the group generated by this transformation and those of (i) there exists a UMP invariant test, with rejection region |R| ≥ C. [(i): To show that the probability density pρ (r) of R has monotone likelihood ratio, apply the condition of Problem 3.27(i), to the expression 5.87 given for this density. Putting t = ρr + 1, the second derivative ∂ 2 log pρ (r)/∂ρ∂r up to a positive factor is ∞    ci cj ti+j−2 (j − i)2 (t − 1) + (i + j) i,j=0 . ∞   i 2 2 ci t i=0 To see that the numerator is positive for all t > 0, note that it is greater than 2 ∞  i=0 ci ti−2 ∞    cj tj (j − i)2 (t − 1) + (i + j) . j=i+1 Holding i fixed and using the inequality cj+1 < interior sum is ≥ 0.] 1 c , 2 j the coefficient of tj in the Problem 6.20 For testing the hypothesis that the correlation coefficient ρ of a bivariate normal distribution is ≤ ρ0 , determine the power against the alternative ρ = ρ1 , when the level of significance α is .05, ρ0 = .3, ρ1 = .5, and the sample size n is 50, 100, 200. Section 6.5 Problem 6.21 Almost invariance of a test φ with respect to the group G of either Problem 6.10(i) or Example 6.3.4 implies that φ is equivalent to an invariant test. Problem 6.22 The totality of permutations of K distinct numbers a1 , . . . , aK , for varying a1 , . . . , aK can be represented as a subset CK of Euclidean K-space RK , and the group G of Example 6.5.1 as the union of C2 , C3 , . . . . Let ν be the measure over G which assigns to a subset B of G the value ∞ k=2 µK (B ∩ CK ), 262 6. Invariance where µK denotes Lebesgue measure in EK . Give an example of a set B ⊂ G and an element g ∈ G such that ν(B) > 0 but ν(Bg) = 0. [If a, b, c, d are distinct numbers, the permutations g, g  taking (a, b) into (b, a) and (c, d) into (d, c) respectively are points in C2 , but gg  is a point in C4 .] Section 6.6 Problem 6.23 Show that (i) G1 of Example 6.6.11 is a group; 2 2 (ii) the test which rejects when X21 /X11 > C is UMP invariant under G1 ; (iii) the smallest group containing G1 and G2 is the group G of Example 6.6.11. Problem 6.24 Consider a testing problem which is invariant under a group G of transformations of the sample space, and let C be a class of tests which is closed under G, so that φ ∈ C implies φg ∈ C, where φg is the test defined by φg(x) = φ(gx). If there exists an a.e. unique UMP member φ0 of C, then φ0 is almost invariant. Problem 6.25 Envelope power function. Let S(α) be the class of all level-α tests of a hypothesis H, and let βα∗ (θ) be the envelope power function, defined by βα∗ (θ) = sup βφ (θ), φ∈S(α) where βφ denotes the power function of φ. If the problem of testing H is invariant under a group G, then βα∗ (θ) is invariant under the induced group Ḡ. Problem 6.26 (i) A generalization of equation (6.1) is   f (x) dPθ (x) = f (g −1 x) dPḡθ (x). A gA (ii) If Pθ1 is absolutely continuous with respect to Pθ0 , then Pḡθ1 is absolutely continuous with respect to Pḡθ0 and dPθ1 dPḡθ1 (x) = (gx) dPθ0 dPḡθ0 (a.e. Pθ0 ) . (iii) The distribution of dPθ1 /dPθ0 (X) when X is distributed as Pθ0 is the same as that of dPḡθ1 /dPḡθ0 (X  ) when X  is distributed as Pḡθ0 . Problem 6.27 Invariance of likelihood ratio. Let the family of distributions P = {Pθ , θ ∈ Ω} be dominated by µ, let pθ = dPθ /dµ, let µg −1 be the measure defined by µg −1 (A) = µ[g −1 (A)], and suppose that µ is absolutely continuous with respect to µg −1 for all g ∈ G. (i) Then pθ (x) = pḡθ (gx) dµ (gx) dµg −1 (a.e. µ). 6.14. Problems 263 (ii) Let Ω and ω be invariant under Ḡ, and countable. Then the likelihood ratio supΩ pθ (x)/ supω pθ (x) is almost invariant under G. (iii) Suppose that pθ (x) is continuous in θ for all x, that Ω is a separable pseudometric space, and that Ω and ω are invariant. Then the likelihood ratio is almost invariant under G. Problem 6.28 Inadmissible likelihood-ratio test. In many applications in which a UMP invariant test exists, it coincides with the likelihood-ratio test. That this is, however, not always the case is seen from the following example. Let P1 , . . . , Pn be n equidistant points on the circle x2 + y 2 = 4, and Q1 , . . . , Qn on the circle x2 + y 2 = 1. Denote the origin in the (x, y) plane by O, let 0 < α ≤ 12 be fixed, and let (X, Y ) be distributed over the 2n + 1 points P1 , . . . , Pn , Q1 , . . . , Qn , O with probabilities given by the following table: H K Pi α/n pi /n Qi (1 − 2α)/n 0 O α (n − 1)/n  where pi = 1. The problem remains invariant under rotations of the plane by the angles 2kπ/n (k = 0, 1, . . . , n − 1). The rejection region of the likelihood-ratio test consists of the points P1 , . . . , Pn , and its power is 1/n. On the other hand, the UMP invariant test rejects when X = Y = 0, and has power (n − 1)/n. Problem 6.29 Let G be a group of transformations of X , and let A be a σ-field of subsets of X , and µ a measure over (X , A). Then a set A ∈ A is said to be almost invariant if its indicator function is almost invariant. (i) The totality of almost invariant sets forms a σ-field A0 , and a critical function is almost invariant if and only if it is A0 -measurable. (ii) Let P = {Pθ , θ ∈ Ω} be a dominated family of probability distributions over (X , A), and suppose that ḡθ = θ for all ḡ ∈ Ḡ, θ ∈ Ω. Then the σ-field A0 of almost invariant sets is sufficient for P.  [Let λ = ci Pθi , be equivalent to P. Then dPg−1 θ dPθ dPθ (x) = (gx) =  (x) ci dPg−1 θi dλ dλ (a.e. λ), so that dPθ /dλ is almost invariant and hence A0 -measurable.] Problem 6.30 The UMP invariant test of Problem 6.13 is also UMP similar. [Consider the problem of testing α = 0 vs. α > 0 in the two-parameter exponential family with density  α  2 1−α  C(α, τ ) exp − 2 xi − 0 ≤ α < 1.] |xi | , 2τ τ Note. For the analogous result for the tests of Problem 6.14, 6.15, see Quesenberry and Starbuck (1976). Problem 6.31 The following UMP unbiased tests of Chapter 5 are also UMP invariant under change in scale: 264 6. Invariance (i) The test of g ≤ g0 in a gamma distribution (Problem 5.30). (ii) The test of b1 ≤ b2 in Problem 5.18(i). Section 6.7 Problem 6.32 The definition of d-admissibility of a test coincides with the admissibility definition given in Section 1.8 when applied to a two-decision procedure with loss 0 or 1 as the decision taken is correct or false. Problem 6.33 (i) The following example shows that α-admissibility does not always imply d-admissibility. Let X be distributed as U (0, θ), and consider the tests ϕ1 and ϕ2 which reject when respectively X < 1 and X < 32 for testing H : θ = 2 against K : θ = 1. Then for α = 34 , ϕ1 and ϕ2 are both α-admissible but ϕ2 is not d-admissible. (ii) Verify the existence of the test ϕ0 of Example 6.7.12. √ Problem 6.34 (i) The acceptance region T1 / T2 ≤ C of Example 6.7.13 is a convex set in the (T1 , T2 ) plane. (ii) In Example √ 6.7.13, the conditions of Theorem 6.7.1 are not satisfied for the sets A : T1 / T2 ≤ C and Ω : ξ > k. Problem 6.35 (i) In Example 6.7.13 (continued) show that there exist CO , C1 such that λ0 (η) and λ1 (η) are probability densities (with respect to Lebesgue measure). (ii) Verify the densities h0 and h1 . Problem 6.36 Verify (i) the admissibility of the rejection region (6.24); (ii) the expression for I(z) given in the proof of Lemma 6.7.1. Problem 6.37 Let X1 , . . . , Xm ; Y1 , . . . , Yn be independent N (ξ, σ 2 ) and N (η, σ 2 ) respectively. The one-sided t-test of H : δ = ξ/σ ≤ 0 is admissible against the alternatives (i) 0 < δ < δ1 for any δ1 > 0; (ii) δ > δ2 for any δ2 > 0. Problem 6.38 For the model of the preceding problem, generalize Example 6.7.13 (continued) to show that the two-sided t-test is a Bayes solution for an appropriate prior distribution. Problem 6.39 Suppose X = (X1 , . . . , Xk )T is multivariate normal with unknown mean vector (θ1 , . . . , θk )T and known nonsingular covariance matrix Σ. Consider testing the null hypothesis θi = 0 for all i against θi = 0 for some i. Let C be any closed convex subset of k-dimensional Euclidean space, and let φ be the test that accepts the null hypothesis if X falls in C. Show that φ is admissible. Hint: First assume Σ is the identity and use Theorem 6.7.1. [An alternative proof is provided by Strasser (1985, Theorem 30.4).] 6.14. Problems 265 Section 6.9 Problem 6.40  Wilcoxon two-sample test. Let Uij = 1 or 0 as Xi < Yj or Xi >  Yj , and let U = Uij be the number of pairs Xi , Yj with Xi < Yj .  (i) Then U = Si − 12 n(n + 1), where S1 < · · · < Sn are the ranks of the Y ’s so that the test with rejection region U > C is equivalent to the Wilcoxon test. (ii) Any given arrangement of x’s and y’s can be transformed into the arrangement x . . . xy . . . y through a number of interchanges of neighboring elements. The smallest number of steps in which this can be done for the observed arrangement is mn − U . Problem 6.41 Expectation and variance of Wilcoxon statistic. If the X’s and Y ’s are samples from continuous distributions F and G respectively, the expectation and variance of the Wilcoxon statistic U defined in the preceding problem are given by    U E = P {X < Y } = F dG (6.59) mn and  mnV ar U mn   =  F dG + (n − 1)  +(m − 1) (1 − G)2 dF F 2 dG − (m + n − 1) (6.60)  Under the hypothesis G = F , these reduce to     U 1 m+n+1 U = , V ar = . E mn 2 mn 12mn 2 F dG . (6.61) Problem 6.42 (i) Let Z1 , . . . , ZN be independently distributed with densities f1 , . . . , fN , and let the rank of Zi be denoted by Ti . If f is any probability density which is positive whenever at least one of the fi is positive, then      fN V(tN ) f1 V(t1 ) 1   ···   . (6.62) P {T1 = t1 , . . . , TN = tn } = E N! f V(t1 ) f V(tN ) where V(1) < · · · < V(N ) is an ordered sample from a distribution with density f . (ii) If N = m + n, f1 = · · · = fm = f , fm+1 = · · · = fm+n = g, and S1 < · · · < Sn denote the ordered ranks of Zm+1 , . . . , Zm+n among all the Z’s, the probability distribution of S1 , . . . , Sn is given by (6.27). [(i): The probability in question is . . . f1 (z1 ) . . . fN (zN ) dz1 · · · dzN integrated over the set in which zi is the ti th smallest of the z’s for i = 1, . . . , N . Under the transformation wti = zi the integral becomes . . . f1 (wt1 ) . . . fN (wtN ) dw1 · · · dwN integrated over the set w1 < · · · < wN . The desired result now follows from the fact that the probability density of the order statistics V(1) < · · · < V(N ) is N !f (w1 ) · · · f (wN ) for w1 < . . . < wN .] 266 6. Invariance Problem 6.43 (i) For any continuous cumulative distribution function F , define F −1 (0) = −∞, F −1 (y) = inf{x : F (x) = y} for 0 < y < 1, F −1 (1) = ∞ if F (x) < 1 for all finite x, and otherwise inf{x : F (x) = 1}. Then F [F −1 (y)] = y for all 0 ≤ y ≤ 1, but F −1 [F (y)] may be < y. (ii) Let Z have a cumulative distribution function G(z) = h[F (z)], where F and h are continuous cumulative distribution functions, the latter defined over (0,1). If Y = F (Z), then P {Y < y} = h(y) for all 0 ≤ y ≤ 1. (iii) If Z has the continuous cumulative distribution function F , then F (Z) is uniformly distributed over (0, 1). [(ii): P {F (Z) < y} = P {Z < F −1 (y)} = F [F −1 (y)] = y.] Problem 6.44 Let Zi have a continuous cumulative distribution function Fi (i = 1, . . . , N ), and let G be the group of all transformations Zi = f (Zi ) such that f is continuous and strictly increasing. (i) The transformation induced by f in the space of distributions is Fi = Fi (f −1 ).  ) belong to (ii) Two N -tuples of distributions (F1 , . . . , FN ) and (F1 , . . . , FN the same orbit with respect to Ḡ if and only if there exist continuous distribution functions h1 , . . . , hN defined on (0,1) and strictly increasing continuous distribution functions F and F ’ such that Fi = hi (F ) and Fi = hi (F  ). [(i): P {f (Zi ) ≤ y} = P {Zi ≤ f −1 (y)} = Fi [f −1 (y)]. (ii): If Fi = hi (F ) and the Fi are on the same orbit, so that Fi = Fi (f −1 ), then Fi = hi (F  ) with F  = F (f −1 ). Conversely, if Fi = hi (F ), Fi = hi (F  ), then Fi = Fi (f −1 ) with f = F −1 (F ).] Problem 6.45 Under the assumptions of the preceding problem, if Fi = hi (F ), the distribution of the ranks T1 , . . . , TN of Z1 , . . . , ZN depends only on the hi , not on F . If the hi are differentiable, the distribution of the Ti is given by     E h1 U(t1 ) . . . hN U(tN ) P {T1 = t1 , . . . , TN = tn } = , (6.63) N! where U(1) < · · · < U(N ) is an ordered sample of size N from the uniform distribution U (0, 1). [The left-hand side of (6.63) is the probability that of the quantities F (Z1 ), . . . , F (ZN ), the ith one is the ti th smallest for i = 1, . . . , N . This is given by . . . h1 (y1 ) . . . hN (yN ) dy integrated over the region in which yi is the ti th smallest of the y’s for i = 1, . . . , N . The proof is completed as in Problem 6.42.] Problem 6.46 Distribution of order statistics. (i) If Z1 , . . . , ZN is a sample from a cumulative distribution function F with density f , the joint density of Yi = Z(si ) , i = 1, . . . , n, is N !f (y1 ) . . . f (yn ) (s1 − 1)!(s2 − s1 − 1)! . . . (N − sn )! (6.64) ×[F (y1 )]s1 −1 [F (y2 ) − F (y1 )]s2 −s1 −1 . . . [1 − F (yn )]N −sn for y1 < · · · < yn . 6.14. Problems 267 (ii) For the particular case that the Z’s are a sample from the uniform distribution on (0,1), this reduces to N! (s1 − 1)!(s2 − s1 − 1)! . . . (N − sn )! (6.65) y1s1 −1 (y2 − y1 )s2 −s1 −1 . . . (1 − yn )N −sn . For n = 1, (6.65) is the density of the beta-distribution Bs,N −s+1 , which therefore is the distribution of the single order statistic Z(s) from U (0, 1). (iii) Let the distribution of Y1 , . . . , Yn be given by (6.65), and let Vi be defined by Yi = Vi Vi+1 . . . Vn for i = 1, . . . , n. Then the joint distribution of the Vi is n  N! v si −1 (1 − vi )si+1 −si −1 (sn+1 = N + 1), (s1 − 1)! . . . (N − sn )! i=1 i so that the Vi are independently distributed according to the betadistribution Bsi ,si+1 −si . [(i): If Y1 = Z(s1 ) , . . . , Yn = Z(sn ) and Yn+1 , . . . , YN are the remaining Z’s in the original order of their subscripts, the joint density of Y1 , . . . , Yn is N (N − 1) . . . (N −n+1) . . . f (yn+1 ) . . . f (yN ) dyn+1 . . . dyN integrated over the region in which s1 − 1 of the y’s are < y1 , s2 − s1 − 1 between y1 and y2 , . . ., and N − sn > yn . Consider any set where a particular s1 − 1 of the y’s is < y1 , a particular s2 − s1 − 1 of them is between y1 and y2 , and so on, There are N !/(s1 − 1)! . . . (N − sn )! of these regions, and the integral has the same value over each of them, namely [F (y1 )]s1 −1 [F (y2 )−F (y1 )]s2 −s1 −1 . . . [1−F (yn )]N −sn .] Problem 6.47 (i) If X1 , . . . , Xm and Y1 , . . . , Yn are samples with continuous cumulative distribution functions F and G = h(F ) respectively, and if h is differentiable, the distribution of the ranks S1 < . . . < Sn of the Y ’s is given by     E h U(s1 ) . . . h U(sn ) m+n (6.66) P {S1 = s1 , . . . , Sn = sn } = m where U(1) < · · · < U(m+n) is an ordered sample from the uniform distribution U (0, 1). (ii) If in particular G = F k , where k is a positive integer, (6.66) reduces to P {S1 = = s1 , . . . , Sn = sn } n Γ (sj+1 ) kn  Γ (sj + jk − j) m+n · . Γ (s ) Γ (s + jk − j) j j+1 m j=1 (6.67) Problem 6.48 For sufficiently small θ > 0, the Wilcoxon test at level 3  N α=k , k a positive integer, n maximizes the power (among rank tests) against the alternatives (F, G) with G = (1 − θ)F + θF 2 . 268 6. Invariance Problem 6.49 An alternative proof of the optimum property of the Wilcoxon test for detecting a shift in the logistic distribution is obtained from the preceding problem by equating F (x − θ) with (1 − θ)F (x) + θF 2 (x), neglecting powers of θ higher than the first. This leads to the differential equation F − θF  = (1 − θ)F + θF 2 , the solution of which is the logistic distribution. Problem 6.50 Let F0 be a family of probability measures over (X , A), and let C be a class of transformations of the space X . Define a class F1 of distributions by F1 ∈ F1 if there exists F0 ∈ F0 and f ∈ C such that the distribution of f (X) is F1 when that of X is F0 . If φ is any test satisfying (a) EF0 φ(X) = α for all F0 ∈ F0 , and (b) φ(x) ≤ φ[f (x)] for all x and all f ∈ C, then φ is unbiased for testing F0 against F1 Problem 6.51 Let X1 , . . . , Xm ; Y1 , . . . , Yn be samples from a common continuous distribution F . Then the Wilcoxon statistic U defined in Problem 6.40 is distributed symmetrically about 12 mn even when m = n. Problem 6.52 (i) If X1 , . . . , Xm and Y1 , . . . , Yn are samples from F (x) and G(y) = F (y − ∆) respectively (F continuous), and D(1) < · · · < D(mn) denote the ordered differences Yj − Xi , then   P D(k) < ∆ < D(mn+1−k) = P0 [k ≤ U ≤ mn − k], where U is the statistic defined in Problem 6.40 and the probability on the right side is calculated for ∆ = 0. (ii) Determine the above confidence interval for ∆ when m = n = 6, the confidence coefficient is 20 , and the observations are x : .113, .212, .249, 21 .522, .709, .788, and y : .221, .433, .724, .913, .917, 1.58. (iii) For the data of (ii) determine the confidence intervals based on Student’s t for the case that F is normal. Hint: D(i) ≤ ∆ < D(i+1) if and only if U∆ = mn − i, where U∆ is the statistic U of Problem 6.40 calculated for the observations X1 , . . . , Xm ; Y1 − ∆, . . . , Yn − ∆. [An alternative measure of the amount by which G exceeds F (without assuming a location model) is p = P {X < Y }. The literature on confidence intervals for p is reviewed in Mee (1990).] Problem 6.53 (i) Let X, X  and Y , Y ’ be independent samples of size 2 from continuous distributions F and G respectively. Then p = P {max(X, X  ) < min(Y, Y  )} + P {max(Y, Y  ) < min(X, X  )} = 1 3 + 2∆, where ∆ = (F − G)2 d[(F + G)/2]. (ii) ∆ = 0 if and only if F = G. [(i): p = (1 − F )2 dG2 + (1 − G)2 dF 2 which after some computation reduces to the stated form. 6.14. Problems 269 (ii): ∆ = 0 implies F (x) = G(x) except on a set N which has measure zero both under F and G. Suppose that G(x1 ) − F (x1 ) = η > 0. Then there exists x0 such that G(x0 ) = F (x0 ) + 12 η and F (x) < G(x) for x0 ≤ x ≤ x1 . Since G(x1 ) − G(x0 ) > 0, it follows that ∆ > 0.] Problem 6.54 Continuation. (i) There exists at every significance level α a test of H : G = F which has power > α against all continuous alternatives (F, G) with F = G. (ii) There does not exist a nonrandomized unbiased rank test of H against all G = F at level 3  m+n α=1 . n [(i): let Xi , Xi ; Yi , Yi (i = 1, . . . , n) be independently distributed, the X’s with  distribution F , the Y ’s with distribution G, and let Vi = 1 if max(X i , X1 ) <    min(Yi , Yi ) or max(Yi , Yi ) < min(Xi , Xi ), and Vi = 0 otherwise. Then Vi has a binomial distribution with the probability p defined in Problem 6.53, and the problem reduces to that of testing p = 13 against p > 13 . (ii): Consider the particular alternatives for which P {X < Y } is either 1 or 0.] Problem 6.55 (i) Let X1 , . . . , Xm ; Y1 , . . . , Yn be i.i.d. according to a continuous distribution F , let the ranks of the Y ’s be S1 < · · · < Sn , and let T = h(S1 ) + · · · + h(Sn ). Then if either m = n or h(s) + h(N  + 1 − s) is independent of s, the distribution of T is symmetric about n N i=1 h(i)/N . (ii) Show that the two-sample Wilcoxon and normal-scores statistics are symmetrically distributed under H, and determine their centers of symmetry.  [(i): Let Si = N + 1 − Si , and use the fact that T  = h(Sj ) has the same distribution under H as T .] Section 6.10 Problem 6.56 (i) Let m and n be the numbers of negative and positive observations among Z1 , . . . , ZN , and let S1 < · · · < Sn denote the ranks of the positive Z’s among |Z1 |, . . . |ZN |. Consider the N + 12 N (N − 1) distinct sums  Zi +Zj with i = j as well as i = j. The Wilcoxon signed rank statistic Sj , is equal to the number of these sums that are positive. (ii) If the common distribution of the Z’s is D, then    E Sj = 12 N (N + 1) − N D(0) − 12 N (N − 1) D(−z) dD(z). [(i) Let K be the required number of positive sums. Since Zi + Zj is positive if and  only  if the Z corresponding to the larger of |Zi | and |Zj | is positive, N N K = i=1 j=1 Uij where Uij = 1 if Zj > 0 and |Zi | ≤ Zj and Uij = 0 otherwise.] 270 6. Invariance Problem 6.57 Let Z1 , . . . , ZN be a sample from a distribution with density f (z − θ), where f (z) is positive for all z and f is symmetric about 0, and let m, n, and the Sj be defined as in the preceding problem. (i) The distribution of n and the Sj is given by P {the number of positive Z’s is n and S1 = s1 , . . . , Sn = sn } (6.68)          f V(r1 ) + θ . . . f V(rm ) + θ f V(s1 ) − θ . . . f V(sn ) − θ 1     , = NE 2 f V(1) . . . f V(N ) where V(1) < · · · < V(N ) , is an ordered sample from a distribution with density 2f (v) for v > 0, and 0 otherwise. (ii) The rank test of the hypothesis of symmetry with respect to the origin, which maximizes the derivative of the power function at θ = 0 and hence maximizes the power for sufficiently small θ > 0, rejects, under suitable regularity conditions, when  n    f (V(sj ) −E > C. f (V(sj ) j=1 (iii) In the particular case that f (z)  is a normal density with zero mean, the rejection region of (ii) reduces to E(V (sj ) > C, where V(1) < · · · < V(N ) is an ordered sample from a χ-distribution with 1 degree of freedom. (iv) Determine a density f such that the one-sample Wilcoxon test is most powerful against the alternatives f (z − θ) for sufficiently small positive θ. [(i): Apply Problem 6.42(i) to find an expression for P {S1 = s1 , . . . , Sn = sn given that the number of positive Z’s is n}.] Problem 6.58 An alternative expression for (6.68) is obtained if the distribution of Z is characterized by (ρ, F, G). If then G = h(F ) and h is differentiable, the distribution of n and the Sj is given by   ρm (1 − ρ)n E h (U(s1 ) ) · · · h (U(sn ) ) , (6.69) where U(1) , < · · · < U(N ) is an ordered sample from U (0, 1). Problem 6.59 Unbiased tests of symmetry. Let Z1 , . . . , ZN , be a sample, and let φ be any rank test of the hypothesis of symmetry with respect to the origin such that zi ≤ zi for all i implies φ(z1 , . . . , zN ) ≤ φ(z1 , . . . , z  N ). Then φ is unbiased against the one-sided alternatives that the Z’s are stochastically larger than some random variable that has a symmetric distribution with respect to the origin. Problem 6.60 The hypothesis of randomness.7 Let Z1 , . . . , ZN be independently distributed with distributions F1 , . . . , FN , and let Ti denote the rank of Zi among the Z’s For testing the hypothesis of randomness F1 = · · · = FN against 7 Some tests of randomness are treated in Diaconis (1988). 6.14. Problems 271 the alternatives K of an upward trend, namely that Zi is stochastically increasing with i, consider the rejection regions  iti > C (6.70) and  iE(V(ti ) ) > C, (6.71) where V(1) < · · · < V(N ) is an ordered sample from a standard normal distribution and where ti is the value taken on by Ti . (i) The second of these tests is most powerful among rank tests against the normal alternatives F = N (γ + iδ, σ 2 ) for sufficiently small δ. (ii) Determine alternatives against which the first test is a most powerful rank test. (iii) Both tests are unbiased against the alternatives of an upward trend; so is  any rank test φ satisfying φ(z1 , . . . , zN ) ≤ φ(z1 , . . . , zN ) for any two points   for which i < j, zi < zj implies zi < zj for all i and j. [(iii): Apply Problem 6.50 with C the class of transformations z1 = z1 , zi = fi (zi ) for i > 1, where z < f2 (z) < · · · < fN (z) and each fi is nondecreasing. If F0 is the class of N -tuples (F1 , . . . , FN ) with F1 = · · · = FN , then F1 coincides with the class K of alternatives.] Problem 6.61 In the preceding problem let Uij = 1 if (j − i)(Zj − Zi ) > 0, and = 0 otherwise.  (i) The test statistic iTi , can be expressed in terms of the U ’s through the relation N  i=1 iTi =  N (N + 1)(N + 2) (j − i)Uij + , 6 i C as another rejection region for the preceding problem.  [(i): Let Vij = 1 or 0 as Zi ≤ Zi or Zi > Zj . Then Tj = N i=1 Vij , and Vij = Uij or N N  1 − Uij as i < j or i ≥ j. Expressing j=1 jTj = j=1 j N i=1 Vij in terms of the U ’s and using the fact that Uij = Uji , the result follows by a simple calculation.] Problem 6.62 The hypothesis of independence. Let (X1 , Y1 ), . . . , (XN , YN ) be a sample from a bivariate distribution, and (X(1) , Z1 ), . . . , (X(N ) , ZN ) be the same sample arranged according to increasing values of the X’s so that the Z’s are a permutation of the Y ’s. Let Ri be the rank of Xi among the X’s, Si the rank of Yi among the Y ’s, and Ti the rank of Zi among the Z’s, and consider the hypothesis of independence of X and Y against the alternatives of positive regression dependence. 272 6. Invariance (i) Conditionally, given (X(1) , . . . , X(N ) ), this problem is equivalent to testing the hypothesis of randomness of the Z’s against the alternatives of an upward trend. (ii) The test (6.70) is equivalent to rejecting when the rank correlation coefficient     (Ri − R̄)(Si − S̄) 12 N +1 N +1 = − − S R i i   N3 − N 2 2 (Ri − R̄2 ) (Si − S̄)2 is too large. (iii) An alternative expression for the rank correlation coefficient8 is   6 6 1− 3 (Si − Ri )2 = 1 − 3 (Ti − i)2 . N −N N −N (iv) The test U > C ofProblem 6.61(ii) is equivalent to rejecting when Kendall’s t-statistic i θ0 (in the presence of nuisance parameters ϑ) remains invariant under a group Gθ0 and that A(θ0 ) is a UMP invariant acceptance region for this hypothesis at level α. Let the associated confidence sets S(x) = {θ : x ∈ A(θ)} 8 For further material on these and other tests of independence, see Kendall (1970), Aiyar, Guillier, and Albers (1979), Kallenberg and Ledwina (1999). 6.14. Problems 273 be one-sided intervals S(x) = {θ : θ(x) ≤ θ}, and suppose they are equivariant under all Gθ and hence under the group G generated by these. Then the lower confidence limits θ(X) are uniformly most accurate equivariant at confidence level 1 − α in the sense of minimizing Pθ,ϑ {θ(X) ≤ θ } for all θ < θ. (ii) Let X1 , . . . , Xn be independently distributed as N (ξ, σ 2 ). The upper con 2 2 fidence limits σ ≤ (Xi − X̄) /C0 of Example 5.5.1 are uniformly most accurate equivariant under the group Xi = Xi + c, −∞ < c < ∞. They are also equivariant (and hence uniformly most accurate equivariant) under the larger group Xi = aXi + c, −∞ < a, c < ∞. Problem 6.66 Counterexample. The following example shows that the equivariance of S(x) assumed in the paragraph following Lemma 6.11.1 does not follow from the other assumptions of this lemma. In Example 6.5.1, let n = 1, let G(1) be the group G of Example 6.5.1, and let G(2) be the corresponding group when the roles of Z and Y = Y1 are reversed. For testing H(θ0 ) : θ = θ0 against θ = θ0 let Gθ0 be equal to G(1) augmented by the transformation Y  = θ0 − (Y1 − θ0 ) when θ ≤ 0, and let Gθ0 be equal to G(2) augmented by the transformation Z  = θ0 − (Z − θ0 ) when θ > 0. Then there exists a UMP invariant test of H(θ0 ) under Gθ0 for each θ0 , but the associated confidence sets S(x) are not equivariant under G = {Gθ , −∞ < θ < ∞}. Problem 6.67 (i) Let X1 , . . . , Xn be independently distributed as N (ξ, σ 2 ), and let θ = ξ/σ. The lower confidence bounds θ for θ, which at confidence level 1−α are uniformly most accurate invariant under the transformations Xi = aXi , are   √ nX̄ −1 θ=C  (Xi − X̄)2 /(n − 1) where the function C(θ) is determined from a table of noncentral t so that 0 / √ nX̄ ≤ C(θ) = 1 − α. Pθ  (Xi − X̄)2 /(n − 1) (ii) Determine θ when the x’s are 7.6, 21.2, 15.1, 32.0, 19.7, 25.3, 29.1, 18.4 and the confidence level is 1 − α = .95. Problem 6.68 (i) Let (X1 , Y1 ), . . . , (Xn , Yn ) be a sample from a bivariate normal distribution, and let    (Xi − X̄)(Yi − Ȳ ) −1 , ρ=C   (Xi − X̄)2 (Yi − Ȳ )2 where C(ρ) is determined such that 0 /  (Xi − X̄)(Yi − Ȳ ) Pθ ≤ C(ρ) = 1 − α.   (Xi − X̄)2 (Yi − Ȳ )2 Then ρ is a lower confidence limit for the population correlation coefficient ρ at confidence level 1 − α; it is uniformly most accurate invariant with 274 6. Invariance respect to the group of transformations Xi = aXi + b, Yi = cYi + d, with ac > 0, −∞ < b, d < ∞. (ii) Determine ρ at level 1 − α = .95 when the observations are (12.9,.56), (9.8,.92), (13.1,.42), (12.5,1.01), (8.7,.63), (10.7,.58), (9.3,.72), (11.4,.64). Note. The following problems explore the relationship between pivotal quantities and equivariant confidence sets. For more details see Arnold (1984). Let X be distributed according Pθ,ϑ , and consider confidence sets for θ that are equivariant under a group G∗ , as in Section 6.11. If w is the set of possible θ-values, define a group G̃ on X × w by g̃(θ, x) = (gx, ḡθ). Problem 6.69 Let V (X, θ) be any pivotal quantity [i.e. have a fixed probability distribution independent of (θ, ϑ)], and let B be any set in the range space of V with probability P (V ∈ B) = 1 − α. Then the sets S(x) defined by θ ∈ S(x) if and only if V (θ, x) ∈ B (6.72) are confidence sets for θ with confidence coefficient 1 − α. Problem 6.70 (i) If G̃ is transitive over X × w and V (X, θ) is maximal invariant under G̃, then V (X, θ) is pivotal. (ii) By (i), any quantity W (X, θ) which is invariant under G̃ is pivotal; give an example showing that the converse need not be true. Problem 6.71 Under the assumptions of the preceding problem, the confidence set S(x) is equivariant under G∗ . Problem 6.72 Under the assumptions of Problem 6.70, suppose that a family of confidence sets S(x) is equivariant under G∗ . Then there exists a set B in the range space of the pivotal V such that (6.72) holds. In this sense, all equivariant confidence sets can be obtained from pivotals. [Let A be the subset of X × w given by A = {(x, θ) : θ ∈ S(x)}. Show that g̃A = A, so that any orbit of G̃ is either in A or in the complement of A. Let the maximal invariant V (x, θ) be represented as in Section 6.2 by a uniquely defined point on each orbit, and let B be the set of these points whose orbits are in A. Then V (x, θ) ∈ B if and only if (x, θ) ∈ A.] Note. Problem 6.72 provides a simple check of the equivariance of confidence sets. In Example 6.12.2, for instance, the confidence sets (6.43) are based on the pivotal vector (X1 − ξ1 , . . . , Xr − ξr ), and hence are equivariant. Section 6.12 Problem 6.73 In Examples 6.12.1 and 6.12.2 there do not exist equivariant sets that uniformly minimize the probability of covering false values. Problem 6.74 In Example 6.12.1, the density p(v) of V = 1/S 2 is unimodal. Problem 6.75 Show that in Example 6.12.1, 6.14. Problems 275 (i) the confidence sets σ 2 /S 2 ∈ A∗∗ with A∗∗ given by (6.42) coincide with the uniformly most accurate unbiased confidence sets for σ 2 ; (ii) if (a, b) is best with respect to (6.41) for σ, then (ar , br ) is best for σ r (r > 0). Problem 6.76 Let X1 , . . . , Xr be i.i.d. N (0, 1), and let S 2 be independent of √ √ the X’s and distributed as χ2ν . Then the distribution of (X1 /S ν, . . . , Xr /S ν) is a central multivariate t-distribution, and its density is − 1 (ν+r) 2 Γ( 12 (ν + r)) 1 2 . 1 + v p(v1 , . . . , vr ) = i r/2 ν (πν) Γ(ν/2) Problem 6.77 The confidence sets (6.49) are uniformly most accurate equivariant under the group G defined at the end of Example 6.12.3. Problem 6.78 In Example 6.12.4, show that (i) both sets (6.57) are intervals; (ii) the sets given by vp(v) > C coincide with the intervals (5.41). Problem 6.79 Let X1 , . . . , Xm ; Y1 , . . . , Yn be independently normally distributed as N (ξ, σ 2 ) and N (η, σ 2 ) respectively. Determine the equivariant confidence sets for η − ξ that have smallest Lebesgue measure when (i) σ is known; (ii) σ is unknown. Problem 6.80 Generalize the confidence sets of Example 6.11.3 to the case that the Xi are N (ξi , di σ 2 ) where the d’s are known constants. Problem 6.81 Solve the problem corresponding to Example 6.12.1 when (i) X1 , . . . , Xn is a sample from the exponential density E(ξ, σ), and the parameter being estimated is σ; (ii) X1 , . . . , Xn is a sample from the uniform density U (ξ, ξ + τ ), and the parameter being estimated is τ . Problem 6.82 Let X1 , . . . , Xn be a sample from the exponential distribution E(ξ, σ). With respect to the transformations Xi = bXi +a determine the smallest equivariant confidence sets (i) for σ, both when size is defined by Lebesgue measure and by the equivariant measure (6.41); (ii) for ξ. Problem 6.83 Let Xij (j = 1, . . . , ni ; i = 1, . . . , s) be samples from the exponential distribution E(ξi , σ). Determine the smallest equivariant confidence sets  for (ξ1 , . . . , ξr ) with respect to the group Xij = bXij + ai . 276 6. Invariance Section 6.13 Problem 6.84 If the confidence sets S(x) are equivariant under the group G, then the probability Pθ {θ ∈ S(X)} of their covering the true value is invariant under the induced group Ḡ. Problem 6.85 Consider the problem of obtaining a (two-sided) confidence band for an unknown continuous cumulative distribution function F . (i) Show that this problem is invariant both under strictly increasing and strictly decreasing continuous transformations Xi = f (Xi ), i = 1, . . . , n, and determine a maximal invariant with respect to this group. (ii) Show that the problem is not invariant under the transformation ⎧ if |Xi | ≥ 1, ⎨ Xi  Xi − 1 if 0 < Xi < 1, Xi = ⎩ Xi + 1 if − 1 < Xi < 0. [(ii): For this transformation g, the set g ∗ S(x) is no longer a band.] 6.15 Notes Invariance considerations were introduced for particular classes of problems by Hotelling (1936) and Pitman (1939b). The general theory of invariant and almost invariant tests, together with its principal parametric applications, was developed by Hunt and Stein (1946) in an unpublished paper. In their paper, invariance was not proposed as a desirable property in itself but as a tool for deriving most stringent tests (cf. Chapter 8). Apart from this difference in point of view, the present account is based on the ideas of Hunt and Stein, about which E. L. Lehmann learned through conversations with Charles Stein during the years 1947–1950. Of the admissibility results of Section 6.7, Theorem 6.7.1 is due to Birnbaum (1955) and Stein (1956a); Example 6.7.13 (continued) and Lemma 6.7.1, to Kiefer and Schwartz (1965). The problem of minimizing the volume or diameter of confidence sets is treated in DasGupta (1991). Deuchler (1914) appears to contain the first proposal of the two-sample procedure known as the Wilcoxon test, which was later discovered independently by many different authors. A history of this test is given by Kruskal (1957). Hoeffding (1951) derives a basic rank distribution of which (6.20) is a special case, and from it obtains locally optimum tests of the type (6.21). 7 Linear Hypotheses 7.1 A Canonical Form Many testing problems concern the means of normal distributions and are special cases of the following general univariate linear hypothesis. Let X1 , . . . , Xn be independently normally distributed with means ξ1 , . . . , ξn and common variance σ 2 . The vector of means1 ξ is known to lie in a given s-dimensional linear subspace Ω (s < n), and the hypothesis H to be tested is that ξ lies in a given (s − r)-dimensional subspace ω of Ω (r ≤ s). Example 7.1.1 In the two-sample problem of testing equality of two normal means (considered with a different notation in Section 5.3), it is given that ξi = ξ for i = 1, . . . , n1 and ξi = η for i = n1 + 1, . . . , n1 + n2 , and the hypothesis to be tested is η = ξ. The space Ω is then the space of vectors (ξ, . . . , ξ, η, . . . , η) = ξ(1, . . . , 1, 0, . . . , 0) + η(0, . . . , 0, 1, . . . , 1) spanned by (1, . . . , 1, 0, . . . , 0) and (0, . . . , 0, 1, . . . , 1), so that s = 2. Similarly, ω is the set of all vectors (ξ, . . . , ξ) = ξ(1, . . . , 1) and hence r = 1. Another hypothesis that can be tested in this situation is η = ξ = 0. The space ω is then the origin, s − r = 0 and hence r = 2. The more general hypothesis ξ = ξ0 , η = η0 is not a linear hypothesis, since ω does not contain the origin. However, it reduces to the previous case through the transformation Xi = Xi − ξ0 (i = 1, . . . , n1 ), Xi = Xi − η0 (i = n1 + 1, . . . , n1 + n2 ). 1 Throughout this chapter, a fixed coordinate system is assumed given in n-space. A vector with components ξ1 , . . . , ξn is denoted by ξ, and an n × 1 column matrix with elements ξ1 , . . . , ξn by ξ. 278 7. Linear Hypotheses Example 7.1.2 The regression problem of Section 5.6 is essentially a linear hypothesis. Changing the notation to make it conform with that of the present section, let ξi = α + βti , where α, β are unknown, and the ti known and not all equal. Since Ω is the space of all vectors α(1, . . . , 1) + β(t1 , . . . , tn ), it has dimension s = 2. The hypothesis to be tested may be α = β = 0 (r = 2) or it may only specify that one of the parameters is zero (r = 1). The more general hypotheses α = α0 , β = β0 can be reduced to the previous case by letting Xi = Xi − α0 , −β0 ti , since then E(Xi ) = α + β  ti with α = α − α0 , β  = β − β0 . Higher polynomial regression and regression in several variables also fall under the linear-hypothesis scheme. Thus if ξi = α + βti + γt2i or more generally ξi = α + βti + γui , where the ti and ui are known, it can be tested whether one or more of the regression coefficients α, β, γ are zero, and by transforming to the variables Xi = Xi − α0 − β0 ti − γ0 ui also whether these coefficients have specified values other than zero. In the general case, the hypothesis can be given a simple form by making an orthogonal transformation to variables Y1 , . . . , Yn Y = CX, C = (cij ) i, j = 1, . . . , n, (7.1) such that the first s row vectors c1 , . . . , cs of the matrix C span Ω , with cr+1 , . . . , cs , spanning ω . Then Ys+1 = · · · = Yn = 0 if and only if X is in Ω , and Y1 = · · · = Yr = Ys+1 = · · · = Yn = 0 if and only if X is in ω. Let ηi = E(Yi ), so that η = Cξ. Then since ξ lies in Ω a priori and in ω under H, it follows that ηi = 0 for i = s + 1, . . . , n in both cases, and ηi = 0 for i = 1, . . . , r when H is true. Finally, since the transformation is orthogonal, the variables Y1 , . . . , Yn are again independent and normally distributed with common variance σ 2 , and the problem reduces to the following canonical form. The variables Y1 , . . . , Yn are independently, normally distributed with common variance σ 2 and means E(Yi ) = ηi for i = 1, . . . , s and E(Yi ) = 0 for i = s + 1, . . . , n, so that their joint density is !  s " n   1 1 2 2 √ exp − 2 (yi − ηi ) + yi . (7.2) 2σ ( 2πσ)n i=1 i=s+1 The η’s and σ 2 are unknown, and the hypothesis to be tested is H : η 1 = · · · = ηr = 0 (r ≤ s < n). (7.3) Example 7.1.3 To illustrate the determination of the transformation (7.1), consider once more the regression model ξi = α + βti , of Example 7.1.2. It was seen there that Ω is spanned by (1, . . . , 1) and (t1 , . . . , tn ). If the hypothesis being tested is β = 0, ω is the one-dimensional space spanned by the first of these vectors. The row vector c2 is in ω and of length 1, and hence √ √ c2 = (1/ n, . . . , 1/ n). Since c1 is in Ω , of length 1, and orthogonal to c2 , its coordinates are i = 1, . . . , n, where a and b are determined by the  of the form a+bti , conditions (a + bti )  = 0 and (a + bti )2 = 1. The solutions of theseequations are a = −bt̄, b = 1/ (tj − t̄)2 , and therefore a + bti = (ti − t̄)/ (tj − t̄)2 , and   Xi (ti − t̄) (Xi − X̄)(ti − t̄) Y1 =  = .  2 (tj − t̄) (tj − t̄)2 7.1. A Canonical Form 279 The remaining row vectors of C can be taken to be any set of orthogonal unit vectors that are orthogonal to Ω ; it turns out not to be necessary to determine them explicitly. If the hypothesis to be tested is α = 0, ω is spanned by (t1 , . . . , tn ), so that , t2j . The coordinates of c1 are again of the form the ith coordinate of c2 is ti /  (a + bti )ti = 0 and a + bti with a and b now determined by the equations ,    2  (a + bti )2 = 1. The solutions are b = −ant̄/ t2j , a = tj /n (tj − t̄)2 , and therefore 6   n t2j t̄   Y1 =  X t X̄ − i i . (tj − t̄)2 t2j In the case of the hypothesis α = β = 0, ω is the origin, and c1 , c2 can be taken as any two orthogonal unit vectors in Ω . One possible choice is that appropriate to the hypothesis β = 0, in which case Y1 is the linear function given there and √ Y2 = xX̄. The general linear-hypothesis problem in terms of the Y ’s remains invariant under the group G1 of transformations Yi = Yi + ci for i = r + 1, . . . , s; Yi = Yi for i = 1, . . . , r; s + 1, . . . , n. This leaves Y1 , . . . , Yr and Ys+1 , . . . , Yn as maximal invariants. Another group of transformations leaving the problem invariant is the group G2 of all orthogonal transformations of Y1 , . . . , Yr . The middle set of variables having been eliminated, from Example 6.2.1(iii) that a maximal r it follows 2 invariant under G Y , Y , . . . , Yn . This can be reduced to U and 2 is U = s+1 i i=1  2 V = n i=s+1 Yi by sufficiency. Finally, the problem also remains invariant under the group G3 of scale changes Yi = cYi , c = 0, for i = 1, . . . , n. In the space of U and V this induces the transformation U ∗ = c2 U, V ∗ = c2 V , under which W = U/V is maximal invariant. Thus the principle of invariance reduces the data to the single statistic 2 r  Yi2 i=1 W =  . (7.4) n Yi2 i=s+1 Each of the three transformation groups Gi (i = 1, 2, 3) which lead to the above reduction induces a corresponding group Ḡi in the parameter space. The group Ḡ1 consists of the translations ηi = ηi +ci (i = r +1, . . . , s), ηi = ηi (i = 1, . . . , r), σ  = σ, which leaves (η1 , . . . , ηr , σ) as maximal invariants. Since any orthogonal transformation of Y1 , . . . , Yr induces the same transformation on η1 , . . . , ηr and  r 2 2 η . Finally the leaves σ 2 unchanged, a maximal invariant under Ḡ2 is i=1 i , σ elements of Ḡ3 are the transformations ηi = cηi , σ  = |c|σ, and hence a maximal invariant with respect to the totality of these transformations is r  ηi2 i=1 2 ψ = . (7.5) σ2 2 A corresponding reduction without assuming normality is discussed by Jagers (1980). 280 7. Linear Hypotheses It follows from Theorem 6.3.2 that the distribution of W depends only on ψ 2 , so that the principle of invariance reduces the problem to that of testing the simple hypothesis H : ψ = 0. More precisely, the probability density of W is (cf. Problems 7.2 and 7.3) 1 pψ (w) = e− 2 ψ 2 ∞  1 ck k=0 where ( 12 ψ 2 )k w 2 r−1+k , k! (1 + w) 12 (r+n−s)+k (7.6)   Γ 12 (r + n − s) + k   . Γ 12 r + k Γ[ 12 (n − s)] ck = For any ψ1 the ratio pψ1 (w)/po (w) is an increasing function of w, and it follows from the Neyman-Pearson fundamental lemma that the most powerful invariant test for testing ψ = 0 against ψ = ψ1 rejects when W is too large, or equivalently when r  Yi2 /r i=1 ∗ W =  > C. (7.7) n Yi2 /(n − s) i=s+1 The cutoff point C is determined so that the probability of rejection is α when ψ = 0. Since in this case W ∗ is the ratio of two independent χ2 variables, each divided by the number of its degrees of freedom, the distribution of W ∗ is the F -distribution with r and n − s degrees of freedom, and hence C is determined by  ∞ Fr,n−s (y)dy = α. (7.8) C The test is independent of ψ1 , and hence is UMP among all invariant tests. By Theorem 6.5.2, it is also UMP among all tests whose power function depends only on ψ 2 . The rejection region (7.7) can also be expressed in the form r  Yi2 i=1 > C. (7.9) r n   2 2 Yi + Yi i=1 i=s+1 When ψ = 0, the left-hand side is distributed according to the beta-distribution with r and n − s degrees of freedom [defined through (5.24)], so that C  is determined by  1 B 1 r, 1 (n−s) (y) dy = α. (7.10) C 2 2 For an alternative value of ψ, the left-hand side of (7.9) is distributed according to the noncentral beta-distribution with noncentrality parameter ψ, the density of which is (Problem 7.3)  k 1 2 ∞ ψ  1 2 2 gψ (y) = e− 2 ψ (7.11) B 1 r+k, 1 (n−s) (y). 2 2 k! k=0 7.2. Linear Hypotheses and Least Squares The power of the test against an alternative ψ is therefore  1 β(ψ) = gψ (y) dy. 281 3 C In the particular case r = 1 the rejection region (7.7) reduces to 6 |Y1 | n  i=s+1 > C0 . Yi2 /(n (7.12) − s) This is a two-sided t-test which by the theory of Chapter 5 (see for example Problem 5.5) is UMP unbiased. On the other hand, no UMP unbiased test exists for r > 1. The F -test (7.7) shares the admissibility properties of the two-sided t-test discussed in Section 6.7. In particular, the test is admissible against distant alternatives ψ 2 ≥ ψ12 (Problem 7.6) and against nearby alternatives ψ 2 ≤ ψ22 (Problem 7.7). It was shown by Lehmann and Stein (1953) that the test is in fact admissible against the alternatives ψ 2 ≤ ψ12 for any ψ1 and hence against all invariant alternatives. 7.2 Linear Hypotheses and Least Squares In applications to specific problems it is usually not convenient to carry out the reduction to canonical form explicitly. The teststatistic W can be expressed in 2 terms of the original variables by noting that n i=s+1 Yi is the minimum value of s n n    (Yi − ηi )2 + Yi2 = [Yi − E(Yi )]2 i=1 i=s+1 i=1 under unrestricted variation of the η’s. Also, since the transformation Y = CX is orthogonal and orthogonal transformations leave distances unchanged, n  [Yi − E(Yi )]2 = i=1 n  (Xi − ξi )2 . i=1 Furthermore, there is a 1 : 1 correspondence between the totality of s-tuples (η1 , . . . , ηs ) and the totality of vectors ξ in Ω . Hence n  i=s+1 Yi2 = n  (Xi − ξˆi )2 , (7.13) i=1 ˆ are the least-squares estimates of the ξ’s under Ω, that is, the values where the ξ’s  2 that minimize n i=1 (Xi − ξi ) subject to ξ in Ω. 3 Tables of the power of the F-test are provided by Tiku (1967, 1972) [reprinted in Graybill (1976)] and Cohen (1977); charts are given in Pearson and Hartley (1972). Various approximations are discussed by Johnson, Kotz and Balakrishnan (1995). 282 7. Linear Hypotheses In the same way it is seen that r  n  Yi2 + i=1 Yi2 = i=s+1 n  ˆ (Xi − ξˆi )2 i=1  ˆˆ where the ξ’s are the values that minimize (Xi − ξi )2 subject to ξ in test (7.7) therefore becomes 7  n n   ˆ (Xi − ξˆi )2 − (Xi − ξˆi )2 r i=1 i=1 W∗ = > C, n  (Xi − ξˆi )2 /(n − s) ω. The (7.14) i=1 ˆ where C is determined by (7.8). Geometrically the vectors ξˆ and ξˆ are the proˆ and ˆ ξˆ has a jections of X on Ω and ω , so that the triangle formed by X, ξ, ˆ right angle at ξ (see Figure 7.1). X _ • 0 ^^ ␰_ ^␰ _ • ⌸␻ ⌸⍀ Figure 7.1. Thus the denominator and numerator of W ∗ , except for the factors 1/(n − s) ˆ and 1/r, are the squares of the distances between X and ξˆ and between ξˆ and ξˆ ∗ respectively. An alternative expression for W is therefore 7 n  ˆ (ξˆi − ξˆi )2 r i=1 W∗ =  . (7.15) n (Xi − ξˆi )2 /(n − s) i=1 It is desirable to express also the noncentrality parameter ψ 2 = terms of the ξ’s. Now X = C −1 Y , ξ = C −1 η, and r  i=1 Yi2 = r i=1 n n   ˆ (Xi − ξˆi )2 − (Xi − ξˆi )2 . i=1 i=1 If the right-hand side of (7.16) is denoted by f (X), it follows that ηi2 /σ 2 in (7.16) r i=1 ηi2 = f (ξ). 7.2. Linear Hypotheses and Least Squares 283 A slight generalization of a linear hypothesis is the inhomogeneous hypothesis which specifies for the vector of means ξ a subhyperplane ω of Ω not passing through the origin. Let ω denote the subspace of Ω which passes through the origin and is parallel to ω . If ξ 0 is any point of ω , the set ω consists of the totality of points ξ = ξ ∗ + ξ 0 as ξ ∗ ranges over ω . Applying the transformation (7.1) with respect to ω , the vector of means η for ξ ∈ ω is then given by η = Cξ = Cξ ∗ + Cξ 0 in the canonical form (7.2), and the totality of these vectors is therefore characterized by the the equations η1 = η10 , . . . , ηr = ηr0 , ηs+1 = · · · = ηn = 0, where ηi0 is the ith coordinate of Cξ 0 . In the canonical form, the inhomogeneous hypothesis ξ ∈ ω therefore becomes ηi = ηi0 (i = 1, . . . , r). This reduces to the homogeneous case on replacing Yi with Yi − ηi0 , and it follows from (7.7) that the UMP invariant test has the rejection region r  (Yi − ηio )2 /r i=1 n  i=s+1 >C , (7.17) Yi2 /(n − s)  and that the noncentrality parameter is ψ 2 = ri=1 (ηi − ηi0 )2 /σ 2 . In applications it is usually most convenient to apply the transformation Xi −ξi0 directly to (7.14) or (7.15). It follows from (7.17) that such a transformation always leaves the denominator unchanged. This can also be seen geometrically, since the transformation is a translation of n-space parallel to Ω and therefore  leaves the distance (Xi − ξˆi )2 from X to Ω unchanged. The noncentrality parameter can be computed as before by replacing X with ξ in the transformed numerator (7.16). Some examples of linear hypotheses, all with r = 1, were already discussed in Chapter 5. The following treats two of these from the present point of view. Example 7.2.1 Let X1 , . . . , Xn be independently, normally distributed with common mean µ and variance σ 2 , and consider the hypothesis H : µ = 0. Here −1  Ω is the line ξi = · · · = ξn , ω is the origin, and s = r = 1. Let X̄ = n i Xi . From the identity   (Xi − µ)2 = (Xi − X̄)2 + n(X̄ − µ)2 , ˆ it is seen that ξˆi = X̄, while ξˆi = 0. The test statistic and ψ 2 are therefore given by nX̄ 2 W =  (Xi − X̄)2 and ψ2 = nµ2 . σ2 Under the hypothesis, the distribution of (n − 1)W is that of the square of a variable having Student’s t-distribution with n − 1 degrees of freedom. Example 7.2.2 In the two-sample problem considered in Example 7.1.1 with n = n1 + n2 , the sum of squares n1 n   (Xi − ξ)2 + (Xi − η)2 i=1 i=n1 +1 284 7. Linear Hypotheses is minimized by ξˆ = X·(1) = n1  Xi , n1 i=1 n  η̂ = X·(2) = i=n1 +1 Xi , n2 while, under the hypothesis η − ξ = 0, (1) n1 X· ˆˆ ˆ ξ = η̂ = X̄ = (2) + n2 X· n . The numerator of the test statistic (7.15) is therefore n1 (X·(1) − X̄)2 + n2 (X·(2) − X̄)2 = $2 n1 n2 # (2) X· − X·(1) . n1 + n2 The more general hypothesis η − ξ = θ0 reduces to the previous case on replacing Xi with Xi − θ0 for i = n1 + 1, . . . , n, and is therefore rejected when 2 7    (2) (1) 1 + n12 X· − X· − θ0 n1 ! " > C. 2  2 7 n1  n   (1) (2) + Xi − X· Xi − X· (n1 + n2 − 2) i=n1 +1 i=1 The noncentrality parameter is ψ 2 = (η − ξ − θ0 )2 /(1/n1 + 1/n2 )σ 2 . Under the hypothesis, the square root of the test statistic has the t-distribution with n1 + n2 − 2 degrees of freedom. ˆ Explicit formulae for the ξˆi and ξˆi can be obtained by introducing a coordinate system into the parameter space. Suppose that, in such a system, Ω is defined by the equations ξi = s  aij βj , i = 1, . . . , n, j=1 or, in matrix notation, ξ = A n×1 B, (7.18) n×s s×1 unknownparameters. If where A is known and of rank s, and β1 , . . . , βs are β̂1 , . . . , β̂s are the least-squares estimators minimizing i (Xi − j aij βj )2 , it is seen by differentiation that the β̂j are the solutions of the equations AT Aβ = AT X and hence are given by β̂ = (AT A)−1 AT X. (That AT A is nonsingular follows by Problem 6.3.) Thus, we obtain ξˆ = A(AT A)−1 AT X. ˆ Since ξˆ = ξ(X) is the projection of X into the space Ω spanned by the s columns of A, the formula ξˆ = A(AT A)−1 AT X shows that P = A(AT A)−1 AT has the property claimed for it in Example 6.2.3, that for any X in Rn , P X is the projection of X into Ω . 7.3. Tests of Homogeneity 285 7.3 Tests of Homogeneity The UMP invariant test obtained in the preceding section for testing the equality of the means of two normal distributions with common variance is also UMP unbiased (Section 5.3). However, when a number of populations greater than 2 is to be tested for homogeneity of means, a UMP unbiased test no longer exists, so that invariance considerations lead to a new result. Let Xij (j = 1, . . . , ni ; i = 1, . . . , s) be independently distributed as N (µi , σ 2 ), and consider the hypothesis H : µ 1 = · · · = µs . This arises, for example, in the comparison of a number of different treatments, processes, varieties, or locations, when one wishes to test whether these differences have any effect on the outcome X. It may arise more generally in any situation involving a one-way classification of the outcomes, that is, in which the outcomes are classified according to a single factor. In such situations, when rejecting H one will frequently want to know more about the µs than just that they are unequal. The resulting multiple comparison problem will be discussed in Section 9.3. The hypothesis H is a linear hypothesis with r = s − 1, with Ω given by the equations  ξij = ξik for j, k = 1, . . . , n, i = 1, . . . , s and with ω the line on which all n = ni coordinates ξij are equal. We have    (Xij − µi )2 = (Xij − Xi· )2 + ni (Xi· − µi )2 with Xi· = ni Xij /ni , and hence ξˆij = Xi· . Also,   (Xij − µ)2 = (Xij − X·· )2 + n(X·· − µ)2 j=1  ˆ with X·· = Xij /n, so that ξˆij = X·· . Using the form (7.15) of W ∗ , the test therefore becomes  ni (Xi· − X·· )2 /(s − 1) W∗ =  > C. (7.19) (Xij − Xi· )2 /(n − s) The noncentrality parameter is ψ2 =  with ni (µi − µ· )2 σ2  µ· = ni µi . n The sum of squares in both numerator and denominator of (7.19) admits three interpretations, which are closely related: (i) as the two components in the decomposition of the total variation    (Xij − X·· )2 = (Xij − Xi· )2 + ni (Xi· − X·· )2 , of which the first represents the variation within, and the second the variation between populations; (ii) as a basis, through the test (7.19), for comparing these two sources of variation; (iii) as estimates of their expected values, (n − s)σ 2 and 2 (s − 1)σ + ni (µi − µ· )2 (Problem 7.11). This breakdown of the total variation, together with the various interpretations of the components, is an example of 286 7. Linear Hypotheses an analysis of variance,4 which will be applied to more complex problems in the succeeding sections. When applying the principle of invariance, it is important to make sure that the underlying symmetry assumptions really are satisfied. In the problem of testing the equality of a number of normal means  µ1 , . . . , µs , for example, all parameter points, which have the same value of ψ 2 = ni (µi −µ· )2 /σ 2 , are identified under the principle of invariance. This is appropriate only when these alternatives can be considered as being equidistant from the hypothesis. In particular, it should then be immaterial whether the given value of ψ 2 is built up by a number of small contributions or a single large one. Situations where instead the main emphasis is on the detection of large individual deviations do not possess the required symmetry, and the test based on (7.19) need no longer be optimum. The robustness properties against nonnormality of the F -test for testing equality of means will be discussed using a large sample approach in Section 11.3, as well as the corresponding test for equality of variances. Alternatively, permutation tests will be applied in Section 15.2. Instead of assuming Xij is normally distributed, suppose that Xij has distribution F (x − µi ), where F is an arbitrary distribution with finite variance. If F has heavy tails, the test (7.19) tends to be inefficient. More efficient tests can be obtained by generalizing the considerations of Sections 6.8 and 6.9. Suppose the Xij are samples of size ni from continuous distributions Fi (i = 1, . . . , s) and that we wish to test H : F1 = · · · = Fs . Invariance, by the argument of Section 6.8, then reduces the data to the ranks Rij of the Xij in the combined sample of n = ni observations. A natural analogue of the test  two-sample Wilcoxon is the Kruskal–Wallis test, which rejects H when ni (Ri· − R·· )2 is too large. For the shift model Fi (y) = F (y − µi ), the performance of this test relative to (7.19) is similar to that of the Wilcoxon to the t-test in the case s = 2; the notion of asymptotic relative efficiency will be developed in Section 13.2. The theory of this and related rank tests is developed in books on nonparametric statistics such as Randles and Wolfe (1979), Hettmansperger (1984), Gibbons and Chakraborti (1992), Lehmann (1998) and Hájek, Sidák and Sen (1999). Unfortunately, such rank tests are available only for the simplest linear models. An alternative approach capable of achieving similar efficiencies for much wider classes of linear models can be obtained through large-sample theory, which will be studied in Chapters 11-15. Briefly, the least-squares estimators may be replaced by estimators with better efficiency properties for nonnormal distributions. Furthermore, asymptotically valid significance levels can be obtained through “Studentization”,5 that is, by dividing the statistic by a suitable estimator of its standard deviation; see Section 11.3. Different ways of implementing such a program are reviewed, for example, by Draper (1981, 1983), McKean and 4 For conditions under which such a breakdown is possible, see Albert (1976). term (after Student, the pseudonym of W. S. Gosset) is a misnomer. The procedure of dividing the sample mean X̄ by its estimated standard deviation and referring the resulting statistic to the standard normal distribution (without regard to the distribution of the X ’s) was used already by Laplace. Student’s contribution consisted of pointing out that if the X ’s are normal, the approximate normal distribution of the t-statistic can be replaced by its exact distribution—Student’s t. 5 This 7.4. Two-Way Layout: One Observation per Cell 287 Schrader (1982), Ronchetti (1982) and Hettmansperger, McKean and Sheather (2000). [For a simple alternative of this kind to Student’s t-test, see Prescott (1975).] Sometimes, it is of interest to test the hypothesis H : µ1 = · · · = µs considered at the beginning of the section, against only the ordered alternatives µ1 ≤ · · · ≤ µs rather than against the general alternatives of any inequalities among the µ’s. Then the F -test (7.19) is no longer reasonable; more powerful alternative tests for this and other problems involving ordered alternatives are discussed by Robertson, Wright and Dykstra (1988). The problem of testing H against onesided alternatives such as K : ξi ≥ 0 for all i, with at least one inequality strict, is treated by Perlman (1969) and in Barlow et al. (1972), which gives a survey of the literature; also see Tang (1994), Liu and Berger (1995) and Perlman and Wu (1999). Minimal complete classes and admissibility for this and related problems are discussed by Marden (1982a) and Cohen and Sackrowitz (1992). 7.4 Two-Way Layout: One Observation per Cell The hypothesis of equality of several means arises when a number of different treatments, procedures, varieties, or manifestations of some other factors are to be compared. Frequently one is interested in studying the effects of more than one factor, or the effects of one factor as certain other conditions of the experiment vary, which then play the role of additional factors. In the present section we shall consider the case that the number of factors affecting the outcomes of the experiment is two. Suppose that one observation is obtained at each of a number of levels of these factors, and denote by Xij (i = 1, . . . , a; j = 1, . . . , b) the value observed when the first factor is at the ith and the second at the jth level. It is assumed that the Xij are independently normally distributed with constant variance σ 2 , and for the moment also that the two factors act independently (they are then said to be additive), so that ξij is of the form αi + βj . Putting µ = α· + β· and αi = αi − α· , βj = βj − β· , this can be written as   αi = βj = 0, (7.20) ξij = µ + αi + βj , where the α’s and β’s (the main effects of A and B) and µ are uniquely determined by (7.20) as6 αi = ξi· − ξ·· , βj = ξ·j − ξ·· , µ = ξ·· . (7.21) Consider the hypothesis H : α1 = · · · = αa = 0 (7.22) that the first factor has no effect on the outcome being observed. This arises in two quite different contexts. The factor of interest, corresponding say to a number of treatments, may be β, while α corresponds to a classification according to, 6 The replacing of a subscript by a dot indicates that the variable has been averaged with respect to that subscript. 288 7. Linear Hypotheses for example, the site on which the observations are obtained (farm, laboratory, city, etc.). The hypothesis then represents the possibility that this subsidiary classification has no effect on the experiment so that it need not be controlled. Alternatively, α may be the (or a) factor of primary interest. In this case, the formulation of the problem as one of hypothesis testing would usually be an oversimplification, since in case of rejection of H, one would require estimates of the α’s or at least a grouping according to high and low values. The hypothesis H is a linear hypothesis with r = a−1, s = 1+(a−1)+(b−1) = a + b − 1, and n − s = (a − 1)(b − 1). The least-squares estimates of the parameters under Ω can be obtained from the identity   (Xij − ξij )2 = (Xij − µ − αi − βj )2  = [(Xij − Xi· − X·j + X·· ) + (Xi· − X·· − αi ) =  +a  + (X·j − X·· − βj ) + (X·· − µ)]2 (Xij − Xi· − X·j + X·· )2  +b (Xi· − X·· − αi )2 (X·j − X·· − βj )2 + ab (X·· − µ)2 , which is valid because in the expansion of the third sum of squares the crossproduct terms vanish. It follows that α̂i = Xi· − X·· , and that β̂j = X·j − X·· , µ̂ = X·· , (7.23) 2    Xij − ξˆij = (Xij − Xi· − X·j + X·· )2 . ˆ ˆ = X·· , and hence Under the hypothesis H we still have β̂ j = X·j − X·· and µ̂ ˆ ξˆij − ξˆij = Xi· − X·· . The best invariant test therefore rejects when  b (Xi· − X·· )2 /(a − 1) W∗ =  > C. (7.24) (Xij − Xi· − X·j + X·· )2 /(a − 1)(b − 1) The noncentrality parameter, on which the power of the test depends, is given by   b (ξi· − ξ·· )2 b αi2 = . (7.25) ψ2 = σ2 σ2 This problem provides another example of an analysis of variance. The total variation can be broken into three components,    (Xij − X·· )2 = b (Xi· − X·· )2 + a (X·j − X·· )2  + (Xij − Xi· − X·j + X·· )2 . Of these, the first contains the variation due to the α’s, the second that due to the β’s. The last component, in the canonical form of Section 7.1, is equal to  n 2 i=s+1 Yi . It is therefore the sum of squares of those variables whose means are zero even under Ω. Since this residual part of the variation, which on division by n − s is an estimate of σ 2 , cannot be attributed to any effects such as the α’s or 7.4. Two-Way Layout: One Observation per Cell 289 β’s, it is frequently labeled “error,” as an indication that it is due solely to the randomness of the observations, not to any differences of the means. Actually, the breakdown is not quite as sharp as is suggested by the above description. Any component such as that attributed to the α’s always also contains some “error,” as is seen for example from its expectation, which is   2 E (Xi· − X·· )2 = (a − 1)σ 2 + b αi . Instead of testing whether a certain factor has any effect, one may wish to estimate the size of the effect at the various levels of the factor. Other parameters that are sometimes interesting to estimate are the average outcomes (for example yields) ξ1· , . . . , ξa· when the factor is at the various levels. If θi = µ + αi = ξi· , confidence sets for (θ1 , . . . , θa ) are obtained by considering the hypotheses H(θ0 ) : θi = θi0 (i = 1, . . . , a). For testing θ1 = · · · = θa = 0, the least-squares estimates ˆ of the ξij are ξˆij = Xi· + X·j − X·· and ξˆij = X·j − X·· . The denominator sum of  squares is therefore (Xij − Xi· − X·j + X·· )2 as before, while the numerator sum of squares is   2  ˆ 2 ξˆij − ξˆij = b Xi· . The general hypothesis reduces to this special case on replacing Xij with the variable Xij − θi0 . Since s = a + b − 1 and r = a, the hypothesis H(θ0 ) is rejected when  b (Xi· − θi0 )2 /a  > C. (Xij − Xi· − X·j + X·· )2 /(a − 1)(b − 1) The associated confidence sets for (θ1 , . . . , θa ) are the spheres   aC (Xij − Xi· − X·j + X·· )2 (θi − Xi· )2 ≤ . (a − 1)(b − 1)b When considering confidence sets for the effects α1 , . . . , αa , one must take account of the fact that the α’s are not independent. Since they add up to zero, it would be enough to restrict attention to α1 , . . . , αa−1 . However, an easier and more symmetric solution is found by all the α’s. The rejection region of retaining H : αi = αi0 for i = 1, . . . , a (with αi0 = 0) is obtained from (7.24) by letting  Xij = Xij − αi0 , and hence is given by   C (Xij − Xi· − X·j + X·· )2 b (Xi· − X·· − αi0 )2 > . (b − 1) The associated confidence set consists of the totality of points (α1 , . . . , αa )  satisfying αi = 0 and   C (Xij − Xi· − X·j + X·· )2 . [αi − (Xi· − X·· )]2 ≤ b(b − 1) defines a sphere whose center (X1 . − In the space of (α1 , . . . , αa ), this inequality X·· , . . . , Xa· − X·· ) lies on the hyperplane αi = 0. The confidence sets for the α’s therefore consist of the interior and surface of the great hyperspheres obtained  by cutting the a-dimensional spheres with the hyperplane αi = 0. 290 7. Linear Hypotheses In both this and the previous case, the usual method shows the class of confidence sets to be invariant under the appropriate group of linear transformations, and the sets are therefore uniformly most accurate invariant. A rank test of (7.22) analogous to the Kruskal–Wallis test for the one-way layout is Friedman’s test, obtained by ranking the s observations X1j , . . . , Xsj separately from 1 to s at each level j of the second factor. If theseranks are denoted by R1j , . . . , Rsj , Friedman’s test rejects for large values of (Ri· − R·· )2 . Unless s is large, this test suffers from the fact that comparisons are restricted to observations at the same level of factor 2. The test can be improved by “aligning” the observations from different levels, for example, by subtracting from each observation at the jth level its mean X.j for that level, and then ranking the aligned observations from 1 to ab. For a discussion of these tests and their efficiency see Lehmann (1998, Chapter 6), and for an extension to tests of (7.22) in the model (7.20) when there are several observations per cell, Mack and Skillings (1980). Further discussion is provided by Hettmansperger (1984) and Gibbons and Chakraborti (1992). That in the experiment described at the beginning of the section there is only one observation per cell, and that as a consequence hypotheses about the α’s and β’s cannot be tested without some restrictions on the means ξij , does not of course justify the assumption of additivity. Rather, it is the other way around: the experiment should not be performed with just one observation per cell unless the factors can safely be assumed to be additive. Faced with such an experiment without prior assurance that the assumption holds, one should test the hypothesis of additivity. A number of tests for this purpose are discussed, for example, in Hegemann and Johnson (1976) and Marasinghe and Johnson (1981). 7.5 Two-Way Layout: m Observations Per Cell In the preceding section it was assumed that the effects of the two factors α and β are independent and hence additive. The factors may, however, interact in the sense that the effect of one depends on the level of the other. Thus the effectiveness of a teacher depends for example on the quality or the age of the students, and the benefit derived by a crop from various amounts of irrigation depends on the type of soil as well as on the variety being planted. If the additivity assumption is dropped, the means ξij of Xij are no longer given by (7.20) under Ω but are completely arbitrary. More than ab observations, one for each combination of levels, are then required, since otherwise s = n. We shall here consider only the simple case in which the number of observations is the same at each combination of levels. Let Xijk (i = 1, . . . , a; j = 1, . . . , b; k = 1, . . . , m) be independent normal with common variance σ 2 and mean E(Xijk ) = ξij . In analogy with the previous notation we write ξij = =  ξ·· + (ξi· − ξ·· ) + (ξ·j − ξ·· ) + (ξij − ξi· − ξ·j + ξ·· ) µ + αi + βj + γij    with i αi = j βj = i γij = j γij = 0. Then αi is the average effect of factor 1 at level i, averaged over the b levels of factor 2, and a similar interpretation 7.5. Two-Way Layout: m Observations Per Cell 291 holds for the β’s. The γ’s are called interactions, since γij measures the extent to which the joint effect ξij − ξ·· of factors 1 and 2 at levels i and j exceeds the sum (ξi· − ξ·· ) + (ξ·j − ξ·· ) of the individual effects. Consider again the hypothesis that the α’s are zero. Then r = a − 1, s = ab, and n − s = (m − 1)ab. From the decomposition    (Xijk − ξij )2 = (Xijk − Xij· )2 + m (Xij· − ξij )2 and  (Xij· − ξij )2 =  (Xij· − Xi·· − X·j· + X··· − γij )2   +b (Xi·· − X··· − αi )2 + a (X·j· − X··· − βj )2 +ab(X··· − µ)2 it follows that ˆ = ξˆ·· = X··· , µ̂ = µ̂ α̂i = ξˆi· − ξˆ·· = Xi·· − X··· , ˆ β̂ j = β̂ j = ξˆ·j − ξˆ·· = X·j· − X··· , γ̂ ij = γ̂ˆ ij = Xij· − Xi·· − X·j· + X··· , and hence that   (Xijk − ξˆij )2 = (Xijk − Xij· )2 ,   ˆ (Xi·· − X··· )2 . (ξˆij − ξˆij )2 = mb The most powerful invariant test therefore rejects when  mb (Xi·· − X··· )2 /(a − 1) > C, W∗ =  (Xijk − Xij· )2 /(m − 1)ab and the noncentrality parameter in the distribution of W ∗ is   mb αi2 mb (ξi· − ξ·· )2 = . σ2 σ2 (7.26) (7.27) Another hypothesis of interest is the hypothesis H  that the two factors are additive,7 H  : γij = 0 for all i, j. The least-squares estimates of the parameters are easily derived as before, and the UMP invariant test is seen to have the rejection region (Problem 7.13)  m (Xij· − Xi·· − X·j· + X··· )2 /(a − 1)(b − 1)  W∗ = > C. (7.28) (Xijk − Xij· )2 /(m − 1)ab 7 A test of H  against certain restricted alternatives has been proposed for the case of one observation per cell by Tukey (1949a); see Hegemann and Johnson (1976) for further discussion. 292 7. Linear Hypotheses Under H  , the statistic W ∗ has the F -distribution with (a−1)(b−1) and (m−1)ab degrees of freedom; the noncentrality parameter for any alternative set of γ’s is  2 m γij 2 . (7.29) ψ = σ2 The decomposition of the total variation into its various components, in the present case, is given by    (Xijk − X··· )2 = mb (Xi·· − X··· )2 + ma (X·j· − X··· )2 +m +  (Xij· − Xi·· − X·j· + X··· )2  (Xijk − Xij· )2 . Here the first three terms contain the variation due to the α’s, β’s and γ’s respectively, and the last component corresponds to error. The tests for the hypotheses that the α’s, β’s, or γ’s are zero, the first and third of which have the rejection regions (7.26) and (7.28), are then obtained by comparing the α, β, or γ sum of squares with that for error. An analogous decomposition is possible when the γ’s are assumed a priori to be equal to zero. In that case, the third component which previously was associated with γ represents an additional contribution to error, and the breakdown becomes    (Xijk − X··· )2 = mb (Xi·· − X··· )2 + ma (X·j· − X··· )2 +  (Xijk − Xi·· − X·j· + X··· )2 , with the last term corresponding to error. The hypothesis H : α1 = · · · = αa = 0 is then rejected when  mb (Xi·· − X··· )2 /(a − 1)  > C. (Xijk − Xi·· − X·j· + X··· )2 /(abm − a − b + 1) Suppose now that the assumption of no interaction, under which this test was derived, is not justified. The denominator sum of squares then has a noncentral χ2 -distribution instead of a central one; and is therefore stochastically larger than was assumed (Problem 7.15). follows that the actual rejection probability is   It 2 less than it would be for γij = 0. This shows that the probability of an error of the first kind will not exceed the nominal level of significance, regardless the   of 2 values of the γ’s. However, the power also decreases with increasing γij /σ 2 and tends to zero as this ratio tends to infinity. The analysis of variance and the associated tests derived in this section for two factors extend in a straightforward manner to a larger number of factors (see for example Problem 7.16). On the other hand, if the number of observations is not the same for each combination of levels (each cell ), explicit formulae for the least-squares estimators may no longer be available, but there is no difficulty in computing these estimators and the associated UMP invariant tests numerically. However, in applications it is then not always clear how to define main effects, interactions, and other parameters of interest, and hence what hypothesis to test. These issues are discussed, for example, in Hocking and Speed (1975) and Speed, Hocking, and Hackney (1979). See also TPE2, Chapter 3, Example 4.9, Arnold 7.6. Regression 293 (1981, Section 7.4), Searle (1987), McCulloch and Searle (2001) and Hocking (2003). Of great importance are arrangements in which only certain combinations of levels occur, since they permit reducing the size of the experiment. Thus for example three independent factors, at m levels each, can be analyzed with only m2 observations, instead of the m3 required if 1 observation were taken at each combination of levels, by adopting a Latin-square design (Problem 7.17). The class of problems considered here contains as a special case the two-sample problem treated in Chapter 5, which concerns a single factor with only two levels. The questions discussed in that connection regarding possible inhomogeneities of the experimental material and the randomization required to offset it are of equal importance in the present, more complex situations. If inhomogeneous material is subdivided into more homogeneous groups, this classification can be treated as constituting one or more additional factors. The choice of these groups is an important aspect in the determination of a suitable experimental design.8 A very simple example of this is discussed in Problems 5.49 and 5.50. Multiple comparison procedures for two-way (and higher) layouts are discussed by Spjøtvoll (1974); additional references can be obtained from Miller (1977b, 1986) and Westfall and Young (1993). The more general problem of multiple testing will be treated in Chapter 9. 7.6 Regression Hypotheses specifying one or both of the regression coefficients α, β when X1 , . . . , Xn are independently normally distributed with common variance σ 2 and means ξi = α + βti (7.30) are essentially linear hypotheses, as was pointed out in Example 7.1.2. The hypotheses H1 : α = α0 and H2 : β = β0 were treated in Section 5.6, where they were shown to possess UMP unbiased tests. We shall now consider H1 and H2 , as well as the hypothesis H3 : α = α0 , β = β0 , from the present point of view. By the general theory of Section 7.1, the resulting tests will be UMP invariant under suitable groups of linear transformations. For the first two cases, in which r = 1, this also provides, by the argument of Section 6.6, an alternative proof of their being UMP unbiased. The space Ω is the same for all three hypotheses. It is spanned by the vectors (1, . . . , 1) and (t1 , . . . , tn ) and therefore has dimension s = 2 unless the ti are all 8 For a discussion of various designs and the conditions under which they are appropriate see, for example, Box, Hunter, and Hunter (1978), Montgomery (2001) and Wu and Hamada (2000). Optimum properties of certain designs, proved by Wald, Ehrenfeld, Kiefer, and others, are discussed by Kiefer (1958), Silvey (1980), Atkinson and Donev (1992) and Pukelsheim (1993). The role of randomization, treated for the twosample problem in Section 5.10, is studied by Kempthorne (1955), Wilk and Kempthorne (1955), Scheffé (1959), and others; see, for example, Lorenzen (1984) and Giesbrecht and Gumpertz (2004). 294 7. Linear Hypotheses equal, which we shall assume not to be thecase. The least-squares estimates α and β under Ω are obtained by minimizing (Xi − α − βti )2 . For any fixed value of β, this isachieved by the value α = X̄ − β t̄, for which the sum of squares reduces to [(Xi − X̄) − β(ti − t̄)]2 . By minimizing this with respect to β one finds  (Xi − X̄)(ti − t̄)  , α̂ = X̄ − β̂ t̄; (7.31) β̂ = (tj − t̄)2 and    (Xi − α̂ − β̂ti )2 = (ti − t̄)2 (Xi − X̄)2 − β̂ 2 is the denominator sum of squares for all three hypotheses. The numerator of the test statistic (7.7) for testing the two hypotheses α = 0 and to β = 0 is Y12 , and for testing α = β = 0 is Y12 + Y22 . For the hypothesis α = 0, the statistic Y1 was shown in Example 7.1.3 to be equal to 6   2 6  tj ti Xi (tj − t̄)2 X̄ − t̄  2 n = α̂ n  2 . 2 tj (tj − t̄) tj Since then 6  (tj − t̄)2 E(Y1 ) = α n  2 , tj the hypothesis α = α0 is equivalent to the hypothesis ,   E(Y1 ) = η10 = α0 n (tj − t̄)2 / t2j , for which the rejection region (7.17) is (n − s)(Y1 − η10 )2 / n  Yi2 > C0 i=s+1 and hence ,   |α̂ − α0 | n (tj − t̄)2 / t2j , > C0 .  (Xi − α̂ − β̂ti )2 /(n − 2) (7.32) For the hypothesis β = 0, Y1 was shown to be equal to  , (Xi − X̄)(ti − t̄) (tj − t̄)2 . = β̂  (tj − t̄)2  Since then E(Y1 ) = β (tj − t̄)2 , the hypothesis β = β0 is equivalent to  0 2 E(Y1 ) = η1 = β0 (tj − t̄) and the rejection region is  |β̂ − β0 | (tj − t̄)2 , > C0 . (7.33)  (Xi − α̂ − β̂ti )2 /(n − 2) For testing α = β = 0, it was shown in Example 7.1.3 that , √ √ (tj − t̄)2 , Y1 = β̂ Y2 = nX̄ = n(α̂ + β̂ t̄); 7.6. Regression 295 the numerator of (7.7) is therefore n(α̂ + β̂ t̄)2 + β̂ 2 Y12 + Y22 = 2 2  (tj − t̄)2 . The more general hypothesis  α = α0 , β = β0 is equivalent to E(Y1 ) = η10 , √ E(Y2 ) = η20 , where η10 = β0 (tj − t̄)2 , η20 = n(α0 + β0 t̄); and the rejection region (7.17) can therefore be written as # $  n(α̂ − α0 )2 + 2nt̄(α̂ − α0 )(β̂ − β0 ) + t2i (β̂ − β0 )2 /2 > C. (7.34)  (Xi − α̂ − β̂ti )2 /(n − 2) The associated confidence sets for (α, β) are obtained by reversing this inequality and replacing α0 and β0 by α and β. The resulting sets are ellipses centered at (α̂, β̂). The simple regression model (7.30) can be generalized in many directions; the means ξi may for example be polynomials in t1 of higher than the first degree (see Problem 7.20), or more complex functions such as trigonometric polynomials; or they may be functions of several variables, ti , ui , vi . Some further extensions will now be illustrated by a number of examples. Example 7.6.1 A variety of problems arise when there is more than one regression-line. Suppose that the variables Xij are independently normally distributed with common variance and means ξij = αi + βi tij (j = 1, . . . , ni ; i = 1, . . . , b). (7.35) The hypothesis that these regression lines have equal slopes H : β 1 = · · · = βb may occur for example when the equality of a number of growth rates is to be tested.The parameter space Ω has dimension s = 2b provided none of the sums j (tij − ti· )2 is zero; the number of constraints imposed by the hypothesis  is r = b − 1. of (Xij − ξij )2 under Ω is obtained by The minimum value 2 minimizing j (Xij − αi − βi tij ) for each i, so that by (7.31),  j (Xij − Xi· )(tij − ti· )  , α̂i = Xi· − β̂ i ti· . β̂ i = 2 j (tij − ti· )  Under H, one must minimize (Xij −αi −βtij )2  , which for any fixed β leads to αi = Xi· −βti· and reduces the sum of squares to [(Xij −Xi· )−β(tij −ti· )]2 . Minimizing this with respect to β, one finds  (Xij − Xi· )(tij − ti· ) ˆ ˆ ˆ i = Xi· − β̂  , α̂ β̂ = i· . (tij − ti· )2 Since Xij − ξˆij = Xij − α̂i − β̂ i tij = (Xij − Xi· ) − β̂ i (tij − ti· ) and ˆ ˆ ˆ ˆ i ) + tij (β̂ i − β̂) ξˆij − ξˆij = (α̂i − α̂ = (β̂ i − β̂)(tij − ti· ), 296 7. Linear Hypotheses the rejection region (7.15) is  ˆ 2 2 i (β̂ i − β̂) j (tij − ti· ) /(b − 1) > C, $2  # (Xij − Xi· ) − β̂ i (tij − ti· ) /(n − 2b) (7.36) where the left-hand side under H has the F -distribution with b − 1 and n − 2b degrees of freedom. Since   2 ˆ i βi j (tij − ti· ) E(β̂ i ) = βi and E(β̂) =   , (tij − ti· )2 the noncentrality parameter of the distribution for an alternative set of β’s is   ˆ ψ 2 = i (βi − β̃)2 j (tij − ti· )2 /σ 2 , where β̃ = E(β̂). In the particular case that  βj /b. the ni and the tij are independent of i, β̃ reduces to β̄ = Example 7.6.2 The regression model (7.35) arises in the comparison of a number of treatments when the experimental units are treated as fixed and the unit effects uij (defined in Section 5.9) are proportional to known constants tij . Here tij might for example be a measure of the fertility of the i, jth piece of land or the weight of the i, jth experimental animal prior to the experiment. It is then frequently possible to assume that the proportionality factor βi does not depend on the treatment, in which case (7.35) reduces to ξij = αi + βtij (7.37) and the hypothesis of no treatment effect becomes H : α1 = · · · = αb . The space and Ω coincides with ω of the previous example, so that s = b + 1  (Xij − Xi· )(tij − ti· )  , α̂i = Xi· − β̂ti· . (tij − ti· )2  Minimization of (Xij − α − βtij )2 gives  (Xij − X·· )(tij − t·· ) ˆ ˆ ˆ = X·· − β̂t  , α̂ β̂ = ·· , (tij − t·· )2    where X·· = Xij /n, t·· = tij /n, n = ni . The sum of squares in the ∗ numerator of W in (7.15) is thus # $2    ˆ 2  ˆ ξˆij − ξˆij = (Xi· − X·· ) + β̂(tij − ti· ) − β̂(tij − t·· ) . β̂ = The hypothesis H is therefore rejected when $2  # ˆ (Xi· − X·· ) + β̂(tij − ti· ) − β̂(tij − t·· ) /(b − 1) >C , $2  # (Xij − Xi· ) − β̂(tij − ti· ) /(n − b − 1) (7.38) where under H the left-hand side has the F -distribution with b − 1 and n − b − 1 degrees of freedom. 7.7. Random-Effects Model: One-way Classification 297 The hypothesis H can be tested without first ascertaining the values of the tij ; it is then the hypothesis of no effect in a one-way classification considered in Section 7.3, and the test is given by (7.19). Actually, since the unit effects uij are assumed to be constants, which are now completely unknown, the treatments are assigned to the units either completely at random or at random within subgroups. The appropriate test is then a randomization test for which (7.19) is an approximation. Example 7.6.2 illustrates the important class of situations in which an analysis of variance (in the present case concerning a one-way classification) is combined with a regression problem (in the present case linear regression on the single “concomitant variable” t). Both parts of the problem may of course be considerably more complex than was assumed here. Quite generally, in such combined problems one can test (or estimate) the treatment effects as was done above, and a similar analysis can be given for the regression coefficients. The breakdown of the variation into its various treatment and regression components is the so-called analysis of covariance. 7.7 Random-Effects Model: One-way Classification In the factorial experiments discussed in Sections 7.3, 7.4, and 7.5, the factor levels were considered fixed, and the associated effects (the µ’s in Section 7.3, the α’s, β’s and γ’s in Sections 7.4 and 7.5) to be unknown constants. However, in many applications, these levels and their effects instead are (unobservable) random variables. If all the effects are constant or all random, one speaks of fixed-effects model (model I ) or random-effects model (model II ) respectively, and the term mixed model refers to situations in which both types occur.9 Of course, only the model I case constitutes a linear hypothesis according to the definition given at the beginning of the chapter. In the present section we shall treat as model II the case of a single factor (one-way classification), which was analyzed under the model I assumption in Section 7.3. As an illustration of this problem, consider a material such as steel, which is manufactured or processed in batches. Suppose that a sample of size n is taken from each of s batches and that the resulting measurements Xij (j = 1, . . . , n; i = 1, . . . , s) are independently normally distributed with variance σ 2 and mean ξi . If the factor corresponding to i were constant, with the same effect αi in each replication of the experiment, we would have   ξi = µ + αi αi = 0 and Xij = µ + αi + Uij , where the Uij are independently distributed as N (0, σ 2 ). The hypothesis of no effect is ξ1 = · · · = ξs , or equivalently α1 = · · · = αs = 0. However, the effect is 9 For a recent exposition of random effects models, see Sahai and Ojeda (2004). 298 7. Linear Hypotheses associated with the batches, of which a new set will be involved in each replication of the experiment; the effect therefore does not remain constant. Instead, we shall suppose that the batch effects constitute a sample from a normal distribution, and to indicate their random nature we shall write Ai for αi , so that Xij = µ + Ai + Uij . (7.39) The assumption of additivity (lack of interaction) of batch and unit effect, in the present model, implies that the A’s and U ’s are independent. If the expectation of Ai is absorbed into µ, it follows that the A’s and U ’s are independently normally 2 distributed with zero means and variances σA and σ 2 respectively. The X’s of course are no longer independent. The hypothesis of no batch effect, that the A’s are zero and hence constant, takes the form 2 H : σA =0 This is not realistic in the present situation, but is the limiting case of the hypothesis H(∆0 ) : 2 σA ≤ ∆0 σ2 that the batch effect is small relative to the variation of the material within a batch. correspond respectively to the model I hypotheses  2 These two  hypotheses αi = 0 and αi2 /σ 2 ≤ ∆0 . To obtain a test of H(∆0 ) it is convenient to begin with the same transformation of variables that reduced the corresponding model I problem to canonical form. Each set (Xi1 , . . . , Xin ) is subjected to an orthogonal transformation Yij =  √ √ n nXi· . Since c1k = 1/ n for k = 1, .  . . , n (see Exk=1 cjk Xik such that Yi1 = n ample 7.1.3), it follows from the assumption of orthogonality that k=1 cjk = 0 n for j = 2, . . . , n and hence that Yij = k=1 cjk Uik for j > 1. The Yij with j > 1 are therefore independently normally distributed with zero mean and variance σ 2 . √ They are also independent of Ui· since ( nUi· − Yi2 . . . Yin ) = C(Ui1 Ui2 . . . Uin ) (a prime indicates the transpose of a matrix). On the other hand, the variables √ √ Yi1 = nXi· = n(µ + Ai + Ui· ) are also independently normally distributed √ 2 but with mean nµ and variance σ 2 + nσA . If an additional orthogonal transfor√ mation is made from (Y11 , . . . , Ys1 ) to (Z11 , . . . , Zs1 ) such that Z11 = sY·1 , the 2 2 Z’s are independently normally distributed with common variance σ + nσA and √ means E(Z11 ) = snµ and E(Zi1 ) = 0 for i > 1. Putting Zij = Yij for j > 1 for the sake of conformity, the joint density of the Z’s is then  −s/2 2 (2π)−ns/2 σ −(n−1)s σ 2 + nσA (7.40) ⎤ ⎡   s  s n 2    √ 1 1 2 2  zi1 − 2 zij ⎦ . z11 − snµ + × exp ⎣−  2σ i=1 j=2 2 σ 2 + nσ 2 i=2 A The problem of testing H(∆0 ) is invariant under addition of an arbitrary constant to Z11 , which leaves the remaining Z’s as a maximal set of invariants. These constitute samples of size s(n − 1) and s − 1 from two normal distributions with 2 means zero and variances σ 2 and τ 2 = σ 2 + nσA . 7.7. Random-Effects Model: One-way Classification 299 The hypothesis H(∆0 ) is equivalent to τ 2 /σ 2 ≤ 1 + ∆0 n, and the problem reduces to that of comparing two normal variances, which was considered in Example 6.3.4 without the restriction to zero means. The UMP invariant test, under multiplication of all Zij by a common positive constant, has the rejection region S 2 /(s − 1) 1 · 2A > C, 1 + ∆0 n S /(n − 1)s W∗ = (7.41) where 2 = SA s  2 Zi1 and n s   S2 = i=2 2 Zij = i=1 j=2 n s   Yij2 . i=1 j=2 The constant C is determined by  ∞ Fs−1,(n−1)s (y) dy = α. C Since n  Yij2 − Yi12 = j=1 n  2 Uij − nUi·2 j=1 and s  2 2 Zi1 − Z11 = i=1 s  Yi12 − Y·12 , i=1 the numerator and denominator sums of squares of W ∗ , expressed in terms of the X’s, become 2 =n SA s  (Xi· − X·· )2 and S2 = i=1 s  n  (Xij − Xi· )2 . i=1 j=1 In the particular case ∆0 = 0, the test (7.41) is equivalent to the corresponding model I test (7.19), but they are of course solutions of different problems, and also have different power functions. Instead of being distributed according to a noncentral χ2 -distribution as in model I, the numerator sum of squares of W ∗ is proportional to a central χ2 -variable even when the hypothesis is false, and the power of the test (7.41) against an alternative value of ∆ is obtained from the F -distribution through  ∞ β(∆) = P∆ {W ∗ > C} = Fs−1,(n−1)s (y) dy. 1+∆0 n C 1+∆n The family of tests (7.41) for varying ∆0 is equivalent to the confidence statements   2 SA /(s − 1) 1 ∆= − 1 ≤ ∆. (7.42) n CS 2 /(n − 1)s The corresponding upper confidence bounds for ∆ are obtained from the tests of the hypotheses ∆ ≥ ∆0 . These have the acceptance regions W ∗ ≥ C  , where W ∗ is given by (7.41) and C  is determined by  ∞ Fs−1,(n−1)s = 1 − α . C 300 7. Linear Hypotheses The resulting confidence bounds are   2 SA /(s − 1) 1 ¯ ∆≤ − 1 = ∆. n C  S 2 /(n − 1)s (7.43) Both the confidence sets (7.42) and (7.43) are equivariant with respect to the group of transformations generated by those considered for the testing problems, and hence are uniformly most accurate equivariant. When ∆ is negative, the confidence set (∆, ∞) contains all possible values of the parameter ∆. For small ∆, this will happen with high probability (1 − α for ∆ = 0), as must be the case, since ∆ is then required to be a safe lower bound for a quantity which is equal to or near zero. Even more awkward is the possibility ¯ is negative, so that the confidence set (−∞, ∆) ¯ is empty. An interpretation that ∆ is suggested by the fact that this occurs if and only if the hypothesis ∆ ≥ ∆0 is rejected for all positive values of ∆0 . This may be taken as an indication that the assumed model is not appropriate, 10 although it must be realized that for ¯ < 0 is near α even when the assumptions small ∆ the probability of the event ∆ are satisfied, so that this outcome will occasionally be observed. The tests of ∆ ≤ ∆0 and ∆ ≥ ∆0 are not only UMP invariant but also UMP unbiased, and UMP unbiased tests also exist for testing ∆ = ∆0 against the two-sided alternatives ∆ = ∆0 . This follows from the fact that the joint density of the Z’s constitutes an exponential family. The confidence sets associated with these three families of tests are then uniformly most accurate unbiased (Problem 7.21). That optimum unbiased procedures exist in the model II case but not in the corresponding model I problem is explained by the different structure of the 2 two hypotheses. The model II hypothesis σA = 0 imposes one constraint, since it 2 concerns thesingle parameter σA . On the other hand, the corresponding model I hypothesis si=1 αi2 = 0 specifies the values of the s parameters α1 , . . . , αs , and since s − 1 of these are independent, imposes s − 1 constraints. A UMP invariant test of ∆ ≤ ∆0 does not exist if the sample sizes ni are unequal. An invariant test with a weaker optimum property for this case is obtained by Spjøtvoll (1967). Since ∆ is a ratio of variances, it is not surprising that the test statistic W ∗ is quite sensitive to the assumption of normality; such robustness issues are discussed in Section 11.3.1). More robust alternatives are discussed, for example, by Arvesen and Layard (1975). Westfall (1989) compares invariant variance ratio tests in mixed models. Optimality of standard F tests in balanced ANOVA models with mixed effects is derived in Mathew and Sinha (1988a) and optimal tests in some unbalanced designs are derived in Mathew and Sinha (1988b). 7.8 Nested Classifications The theory of the preceding section does not carry over even to so simple a situation as the general one-way classification with unequal numbers in the different 10 For a discussion of possibly more appropriate alternative models, see Smith and Murray (1984). 7.8. Nested Classifications 301 classes (Problem 7.24). However, the unbiasedness approach does extend to the important case of a nested (hierarchical) classification with equal numbers in each class. This extension is sufficiently well indicated by carrying it through for the case of two factors; it follows for the general case by induction with respect to the number of factors. Returning to the illustration of a batch process, suppose that a single batch of raw material suffices for several batches of the finished product. Let the experimental material consist of ab batches, b coming from each of a batches of raw material, and let a sample of size n be taken from each. Then (7.39) becomes Xijk = µ + Ai + Bij + Uijk (i = 1, . . . , a; j = 1, . . . , b; (7.44) k = 1, . . . , n) where Ai denotes the effect of the ith batch of raw material, Bij that of the jth batch of finished product obtained from this material, and Uijk the effect of the kth unit taken from this batch. All these variables are assumed to be 2 2 independently normally distributed with zero means and with variances σA , σB , 2 and σ respectively. The main part of the induction argument consists of proving the existence of an orthogonal transformation to variables Zijk , the joint density of which, except for a constant, is   ! a 2   √ 1 2 zi11 exp − z111 − abnµ + 2 2 2 (σ 2 + nσB + bnσA ) i=2 " b b n a a   1  2 1 2 zij1 − 2 zijk . (7.45) − 2 2 (σ 2 + nσB 2σ i=1 j=1 ) i=1 j=2 k=2 As a first step, there exists for each fixed i, j an orthogonal transformation from (Xij1 , . . . , Xijn ) to (Yij1 , . . . , Yijn ) such that √ √ √ Yij1 = nXij· = nµ + n(Ai + Bij + Uij .). As in the case of a single classification, the variables Yijk with k > 1 depend only on the U ’s, are independently normally distributed with zero mean and variance σ 2 , and are independent of the Uij· . On the other hand, the variables Yij1 have exactly the structure of the Yij in the one-way classification,  Yij1 = µ + Ai + Uij , √    where µ = nµ, Ai = nAi , Uij = n(Bij + Uij· ), and where the variances of   2 2 2 Ai and Uij are σA = nσA and σ 2 = σ 2 + nσB respectively. These variables can therefore be transformed to variables Zij1 whose density is given by (7.40) with Zij1 in place of Zij . Putting Zijk = Yijk for k > 1, the joint density of all Zijk is then given by (7.45). 2 Two hypotheses of interest can be tested on the basis of (7.45)—H1 : σA /(σ 2 + 2 2 2 nσB ) ≤ ∆0 and H2 : σB /σ ≤ ∆0 . Both state that one or the other of the classifications has little effect on the outcome. Let √ 2 = SA √ a  i=2 2 Zi11 , 2 SB = a  b  i=1 j=2 2 Zij1 , S2 = a  b  n  2 Zijk . i=1 j=1 k=2 2 To obtain a test of H1 , one is tempted to eliminate S through invariance under multiplication of Zijk for k > 1 by an arbitrary constant. However, these 302 7. Linear Hypotheses transformations do not leave (7.45) invariant, since they do not always pre2 , and serve the fact that σ 2 is the smallest of the three variances σ 2 , σ 2 + nσB 2 2 2 σ + nσB + bnσA . We shall instead consider the problem from the point of view of unbiasedness. For any unbiased test of H1 , the probability of rejection is α 2 2 whenever σA /(σ 2 + nσB ) = ∆0 , and hence in particular when the three variances 2 2 are σ , τ0 , and (1 + bn∆0 )τ02 for any fixed τ02 and all σ 2 < τ02 . It follows by the techniques of Chapter 4 that the conditional probability of rejection given S 2 = s2 must be equal to α for almost all values of s2 . With S 2 fixed, the joint distribution of the remaining variables is of the same type as (7.45) after the elimination of Z111 , and a UMP unbiased conditional test given S 2 = s2 has the rejection region 7 2 S (a − 1) A 1 7 W1∗ = ≥ C1 . · (7.46) 1 + bn∆0 S 2 (b − 1)a B 2 2 Since SA and SB are independent of S 2 , the constant C1 is determined by the fact 2 2 that when σA /(σ 2 + nσB ) = ∆0 , the statistic W1∗ is distributed as Fa−1,(b−1)a and hence in particular does not depend on s. The test (7.46) is clearly unbiased and hence UMP unbiased. An alternative proof of this optimality property can be obtained using Theorem 6.6.1. The existence of a UMP unbiased test follows from the exponential family structure of the density (7.45), and the test is the same whether τ 2 is equal to 2 σ 2 + nσB and hence ≥ σ 2 , or whether it is unrestricted. However, in the latter case, the test (7.46) is UMP invariant and therefore is UMP unbiased even when τ 2 ≥ σ2 . The argument with respect to H2 is completely analogous and shows the UMP unbiased test to have the rejection region 7 2 SB (b − 1)a 1 ∗ 7 W2 = ≥ C2 , · (7.47) 1 + n∆0 S 2 (n − 1)ab 2 where C2 is determined by the fact that for σB /σ 2 = ∆0 , the statistic W2∗ is distributed as F(b−1)a,(n−1)ab . 2 2 It remains to express the statistics SA , SB , and S 2 in terms of the X’s. From the corresponding expressions in the one-way classification, it follows that 2 SA = a  2 2 Zi11 − Z111 =b  (Yi·1 − Y··1 )2 , i=1 2 SB = ! b a   i=1 and S 2 = = − 2 Zi11 k=1 " 2 Yijk − 2 Yij1  (Uijk − Uij· )2 . i j =  (Yij1 − Yi·1 )2 , j=1 ! n b a    i=1 j=1 " 2 Zij1 k = ! n   i j k=1 " 2 Uijk − 2 nUij . 7.8. Nested Classifications Hence 2 = bn SA   2 (Xi·· − X··· )2 , SB =n (Xij· − Xi·· )2 ,    2 2 (Xijk − Xij· ) . S = 303 (7.48) It is seen from the expression of the statistics in terms of the Z’s that their 2 2 2 2 2 /(a − 1)] = σ 2 + nσB + bnσA , E[SB /(b − 1)a] = σ 2 + nσB , expectations are E[SA 2 2 and E[S /(n − 1)ab] = σ . The decomposition  2 2 (Xijk − X··· )2 = SA + SB + S2 therefore forms a basis for the analysis of the variance of Xijk , 2 2 + σB + σ2 V ar(Xijk ) = σA 2 2 by providing estimates of the components of variance σA , σB , and σ 2 , and tests of certain ratios of these components. Nested two-way classifications also occur as mixed models. Suppose for example that a firm produces the material of the previous illustrations in different plants. If αi denotes the effect of the ith plant (which is fixed, since the plants do not change in the replication of the experiment), Bij the batch effect, and Uijk the unit effect, the observations have the structure Xijk = µ + αi + Bij + Uijk . (7.49) Instead of reducing the X’s to the fully canonical form in terms of the Z’s as before, it is convenient to carry out only the reduction to the Y ’s (such that √ Yij1 = nXij .) and the first of the two transformations which take the Y√’s into the Z’s. If the resulting variables are denoted by Wijk , they satisfy Wi11 = bYi·1 , Wijk = Yijk for k > 1 and a  2 (Wi11 − W·11 )2 = SA , b a   i=1 i=1 j=2 2 2 Wij1 = SB , b  n a   2 Wijk = S2 , i=1 j=1 k=2 2 2 , SB , and S 2 are given by (7.48). The joint density of the W ’s is, except where SA for a constant, !  a  a  b   1 2 2 exp − (wi11 − µ − αi ) + wij1 (7.50) 2 2(σ 2 + nσB ) i=1 i=1 j=2 " a n b 1  2 wijk . − 2σ 2 i=1 j=1 k=2 This shows clearly the different nature of the problem of testing that the plant effect is small,  2 αi H : α1 = · · · = αa = 0 or H  : 2 ≤ ∆0 , 2 σ + nσB 2 and testing the corresponding hypothesis for the batch effect: σB /σ 2 ≤ ∆0 . The first of these is essentially a model I problem (linear hypothesis). As before, unbiasedness implies that the conditional rejection probability given S 2 = s2 is equal to α a.e. With S 2 fixed, the problem of testing H is a linear hypothesis, and the rejection region of the UMP invariant conditional test given S 2 = s2 has 304 7. Linear Hypotheses the rejection region (7.46) with ∆0 = 0. The constant C1 is again independent of S 2 , and the test is UMP among all tests that are both unbiased and invariant. A test with the same property also exists for testing H  . Its rejection region is 7 2 SA (a − 1) 7 ≥ C, 2 SB (b − 1)a where C  is determined from the noncentral F -distribution instead of, as before, the (central) F -distribution. 2 On the other hand, the hypothesis σB /σ 2 ≤ ∆0 is essentially model II. It is invariant under addition of an arbitrary constant of the variables Wi11 ,      to each 2 2 which leaves ai=1 bj=2 Wij1 and ai=1 bj=1 n W as maximal invariants, ijk k=2 and hence reduces the structure to pure model II with one classification. The test is then given by (7.47) as before. It is both UMP invariant and UMP unbiased. Very general mixed models (containing general type II models as special cases) are discussed, for example, by Harville (1978), J. Miller (1977a), and Brown (1984), but see the note following Problem 7.36. The different one- and two-factor models are discussed from a Bayesian point of view, for example, in Box and Tiao (1973) and Broemeling (1985). In distinction to the approach presented here, the Bayesian treatment also includes inferences concerning the values of the individual random components such as the batch means ξi of Section 7.7. 7.9 Multivariate Extensions The univariate linear models studied so far in this chapter arise in the study of the effects of various experimental conditions (factors) on a single characteristic such as yield, weight, length of life, or blood pressure. This characteristic is assumed to be normally distributed with a mean that depends on the various factors under investigation, and a variance that is independent of these factors. We shall now consider the multivariate analogue of this model, which is appropriate when one is concerned with the effect of one or more factors simultaneously on several characteristics, for example the effect of a change in the diet of dairy cows on both fat content and quantity of milk. A random vector (X1 , . . . , Xp ) has a multivariate normal density if its density is of the form # $  |A| 1 (7.51) aij (xi − ξi )(xj − ξj ) , 1 p exp − 2 (2π) 2 where the matrix A = (aij ) is positive definite, and |A| denotes its determinant. The means and covariance matrix of the X’s are given by E(Xi ) = ξi , E(Xi − ξi )(Xj − ξj ) = σij , (σij ) = A−1 . (7.52) Such a model was previously introduced in Section 3.9.2. Consider now n i.i.d. multivariate normal vectors Xk = (Xk,1 , . . . , Xk,p ), k = 1, . . . , n, with means E(Xk,i ) = ξi and covariance matrix A−1 . A natural extension of the one-sample problem of testing the mean ξ of a normal distribution 7.9. Multivariate Extensions 305 with unknown variance is that of testing the hypothesis ξ1 = ξ1,0 , . . . , ξp = ξp,0 ; without loss of generality, assume ξk,0 = 0 for all k. The joint density of X1 , . . . , Xn is ! " p p n |A|n/2 1  exp − ai,j (xk,i − ξi )(xk,j − ξj ) . 2 (2π)np/2 k=1 i=1 j=1 Writing the exponent as p p   n  (xk,i − ξi )(xk,j − ξj ) , ai,j i=1 j=1 k=1 it is seen that the vector of sample means (X̄1 , . . . , X̄p ) together with Si,j = n  (Xk,i − X̄i )(Xk,j − X̄j ) , i, j = 1, . . . p (7.53) k=1 are sufficient for the unknown mean vector ξ and unknown covariance matrix Σ = A−1 (assumed positive definite). For the remainder of this section, assume n > p, so that the matrix S with (i, j) component Si,j is nonsingular with probability one (Problem 7.38). We shall now consider the group of transformations Xk = CXk (C nonsingular) . This leaves the problem invariant, since it preserves the normality of the variables and their means. It simply replaces the unknown covariance matrix by another one. In the space of sufficient statistics, this group induces the transformations X̄ ∗ = C X̄ and S ∗ = CSC T , where S = (Si,j ) . (7.54) W = X̄ T S −1 X̄ (7.55) Under this group, the statistic is maximal invariant (Problem 7.39). The distribution of W depends only on the maximal invariant in the parameter space; this is found to be ψ2 = p p   aij ξi ξj , (7.56) i=1 j=1 and the probability density of W is given by (Problem 7.40) 1 pψ (w) = e− 2 ψ 2 1 ∞  ( 12 ψ 2 )k w 2 p−1+k . ck 1 k! (1 + w) 2 n+k (7.57) k=0 This is the same as the density of the test statistic in the univariate case, given as (7.6), with r and s there replaced by p. For any ψ0 < ψ1 the ratio pψ1 (w)/pψ0 (w) is an increasing function of w, and it follows from the Neyman–Pearson Lemma that the most powerful invariant test for testing H : ξ1 = · · · = ξp = 0 rejects when W is too large, or equivalently when n−p W > C. (7.58) p 306 7. Linear Hypotheses The quantity (n − 1)W , which for p = 1 reduces to the square of Student’s t, is Hotelling’s T 2 -statistic. The constant C is determined from the fact that for ψ = 0 the statistic (n − p)W/p has the F -distribution with p and n − p degrees of freedom. As in the univariate case, there also exists a UMP invariant test of the more general hypothesis H  : ψ 2 ≤ ψ02 , with rejection region W > C  . The T 2 -test was shown by Stein (1956) to be admissible against the class of alternatives ψ 2 ≥ c for any c > 0 by the method of Theorem 6.7.1. Against the class of alternatives ψ 2 ≤ c admissibility was proved by Kiefer and Schwartz (1965) [see Problem 7.44 and Schwartz (1967, 1969)]. Most accurate equivariant confidence sets for the unknown mean vector (ξ1 , . . . , ξp ) are obtained from the UMP invariant test of H : ξi = ξi0 (i = 1, . . . , p), which has acceptance region  n (X̄i − ξi0 )(n − 1)S i,j (X̄j − ξj0 ) ≤ C , where S i,j are the elements of S −1 . The associated confidence sets are therefore ellipsoids  n (ξi − X̄i )(n − 1)S ij (ξj − X̄j ) ≤ C (7.59) centered at (X̄1 , . . . , X̄p ). These confidence sets are equivariant under the group of transformations considered in this section (Problem 7.41), and by Lemma 6.10.1 are therefore uniformly most accurate among all equivariant confidence sets at the specified level. The result extends to the two-sample problem with equal covariances (Problem 7.43), but the situation becomes more complicated for multivariate generalizations of univariate linear hypotheses with r > 1. Then, the maximal invariant is no longer univariate and a UMP invariant test no longer exists. For a discussion of this case, see Anderson (2003), Section 8.10. 7.10 Problems Section 7.1 Problem 7.1 Expected sums of squares. The expected values of the numerator and denominator of the statistic W ∗ defined by (7.7) are ! n  "  r r  Yi2  Yi2 1 2 2 ηi and E =σ + = σ2 . E r r n − s i=1 i=1 i=s+1 Problem 7.2 Noncentral χ2 -distribution.11 (i) If X is distributed as N (ψ, 1), the probability density of V = X 2 is PψV (v) = ∞ 2 k −(1/2)ψ 2 /k! and where f2k+1 k−0 Pk (ψ)f2k+1 (v), where Pk (ψ) = (ψ /2) e is the probability density of a χ2 -variable with 2k + 1 degrees of freedom. 11 The literature on noncentral χ 2 , including tables, is reviewed in Tiku (1985a), Chou, Arthur, Rosenstein, and Owen (1994), and Johnson, Kotz and Balakrishnan (1995). 7.10. Problems 307 (ii) Let Y1 , . . . , Yr be independently normally  2 distributed with unit variance and means η1 , . . . , ηr . Then U = Yi is distributed according to the noncentral χ2 -distribution with r degrees of freedom and noncentrality  parameter ψ 2 = ri=1 ηi2 , which has probability density pU ψ (u) = ∞  Pk (ψ)fr+2k (u). (7.60) k=0 Here Pk (ψ) and fr+2k (u) have the same meaning as in (i), so that the distribution is a mixture of χ2 -distributions with Poisson weights. [(i): This is seen from 1 pVψ (v) = e− 2 (ψ 2 +v) √ (eψ v + e−ψ √ 2 2πv √ v ) by expanding the expression in parentheses into a power series, and using the √ fact that Γ(2k) = 22k−1 Γ(k)Γ(k + 12 )/ π. (ii): Consider an orthogonal transformation to Z1 , . . . , Zr such that Z1 =  ηi Yi /ψ. Then the Z’s are independent normal with unit variance and means E(Z1 ) = ψ and E(Zi ) = 0 for i > 1.] Problem 7.3 Noncentral F - and beta-distribution.12 Let Y1 , . . . , Yr ; Ys+1 , . . . , Yn be independently normally distributed with common variance σ 2 and means E(Yi ) = ηi (i = 1, . . . , r); E(Yi ) = 0 (i = s + 1, . . . , n).   2 (i) The probability density of W = ri=1 Yi2 / n i=s+1 Yi is given by (7.6). The distribution of the constant multiple (n − s)W/r of W is the noncentral F -distribution.    2 (ii) The distribution of the statistic B = ri=1 Yi2 /( ri=1 Yi2 + n i=s+1 Yi ) is the noncentral beta-distribution, which has probability density ∞  k=0 Pk (ψ)g 1 r+k, 1 (n−s) (b), 2 (7.61) 2 where gp,q (b) = Γ(p + q) p−1 b (1 − b)q−1 , Γ(p)Γ(q) 0≤b≤1 (7.62) is the probability density of the (central) beta-distribution. Problem 7.4 (i) The noncentral χ2 and F distributions have strictly monotone likelihood ratio. (ii) Under the assumptions of Section 7.1, the hypothesis H  : ψ 2 ≤ ψ02 (ψ0 > 0 given) remains invariant under the transformations Gi (i = 1, 2, 3) that were used to reduce H : ψ = 0, and there exists a UMP invariant test with rejection region W > C  . The constant C  is determined by Pψ0 {W > C  } = α, with the density of W given by (7.6). 12 For literature on noncentral F , see Tiku (1985b) and Johnson, Kotz and Balakrishnan (1995). 308 7. Linear Hypotheses  k ∞ k [(i): Let f (z) = ∞ k=0 bk z / k=0 ak z where the constants ak , bk are > 0 and  k k ak z and bk z converge for all z > 0, and suppose that bk /ak < bk+1 /ak+1 for all k. Then  (n − k)(ak bn − an bk )z k+n−1 k 0 for k < n, and hence f is increasing.] Note. The noncentral χ2 and F -distributions are in fact STP∞ [see for example Marshall and Olkin (1979) and Brown, Johnstone and MacGibbon (1981)], and there thus exists a test of H : ψ = ψ0 against ψ = ψ0 which is UMP among all tests that are both invariant and unbiased. Problem 7.5 Best average power. (i) Consider the general linear hypothesis H in the canonical form given by (7.2) and (7.3) of Section 7.1, and for any ηr+1 , . . . , ηs , σ, and ρ let S = S(ηr+1 , . . . , ηs , σ : ρ) denote the sphere {(η1 , . . . , ηr ) : ri=1 ηi2 /σ 2 = ρ2 }. If βφ (η1 , . . . , ηr , σ) denotes the power of a test φ of H, then the test (7.9) maximizes the average power S βφ (η1 , . . . , ηr , σ) dA dA S for every ηr+1 , . . . , ηs , σ, and ρ among all unbiased (or similar) tests. Here dA denotes the differential of area on the surface of the sphere. (ii) The result (i) provides an alternative proof of the fact that the test (7.9) is UMP among all tests whose power function depends only on ri=1 ηi2 /σ 2 .   2 [(i): if U = ri=1 Yi2 , V = n i=s+1 Yi , unbiasedness (or similarity) implies that the conditional probability of rejection given Yr+1 , . . . , Ys , and U + V equals α a.e. Hence for any given ηr+1 , . . . , ηs , σ, and ρ, the average power is maximized by rejecting when the ratio of the average density to the density under H is larger than a suitable constant C(yr+1 , . . . , ys , u + v), and hence when  r    ηi yi g(y1 , . . . , yr ; η1 , . . . , ηr ) = exp dA > C(yr+1 , . . . , ys , u + v). σ2 S i=1 As will be indicated below, the function g depends on y1 , . . . , yr only through u and is an increasing function of u. Since under the hypothesis U/(U + V ) is independent of Yr+1 , . . . , Ys and U + V , it follows that the test  is given by (7.9). The exponent in the integral defining g can be written as ri=1 ηi yi /σ 2 = √ (ρ u cos β)/σ, where β is the angle (0 ≤ β ≤ π) between (η1 , . . . , ηr ) and (y1 , . . . , yr ). Because of the symmetry of the sphere, this is unchanged if β is replaced by the angle γ between (η1 , . . . , ηr ) and an arbitrary fixed vector. This shows that g depends on the y’s only through u: for fixed η1 , . . . , ηr , σ denote it by h(u). Let S  be the subset of S in which 0 ≤ γ ≤ π/2. Then   √ √   ρ u cos γ −ρ u cos γ h(u) = exp + exp dA, σ σ S which proves the desired result.] 7.10. Problems 309 Problem 7.6 Use Theorem 6.7.1 to show that the F -test (7.7) is α-admissible against Ω : ψ ≥ ψ1 for any ψ1 > 0. Problem 7.7 Given any ψ2 > 0, apply Theorem 6.7.2 and Lemma 6.7.1 to obtain the F -test (7.7) as a Bayes test against a set Ω of alternatives contained in the set 0 < ψ ≤ ψ2 . Section 7.2 Problem 7.8 Under the assumptions of Section 7.1 suppose that the means ξi are given by ξi = s  aij βj , j=1 where the constants aij are known and the matrix  A = (aij ) has full rank, and s where the βj are unknown parameters. Let θ = j=1 ej βj be a given linear combination of the βj .  (i)  If β̂j denotes the values of the βj minimizing (Xi − ξi )2 and if θ̂ = s n j=1 ej β̂j = j=1 di Xi , the rejection region of the hypothesis H : θ = θ0 is  2 di |θ̂ − θ0 |/ 1  > C0 , (7.63) 2  Xi − ξˆi /(n − s) where the left-hand side under H has the distribution of the absolute value of Student’s t with n − s degrees of freedom. (ii) The associated confidence intervals for θ are ;  ;  2 2 < < < < Xi − ξˆi Xi − ξˆi = = θ̂ − k ≤ θ ≤ θ̂ + k (7.64) n−s n−s  2 with k = C0 di . These intervals are uniformly most accurate equivariant under a suitable group of transformations. [(i): Consider first the hypothesis θ = 0, and suppose without loss of generality that θ = β1 ; the general case can be reduced to this by making a linear transformation in the space of the β’s. If a1 , . . . , as denote the column vectors of the matrix A which by assumption span ΠΩ , then ξ = β1 a1 +· · ·+βs as , and since ξˆ is in ΠΩ also ξˆ = β̂1 a1 + · · · + β̂s as . The space Πω defined by the hypothesis β1 = 0 is spanned by the vectors a2 , . . . , as and also by the row vectors c2 , . . . , cs of the matrix C of (7.1), while c1 is orthogonal to Πω . By (7.1), the vector X is given  ˆ ˆ s Yi c . by X = n i i=1 Yi ci , and its projection ξ on ΠΩ therefore satisfies ξ = i=1 Equating the two expressions for ξˆ and taking the inner product of both sides of  this equation with ci gives Y1 = β̂1 n i=1 ai1 ci1 , since the c’s are an orthogonal set of unit vectors. This shows that Y1 is proportional to  β̂1 and, since the variance of Y1 is the same as that of the X’s, that |Y1 | = |β̂1 |/ d2i . The result for testing 310 7. Linear Hypotheses β1 = 0 now follows from (7.12) and (7.13). The test for β1 = β10 is obtained by making the transformation Xi∗ = Xi − ai β10 . (ii): The invariance properties of the intervals (7.64) can again be discussed without loss of generality by letting θ be the parameter β1 . In the form of  canonical d21 while η2 , . . . , ηs Section 7.1, one then has E(Y1 ) = η1 = λβ1 with |λ| = 1/ do not involve β1 . The hypothesis β1 = β10 is therefore equivalent to η1 = η10 , with η10 = λβ10 . This is invariant (a) under addition of arbitrary constants to Y2 . . . , Ys ; (b) under the transformations Y1∗ = −(Y1 − η10 ) + η10 ; (c) under the scale changes Yi∗ = cYi (i = 2, . . . , n), Y1∗ − η10 ∗ = c(Y1 − η10 ). The confidence intervals for θ = β1 are then uniformly most accurate equivariant under the group obtained from (a), (b), and (c) by varying η10 .] Problem 7.9 Let Xij (j = 1, . . . , mi ) and Yik (k = 1, . . . , ni ) be independently normally distributed with common variance σ 2 and means E(Xij ) = ξi and E(Yij ) = ξi + ∆. Then the UMP invariant test of H : ∆ = 0 is given by (7.63) with θ = ∆, θ0 = 0 and  θ̂ = i mi ni (Yi· Ni  i mi  − Xi· ) , mi ni Ni ξˆi = Xij + j=1 ni  k=1 Ni (Yik − θ̂) , where Ni = mi + ni . Problem 7.10 Let X1 , . . . , Xn be independently normally distributed with known variance σ02 and means E(Xi ) = ξi , and consider any linear hypothesis with s ≤ n (instead of s < n which is required when the variance is unknown). This remains invariant under a subgroup of that employed when the variance was unknown, and the UMP invariant test has rejection region   2    ˆ 2 ˆ 2 Xi − ξˆi − Xi − ξˆi = ξˆi − ξˆi > Cσ02 (7.65) with C determined by  ∞ χ2r (y) dy = α. (7.66) C Section 7.3 Problem 7.11 If the variables Xij (j = 1, . . . , ni ; i = 1, . . . , s) are independently distributed as N (µi , σ 2 ), then # $  E ni (Xi· − X·· )2 ni (µi − µ· )2 , = (s − 1)σ 2 + $ #  = (n − s)σ 2 . E (Xij − Xi· )2 Problem 7.12 Let Z1 , . . . , Zs be independently distributed as N (ζi , a2i ), i = 1, . . . , s, where the ai are known constants. 7.10. Problems 311 (i) With respect to a suitable group of linear transformations there exists a UMP invariant test of H : ζ1 = · · · = ζs given by the rejection region  2  2  2  1 Zj /a2j Zj /a2j Zi    − = − >C (7.67) Z i 1/a2j a2i ai 1/a2j (ii) The power of this test is the integral from C to ∞ of the noncentral χ2 -density with s − 1 degrees of freedom and noncentrality parameter λ2 obtained by substituting ζi for Zi in the left-hand side of (7.67). Section 7.5 Problem 7.13 The linear-hypothesis test of the hypothesis of no interaction in a two-way layout with m observations per cell is given by (7.28). Problem 7.14 In the two-way layout of  Section 7.5 with a = b = 2, denote the  2 2 2 first three terms in the partition of (Xijk − Xij· )2 by SA , SB , and SAB , corresponding to the A, B, and AB effects (i.e. the α’s, β’s, and γ’s), and denote by HA , HB , and HAB the hypotheses of these effects being zero. Define a new two-level factor B  which is at level 1 when A and B are both at level 1 or both at level 2, and which is at level 2 when A and B are at different levels. Then HB  = HAB , SB  = SAB , HAB  = HB , SAB  = SB , so that the B-effect has become an interaction, and the AB-interaction the effect of the factor B  . [Shaffer (1977b).] Problem 7.15 Let Xλ denote a random variable distributed as noncentral χ2 with f degrees of freedom and noncentrality parameter λ2 . Then Xλ is stochastically larger than Xλ if λ < λ . [It is enough to show that if Y is distributed as N (0, 1), then (Y + λ )2 is stochastically larger than (Y + λ)2 . The equivalent fact that for any z > 0, P {|Y + λ | ≤ z} ≤ P {|Y + λ| ≤ z}, is an immediate consequence of the shape of the normal density function. An alternative proof is obtained by combining Problem 7.4 with Lemma 3.4.2.] Problem 7.16 Let Xijk (i = 1, . . . , a; j = 1, . . . , b; k = 1, . . . , m) be independently normally distributed with common variance σ 2 and mean     E(Xijk ) = µ + αi + βj + γk βj = γk = 0 . αi = Determine the linear hypothesis test for testing H : αi = . . . αa = 0. Problem 7.17 In the three-factor situation of the preceding problem, suppose that a = b = m. The hypothesis H can then be tested on the basis of m2 observations as follows. At each pair of levels (i, j) of the first two factors one observation is taken, to which we refer as being in the ith row and the jth column. If the levels of the third factor are chosen in such a way that each of them occurs once and only once in each row and column, the experimental 312 7. Linear Hypotheses design is a Latin square. The m2 observations are denoted by Xij(k) , where the third subscript indicates the level of the third factor when the first two are at levels i and   j. Itis assumed that E(Xij(k) ) = ξij(k) = µ + αi + βj + γk , with αi = βj = γk = 0. (i) The parameters are determined from the ξ’s through the equations ξi·(·) = µ + αi , ξ·j(·) = µ + βj , ξ··(k) = µ + γk , ξ··(·) = µ. (Summation over j with i held fixed automatically causes summation also over k.) (ii) The least-squares estimates of the parameters may be obtained from the identity   2 xij(k) − ξij(k) i j =  2 2 xi·(·) − x··(·) − αi + m x·j(·) − x··(·) − βj  2 2  +m x··(k) − x··(·) − γk + m2 x··(·) − µ  2 + xij(k) − xi·(·) − x·j(·) − x··(k) + 2x··(·) . m  i k (iii) For testing the hypothesis H : α1 = · · · = αm = 0, the test statistic W ∗ of (7.15) is 2  Xi·(·) − X··(·) m . 2  Xij(k) − Xi·(·) − X·j(·) − X··(k) + 2X··(·) /(m − 2) The degrees of freedom are m − 1 for the numerator and (m − 1)(m  − 2) for the denominator, and the noncentrality parameter is ψ 2 = m αi2 /σ 2 . Section 7.6 Problem 7.18 In a regression situation, suppose that the observed values Xj and Yj of the independent and dependent variable differ from certain true values Xj and Yj by errors Uj , Vj which are independently normally distributed with 2 zero means and variances σU and σV2 . The true values are assumed to satisfy a linear relation: Yj = α + βXj . However, the variables which are being controlled, and which are therefore constants, are the Xj rather than the Xj . Writing xj for Xj , we have xj = Xj + Uj , Yj = Yj + Vj , and hence Yj = α + βxj + Wj , where Wj = Vj − βUj . The results of Section 7.6 can now be applied to test that β or α + βx0 has a specified value. Problem 7.19 Let X1 , . . . , Xm ; Y1 , . . . , Yn be independently normally distributed with common variance σ 2 and means E(Xi ) = α + β(ui − ū), E(Yj ) = γ + δ(vj − v̄), where the u’s and v’s are known numbers. Determine the UMP invariant tests of the linear hypotheses H : β = δ and H : α = γ, β = δ. Problem 7.20 Let X1 , . . . , Xn be independently normally distributed with common variance σ 2 and means ξi = α + βti + γt2i , where the ti are known. If the 7.10. Problems 313 coefficient vectors (tk1 , . . . , tkn ), k = 0, 1, 2, are linearly independent, the parameter space ΠΩ has dimension s = 3, and the least-squares estimates α̂, β̂, γ̂ are the unique solutions of the system of equations  k  k+1  k+2  k α ti + β ti + γ ti = ti Xi (k = 0, 1, 2).  The solutions are linear functions of the X’s, and if γ̂ = ci Xi , the hypothesis γ = 0 is rejected when  2 |γ̂|/ ci 1  > C0 . 2  Xi − α̂ − β̂ti − γ̂t2i /(n − 3) Section 7.7 Problem 7.21 (i) The test (7.41) of H : ∆ ≤ ∆0 is UMP unbiased. (ii) Determine the UMP unbiased test of H : ∆ = ∆0 and the associated uniformly most accurate unbiased confidence sets for ∆. Problem 7.22 In the model (7.39), the correlation coefficient ρ between two observations Xij , Xik belonging to the same class, the so-called intraclass 2 2 correlation coefficient, is given by ρ = σA /(σA + σ 2 ). Section 7.8 Problem 7.23 The tests (7.46) and (7.47) are UMP unbiased. Problem 7.24 If Xij is given by (7.39) but the number ni of observations per batch is not constant, obtain a canonical form corresponding to (7.40) by letting √ Yi1 = ni Xi· . Note that the set of sufficient statistics has more components than when ni is constant. Problem 7.25 The general nested classification with a constant number of observations per cell, under model II, has the structure Xijk··· = µ + Ai + Bij + Cijk + · · · + Uijk··· , i = 1, . . . , a; j = 1, . . . , b; k = 1, . . . , c; . . . . (i) This can be reduced to a canonical form generalizing (7.45). (ii) There exist UMP unbiased tests of the hypotheses HA : 2 σA 2 +d...σ 2 +···+σ 2 cd...σB C HB : 2 σB 2 +···+σ 2 d...σC ≤ ∆0 , ≤ ∆0 . Problem 7.26 Consider the model II analogue of the two-way layout of Section 7.5, according to which Xijk = µ + Ai + Bj + Cij + Eijk (i = 1, . . . , a; j = 1, . . . , b; (7.68) k = 1, . . . , n), 314 7. Linear Hypotheses where the Ai , Bj , Cij , and Eijk are independently normally distributed with 2 2 2 mean zero and with variances σA , σB , σC and σ 2 respectively. Determine tests which are UMP among all tests that are invariant (under a suitable group) and unbiased of the hypotheses that the following ratios do not exceed a given constant (which may be zero): 2 (i) σC /σ 2 ; 2 2 /(nσC + σ 2 ); (ii) σA 2 2 /(nσC + σ 2 ). (iii) σB Note that the  test of (i) requires n > 1, but  those of (ii) 2and2(iii) donot.  2 2 [Let SA = nb (Xi·· − X··· )2 , SB = na (X·j· − X··· ) , SC = n (Xij· −  Xi·· − X·j· + X··· )2 , S 2 = (Xijk − Xij· )2 , and make a transformation to new variables Zijk (independent, normal, and with mean zero except when i = j = k = 1) such that 2 SA = a  2 Zi11 , 2 SB = i=2 S2 = a  b  n  b  j=2 2 Z1j1 , 2 SC = b a   2 Zij1 , i=2 j=2 2 Zijk .] i=1 j=1 k=2 Problem 7.27 Consider the mixed model obtained from (7.68)  by replacing the random variablesAi by unknown constants αi satisfying αi = 0. With (ii) 2 replaced by (ii ) αi2 /(nσC + σ 2 ), there again exist tests which are UMP among an tests that are invariant and unbiased, and in cases (i) and (iii) these coincide with the corresponding tests of Problem 7.26. Problem 7.28 Consider the following generalization of the univariate linear model of Section 7.1. The variables Xi (i = 1, . . . , n) are given by Xi = ξi + Ui , where 1 , . . . , Un ) have a joint density which is spherical, that is, a function of n (U 2 i=1 ui , say   Ui2 . f (U1 , . . . , Un ) = q The parameter spaces ΠΩ and Πω and the hypothesis H are as in Section 7.1. (i) The orthogonal transformation (7.1) reduces (X1 , . . . , Xn ) to canonical variables (Y1 , . . . , Yn ) with Yi = ηi + Vi , where ηi = 0 for i = s + 1, . . . , n, H reduces to (7.3), and the V ’s have joint density q(v1 , . . . , vn ). (ii) In the canonical form of (i), the problem is invariant under the groups G1 , G2 , and G3 of Section 7.1, and the statistic W ∗ given by (7.7) is maximal invariant. Problem 7.29 Under the assumptions of the preceding problem, the null distribution of W ∗ is independent of q and hence the same as in the normal case, namely, F with r and n−s degrees of freedom. [See Problem 5.11]. Note. The analogous multivariate problem is treated by Kariya (1981); also see Kariya (1985) and Kariya and Sinha (1985). For a review of work on spherically and elliptically symmetric distributions, see Chmielewski (1981). 7.10. Problems 315 Problem 7.30 Consider the additive random-effects model Xijk = µ + Ai + Bj + Uijk (i = 1, . . . , a; j = 1, . . . , b; k = 1, . . . , n), where the A’s, B’s, and U ’s are independent normal with zero means and 2 2 , σB , and σ 2 ’ respectively. Determine variances σA (i) the joint density of the X’s, 2 /σ 2 ≤ δ. (ii) the UMP unbiased test of H : σB Problem 7.31 For the mixed model Xij = µ + αi + Bj + Uij (i = 1, . . . , a; j = 1, . . . , n), where the B’s and U ’s are as in Problem 7.30 and the α’s are constants adding to zero, determine (with respect to a suitable group leaving the problem invariant) (i) a UMP invariant test of H : α1 = · · · = αa ; (ii) a UMP invariant test of H : ξ1 = · · · = ξa = 0 (iii) a test of H : 2 /σ 2 σB (ξi = µ + αi ); ≤ δ which is both UMP invariant and UMP unbiased. Problem 7.32 Let (X1j , . . . , Xpj ), j = 1, . . . , n, be a sample from a p-variate normal distribution with mean (ξ1 , . . . , ξp ) and covariance matrix Σ = (σij ), 2 2 where σij = σ 2 when j = i, and σij = ρσ 2 when j = i. Show that the covariance matrix is positive definite if and only if ρ > −1/(p − 1).   2 [For fixed σ and ρ < 0, the quadratic form  (1/σ 2 ) σij yi yj = yi +  2 ρ yi yj takes on its minimum value over yi = 1 when all the y’s are equal.] Problem 7.33 Under the assumptions of the preceding problem, determine the UMP invariant test (with respect to a suitable G) of H : ξi = . . . = ξp . [Show that this model agrees with that of Problem 7.31 if ρ = σb2 /(σb2 +σ 2 ), except that instead of being positive, ρ now only needs to satisfy ρ > −1/(p − 1).] Problem 7.34 Permitting interactions in the model of Problem 7.30 leads to the model Xijk = µ + Ai + Bj + Cij + Uijk (i = 1, . . . , a; j = 1, . . . , b, k = 1, . . . , n). where the A’s, B’s, C’s, and U ’s are independent normal with mean zero and 2 2 2 variances σA , σB , σC and σ 2 . (i) Give an example of a situation in which such a model might be appropriate. (ii) Reduce the model to a convenient canonical form along the lines of Section 7.4. 2 2 = 0; (b) H2 : σC = 0. (iii) Determine UMP unbiased tests of (a) H1 : σB Problem 7.35 Formal analogy with the model of Problem 7.34 suggests the mixed model Xijk = µ + αi + Bj + Cij + Uijk 316 7. Linear Hypotheses with the B’s, C’s, and U ’s as in Problem 7.34. Reduce this model to a canonical form involving X··· and the sums of squares   (Xi·· −X··· −αi )2 , 2 +σ 2 nσC (Xij· −Xi·· −X·j· +X··· )2 2 +σ 2 nσC ,  (X·j· −X··· )2 , 2 +nσ 2 +σ 2 anσB C  (Xijk −Xi·· −X·j· +X··· )2 σ2 . Problem 7.36 Among all tests that are both unbiased and invariant under suitable groups under the assumptions of Problem 7.35, there exist UMP tests of (i) H1 : α1 = · · · = αa = 0; 2 2 (ii) H2 : σB /(nσC + σ 2 ) ≤ C; 2 (iii) H3 : σC /σ 2 ≤ C. Note. The independence assumptions of Problems 7.35 and 7.36 often are not realistic. For alternative models, derived from more basic assumptions, see Scheffé (1956, 1959). Relations between the two types of models are discussed in Hocking (1973), Cohen and Miller (1976), and Stuart and Ord (1991). Problem 7.37 Let (X1j1 , . . . , X1jn ; X2j1 , . . . , X2jn ; . . . ; Xaj1 , . . . , Xajn ), j = 1, . . . , b, be a samplefrom an an-variate normal distribution. Let E(Xijk ) = ξi , and denote by , Xijn ) with ii the matrix of covariances of (Xij1 , . . . (Xi j1 , . . . , Xi jn ). Suppose that for all i, the diagonal elements of ii are = τ 2 and the off-diagonal elements are = ρ1 τ 2 , and that for i = i all n2 elements of  2 ii are = ρ2 τ . (i) Find necessary and sufficient conditions on ρ1 and ρ2 for the overall abn × abn covariance matrix to be positive definite. (ii) Show that this model agrees with that of Problem 7.35 for suitable values of ρ1 and ρ2 . Section 7.9 Problem 7.38 If n ≤ p, the matrix S with (i, j) component Si,j defined in (7.53) is singular. If n > p, it is nonsingular with probability 1. If n ≤ p, the test φ ≡ α is the only test that is invariant under the group of nonsingular linear transformations. Problem 7.39 Show that the statistic W given in (7.55) is maximal invariant. [Hint: If (X̄, S) and (Ȳ , T ) are such that X̄ T S −1 X̄ = Ȳ T T −1 Ȳ , then a transformation C that transforms one to the other is given by C = Y (X T S −1 X)−1 X T S −1 .] Problem 7.40 Verify that the density of W is given by (7.55). Problem 7.41 The confidence ellipsoids (7.59) for (ξ1 , . . . , ξp ) are equivariant under the group of Section 7.9. 7.11. Notes 317 Problem 7.42 For testing a multivariate mean vector ξ is zero in the case where Σ is known, derive a UMPI test. Problem 7.43 Extend the one-sample problem to the two-sample problem for testing whether two multivariate normal distributions with common unknown covariance matrix have the same mean vectors. Problem 7.44 Bayes character and admissibility of Hotelling’s T 2 . (i) Let (Xα1 , . . . , Xαp ), α = 1, . . . , n, be a sample from a p-variate normal distribution with unknown mean ξ = (ξ1 , . . . , ξp ) and covariance matrix Σ = A−1 , and with p ≤ n − 1. Then the one-sample T 2 -test of H : ξ = 0 against K : ξ = 0 is a Bayes test with respect to prior distributions Λ0 and Λ1 which generalize those of Example 6.7.13 (continued). (ii) The test of part (i) is admissible for testing H against the alternatives ψ 2 ≤ c for any c > 0. [If ω is the subset of points (0, Σ) of ΩH satisfying Σ−1 = A + η  η for some fixed positive definite p × p matrix A and arbitrary η = (η1 , . . . , ηp ), and ΩA,b is the subset of points (ξ, Σ) of ΩK satisfying Σ−1 = A + η  η, ξ  = bΣη  for the same A and some fixed b > 0, let Λ0 and Λ1 have densities defined over ω and ΩA,b , respectively by λ0 (η) = C0 |A + η  η|−n/2 and λ1 (η) = C1 |A + η  η|−n/2 exp    nb2  η(A + η  η)−1 η  . 2 (Kiefer and Schwartz, 1965).] Problem 7.45 Suppose (X1 , . . . , Xp ) have the multivariate normal density (7.51), so that E(Xi ) = ξi and A−1 is the known positive definite covariance matrix. The vector of means ξ = (ξ1 , . . . , ξp ) is known to lie in a given s-dimensional linear space ΠΩ with s ≤ p; the hypothesis to be tested is that ξ lies in a given (s − r)-dimensional linear subspace Πω of ΠΩ (r ≤ s). (i) Determine the UMPI test under a suitable group of transformations as explicitly as possible. Find an expression for the power function. (ii) Specialize to the case of a simple null hypothesis. 7.11 Notes The general linear model in the parametric form (7.18) was formulated at the beginning of the 19th century by Legendre and Gauss, who were concerned with estimating the unknown parameters. [For an account of its history, see Seal (1967).] The canonical form (7.2) of the model is due to Kolodziejczyk (1935). The analysis of variance, including the concept of interaction, was developed by Fisher in the 1920s and 1930s, and a systematic account is provided by Scheffé 318 7. Linear Hypotheses (1959) in a book that includes a careful treatment of alternative models and of robustness questions. Different approaches to analysis of variance than that given here are considered in Speed (1987) and the discussion following this paper, and in Diaconis (1988, Section 8C). Rank tests are discussed in Marden and Muyot (1995). Admissibility results for testing homogeneity of variances in a normal balanced one-way layout are given in Cohen and Marden (1989). Linear models have been generalized in many directions. Loglinear models provide extensions to important discrete data. [Both are reviewed in Christensen (2000).] These two classes of models are subsumed in generalized linear models discussed for example in McCullagh and Nelder (1983), Dobson (1990) and Agresti (2002), and they in turn are a subset of additive linear models which are discussed in Hastie and Tibshirani (1990, 1997). Modern treatments of regression analysis can be found, for example, in Weisberg (1985), Atkinson and Riani (2000) and Ruppert, Wand and Carroll (2003). UMPI tests can be constructed for tests of lack of fit in some regression models; see Christensen (1989) and Miller, Neill and Sherfey (1998). Hsu (1941) shows that the test (7.7) is UMP among all tests whose power function depends only on the noncentrality parameter. Hsu (1945) obtains a result on best average power for the T 2 -test analogous to that of Chapter 7, Problem 7.5. Tests of multivariate linear hypotheses and the associated confidence sets have their origin in the work of Hotelling (1931). More details on these procedures and discussion of other multivariate techniques can be found in the comprehensive books by Anderson (2003) and Seber (1984). A more geometric approach stressing invariance is provided by Eaton (1983). For some recent work on using rank tests in multivariate problems, see Choi and Marden (1997), Hettmansperger, Möttönen and Oja (1997), and Akritas, Arnold and Brunner (1997). 8 The Minimax Principle 8.1 Tests with Guaranteed Power The criteria discussed so far, unbiasedness and invariance, suffer from the disadvantage of being applicable, or leading to optimum solutions, only in rather restricted classes of problems. We shall therefore turn now to an alternative approach, which potentially is of much wider applicability. Unfortunately, its application to specific problems is in general not easy, unless there exists a UMP invariant test. One of the important considerations in planning an experiment is the number of observations required to insure that the resulting statistical procedure will have the desired precision or sensitivity. For problems of hypothesis testing this means that the probabilities of the two kinds of errors should not exceed certain preassigned bounds, say α and 1 − β, so that the tests must satisfy the conditions Eθ ϕ(X) ≤ α for θ ∈ ΩH , Eθ ϕ(X) ≥ β for θ ∈ ΩK . (8.1) If the power function Eθ ϕ(X) is continuous and if α < β, (8.2) cannot hold when the sets ΩH and ΩK are contiguous. This mathematical difficulty corresponds in part to the fact that the division of the parameter values θ into the classes ΩH and ΩK for which the two different decisions are appropriate is frequently not sharp. Between the values for which one or the other of the decisions is clearly correct there may lie others for which the relative advantages and disadvantages of acceptance and rejection are approximately in balance. Accordingly we shall assume that Ω is partitioned into three sets Ω = Ω H + ΩI + ΩK , 320 8. The Minimax Principle of which ΩI designates the indifference zone, and ΩK the class of parameter values differing so widely from those postulated by the hypothesis that false acceptance of H is a serious error, which should occur with probability at most 1 − β. To see how the sample size is determined in this situation, suppose that X1 , X2 , . . . constitute the sequence of available random variables, and for a moment let n be fixed and let X = (X1 , . . . , Xn ). In the usual applications (for a more precise statement, see Problem 8.1), there exists a test ϕn which maximizes inf Eθ ϕ(X) (8.2) Ωk among all level-α tests based on X. Let βn = inf ΩK Eθ ϕn (X), and suppose that for sufficiently large n there exists a test satisfying (8.2). [Conditions under which this is the case are given by Berger (1951a) and Kraft (1955).] The desired sample size, which is the smallest value of n for which βn ≥ β, is then obtained by trial and error. This requires the ability of determining for each fixed n the test that maximizes (8.2) subject to Eθ ϕ(X) ≤ α for θ ∈ ΩH . (8.3) A method for determining a test with this maximin property (of maximizing the minimum power over ΩK ) is obtained by generalizing Theorem 3.8.1. It will be convenient in this discussion to make a change of notation, and to denote by ω and ω  the subsets of Ω previously denoted by ΩH and ΩK . Let P = {Pθ , θ ∈ ω ∪ ω  } be a family of probability distributions over a sample space (X , A) with densities pθ = dPθ /dµ with respect to a σ-finite measure µ, and suppose that the densities pθ (x) considered as functions of the two variables (x, θ) are measurable (A × B) and (A × B ), where B and B are given σ-fields over ω and ω  . Under these assumptions, the following theorem gives conditions under which a solution of a suitable Bayes problem provides a test with the required properties. Theorem 8.1.1 For any distributions Λ and Λ over B and B , let ϕΛ,Λ be the most powerful test for testing  pθ (x) dΛ(θ) h(x) = ω at level α against h (x) =  ω pθ (x) dΛ (θ) and let βΛ,Λ , be its power against the alternative h . If there exist Λ and Λ such that sup Eθ ϕΛ,Λ (X) ≤ α, inf Eθ ϕΛ,Λ (X) = βΛ,Λ , ω ω (8.4) then: (i) ϕΛ,Λ maximizes inf ω Eθ ϕ(X) among all level-α tests of the hypothesis H : θ ∈ ω and is the unique test with this property if it is the unique most powerful level-α test for testing h against h . 8.1. Tests with Guaranteed Power 321 (ii) The pair of distributions Λ, Λ is least favorable in the sense that for any other pair ν, ν  we have βΛ,Λ ≤ βν,ν  . Proof. (i): If ϕ∗ is any other level-α test of H, it is also of level α for testing the simply hypothesis that the density of X is h, and the power of ϕ∗ against h therefore cannot exceed βΛ,Λ . It follows that  Eθ ϕ∗ (X) dΛ (θ) ≤ βΛ,Λ = inf Eθ ϕΛΛ (X), inf Eθ ϕ∗ (X) ≤ ω ω ω and the second inequality is strict if ϕΛΛ is unique. (ii): Let ν, ν  be any other distributions over (ω, B) and (ω  , B ), and let   pθ (x)dν(θ), g  (x) = pθ (x) dν  (θ). g(x) = ω ω Since both ϕΛ,Λ and ϕν,ν  are level-α tests of the hypothesis that g(x) is the density of X, it follows that   βν,ν ≥ ϕΛ,Λ (x)g  (x) dµ(x) ≥ inf Eθ ϕΛ,Λ (X) = βΛ,Λ . ω Corollary 8.1.1 Let Λ, Λ that ⎧ ⎨ 1 γ ϕΛ,Λ (x) = ⎩ 0 be two probability distributions and C a constant such if if if pθ (x) dΛ (θ) > C p (x) dΛ (θ) = C ω θ p (x) dΛ (θ) < C ω θ is a size-α test for testing that the density of X is  Λ(ω0 ) = Λ where ω0 ω0 pθ (x) dΛ(θ) p (x) dΛ(θ) ω θ p (x) dΛ(θ) ω θ ω (ω0 ) ω ω (8.5) pθ (x) dΛ(θ) and such that = 1, (8.6) =   θ : θ ∈ ω and Eθ ϕΛ,Λ (X) = sup Eθ ϕΛ,Λ (X) =       θ : θ ∈ ω and Eθ ϕΛ,Λ (X) = inf  Eθ ϕΛ,Λ (X) . θ  ∈ω θ ∈ω Then the conclusions of Theorem 8.1.1 hold. Proof. If h, h , and βΛ,Λ are defined as in Theorem 8.1.1, the assumptions imply that ϕΛ,Λ is a most powerful level-α test for testing h against h , that  Eθ ϕΛ,Λ (X) dΛ(θ) = α, sup Eθ ϕΛ,Λ (X) = ω and that ω  inf Eθ ϕΛ,Λ (X) = ω ω Eθ ϕΛ,Λ (X) dΛ (θ) = βΛ,Λ . The condition (8.4) is thus satisfied and Theorem 8.1.1 applies. 322 8. The Minimax Principle Suppose that the sets ΩH , ΩI , and ΩK are defined in terms of a nonnegative function d, which is a measure of the distance of θ from H, by ΩH = {θ : d(θ) = 0}, ΩK = {0 : d(θ) ≥ ∆}. ΩI = {θ : 0 < d(θ) < ∆}, Suppose also that the power function of any test is continuous in θ. In the limit as ∆ = 0, there is no indifference zone. Then ΩK becomes the set {θ : d(θ) > 0}, and the infimum of β(θ) over ΩK is ≤ α for any level-α test. This infimum is therefore maximized by any test satisfying β(θ) ≥ α for all θ ∈ ΩK , that is, by any unbiased test, so that unbiasedness is seen to be a limiting form of the maximin criterion. A more useful limiting form, since it will typically lead to a unique test, is given by the following definition. A test ϕ0 is said to maximize the minimum power locally1 if, given any other test ϕ, there exists ∆0 such that inf βϕ0 (θ) ≥ inf βϕ (θ) ω∆ ω∆ for all 0 < ∆ < ∆0 , (8.7) where ω∆ is the set of θ’s for which d(θ) ≥ ∆. 8.2 Examples In Chapter 3 it was shown for a family of probability densities depending on a real parameter θ that a UMP test exists for testing H : θ ≤ θ0 against θ > θ0 provided for all θ < θ the ratio pθ (x)/pθ (x) is a monotone function of some real-valued statistic. This assumption, although satisfied for a one-parameter exponential family, is quite restrictive, and a UMP test of H will in fact exist only rarely. A more general approach is furnished by the formulation of the preceding section. If the indifference zone is the set of θ’s with θ0 < θ < θ1 , the problem becomes that of maximizing the minimum power over the class of alternatives ω  : θ ≥ θ1 . Under appropriate assumptions, one would expect the least favorable distributions Λ and Λ of Theorem 8.1.1 to assign probability 1 to the points θ0 and θ1 , and hence the maximin test to be given by the rejection region pθ1 (x)/pθ0 (x) > C. The following lemma gives sufficient conditions for this to be the case. Lemma 8.2.1 Let X1 , . . . , Xn be identically and independently distributed with probability density fθ (x), where θ and x are real-valued, and suppose that for any θ < θ the ratio fθ (x)/fθ (x) is a nondecreasing function of x. Then the level-α test ϕ of H which maximizes the minimum power over ω  is given by ⎧ ⎨ 1 if r(x1 , . . . , xn ) > C, γ if r(x1 , . . . , xn ) = C, (8.8) ϕ(x1 , . . . , x1 ) = ⎩ 0 if r(x1 , . . . , xn ) < C, where r(x1 , . . . , xn ) = fθ1 (x1 ) . . . fθ1 (xn )/fθ0 (x1 ) . . . fθ0 (xn ) and where C and γ are determined by Eθ0 ϕ(X1 , . . . , Xn ) = α. 1A different definition of local minimaxity is given by Giri and Kiefer (1964). (8.9) 8.2. Examples 323 Proof. The function ϕ(x1 , . . . , xn ) is nondecreasing in each of its arguments, so that by Lemma 3.4.2, Eθ ϕ(X1 , . . . , Xn ) ≤ Eθ ϕ(X1 , . . . , Xn )  when θ < 0 . Hence the power function of ϕ is monotone and ϕ is a level-α test. Since ϕ = ϕΛ,Λ , where Λ and Λ are the distributions assigning probability 1 to the points θ0 and θ1 , the condition (8.4) is satisfied, which proves the desired result as well as the fact that the pair of distributions (Λ, Λ ) is least favorable. Example 8.2.1 Let θ be a location parameter, so that fθ (x) = g(x − θ), and suppose for simplicity that g(x) > 0 for all x. We will show that a necessary and sufficient condition for fθ (x) to have monotone likelihood ratio in x is that − log g is convex. The condition of monotone likelihood ratio in x, g(x − θ ) g(x − θ ) ≤ g(x − θ) g(x − θ) for all x < x , θ < θ , is equivalent to log g(x − θ) + log g(x − θ ) ≤ log g(x − θ) + log g(x − θ ). Since x−θ = t(x−θ )+(1−t)(x −θ) and x −θ = (1−t)(x−θ )+t(x −θ), where t = (x − x)/(x − x + θ − θ), a sufficient condition for this to hold is that the function − log g is convex. To see that this condition is also necessary, let a < b be any real numbers, and let x − θ = a, x − θ = b, and x − θ = x − θ. Then x − θ = 12 (x − θ + x − θ ) = 12 (a + b), and the condition of monotone likelihood ratio implies # $ 1 1 log g(a) + log g(b)] ≤ log g (a + b) . 2 2 Since log g is measurable, this in turn implies that − log g is convex.2 A density g for which − log g is convex is called strongly unimodal. Basic properties of such densities were obtained by Ibragimov (1956). Strong unimodality is a special case of total positivity. A density of the form g(x − θ) which is totally positive of order r is said to be a Polya frequency function of order r. It follows from Example 8.2.1 that g(x − θ) is a Polya frequency function of order 2 if and only if it is strongly unimodal. [For further results concerning Polya frequency functions and strongly unimodal densities, see Karlin (1968), Marshall and Olkin (1979), Huang and Ghosh (1982), and Loh (1984a, b).] Two distributions which satisfy the above condition [besides the normal distribution, for which the resulting densities pθ (x1 , . . . , xn ) form an exponential family] are the double exponential distribution with g(x) = 12 e−|x| and the logistic distribution, whose cumulative distribution function is 1 G(x) = , 1 + e−x so that the density is g(x) = e−x /(1 + e−x )2 . 2 See Sierpinski (1920). 324 8. The Minimax Principle Example 8.2.2 To consider the corresponding problem for a scale parameter, let fθ (x) = θ−1 h(x/θ) where h is an even function. Without loss of generality one may then restrict x to be nonnegative, since the absolute values |X1 |, . . . , |Xn | form a set of sufficient statistics for θ. If Yi = log Xi and η = log θ, the density of Yi is h(ey−η )ey−η . By Example 8.2.1, if h(x) > 0 for all x ≥ 0, a necessary and sufficient condition for fθ (x)/fθ (x) to be a nondecreasing function of x for all θ < θ is that − log[ey h(ey )] or equivalently − log h(ey ) is a convex function of y. An example in which this holds—in addition to the normal and double-exponential distributions, where the resulting densities form an exponential family—is the Cauchy distribution with 1 1 h(x) = . π 1 + x2 Since the convexity of − log h(y) implies that of − log h(ey ), it follows that if h is an even function and h(x − θ) has monotone likelihood ratio, so does h(x/θ). When h is the normal or double-exponential distribution, this property of h(x/θ) also follows from Example 8.2.1. That monotone likelihood ratio for the scaleparameter family does not conversely imply the same property for the associated location parameter family is illustrated by the Cauchy distribution. The condition is therefore more restrictive for a location than for a scale parameter. The chief difficulty in the application of Theorem 8.1.1 to specific problems is the necessity of knowing, or at least being able to guess correctly, a pair of least favorable distributions (Λ, Λ ). Guidance for obtaining these distributions is sometimes provided by invariance considerations. If there exists a group G of transformations of X such that the induced group Ḡ leaves both ω and ω  invariant, the problem is symmetric in the various θ’s that can be transformed into each other under Ḡ. It then seems plausible that unless Λ and Λ exhibit the same symmetries, they will make the statistician’s task easier, and hence will not be least favorable. Example 8.2.3 In the problem of paired comparisons considered in Example 6.3.5, the observations Xi (i = 1, . . . , n) are independent variables taking on the values 1 and 0 with probabilities pi and qi = 1 − pi . The hypothesis H to be tested specifies the set ω : max pi ≤ 12 . Only alternatives with pi ≥ 12 for all i are considered, and as ω  we take the subset of those alternatives for which max pi ≥ 12 + δ. One would expect Λ to assign probability 1 to the point p1 = · · · pn = 12 , and Λ to assign positive probability only to the n points (p1 , . . . , pn ) which have n − 1 coordinates equal to 12 and the remaining coordinate equal to 12 + δ. Because of the symmetry with regard to the n variables, it seems plausible that Λ should assign equal probability 1/n to each of these n points. With these choices, the test ϕΛ,Λ rejects when x n 1  +δ i 2 > C. 1 i=1 2  This is equivalent to n i=1 xi > C, which had previously been seen to be UMP invariant for this problem. Since the critical function ϕΛ,Λ (x1 , . . . , xn ) is nonde- 8.2. Examples 325 creasing in each of its arguments, it follows from Lemma 3.4.2 that pi ≤ pi for i = 1, . . . , n implies Ep1 ,...,pn ϕΛ,Λ (X1 , . . . , Xn ) ≤ Ep1 ,...,pn ϕΛ,Λ (X1 , . . . , Xn ) and hence the conditions of Theorem 8.1.1 are satisfied. Example 8.2.4 Let X = (X1 , . . . , Xn ) be a sample from N (ξ, σ 2 ), and consider the problem of testing H : σ = σ0 against the set of alternatives ω  : σ ≤ σ1 or σ ≥ σ2 (σ1 < σ0 < σ2 ). This problem remains invariant under the transformations Xi = Xi +c, which in the parameter space induce the group Ḡ of transformations ξ  = ξ + c, σ  = σ. One would therefore expect the least favorable distribution Λ over the line ω : −∞ < ξ < ∞, σ = σ0 , to be invariant under Ḡ. Such invariance implies that Λ assigns to any interval a measure proportional to the length of the interval. Hence Λ cannot be a probability measure and Theorem 8.1.1 is not directly applicable. The difficulty can be avoided by approximating Λ by a sequence of probability distributions, in the present case for example by the sequence of normal distributions N (0, k), k = 1, 2, . . . . In the particular problem under consideration, it happens that there also exist least favorable distributions Λ and Λ , which are true probability distributions and therefore not invariant. These distributions can be obtained by an examination of the corresponding one-sided problem in Section 3.9, as follows. On ω, where the only variable is ξ, the distribution Λ of ξ is taken as the normal distribution with an arbitrary mean ξ1 and with variance (σ22 − σ02 )/n. Under Λ all probability should be concentrated on the two lines σ = σ1 and σ = σ2 in the (ξ, σ) plane, and we put Λ = pΛ1 + qΛ2 , where Λ1 is the normal distribution with mean ξ1 and variance (σ22 − σ12 )/n, while Λ2 assigns probability 1 to the point (ξ1 , σ2 ). A computation analogous to that carried out in Section 3.9 then shows the acceptance region to be given by p σ1n−1 σ2   −1  n 2 2 (x − x̄) − (x̄ − ξ ) i 1 2 2σ12  2σ2 - . −1 q 2 2 + n exp − x̄) + n(x̄ − ξ ) (x i 1 σ 2σ22  2   −1 1 n (xi − x̄)2 − 2 (x̄ − ξ1 )2 exp 2σ02 2σ2 σ0n−1 σ2 exp a, ≤ a. (i) For all 0 < i < 1, there exist unique constants a and b such that q0 and q1 are probability densities with respect to µ; the resulting qi are members of Pi (i = 0, 1). (ii) There exist δ0 , δ1 such that for all i ≤ δi the constants a and b satisfy a < b and that the resulting q0 and q1 are distinct. (iii) If i ≤ δi for i = 0, 1, the families P0 and P1 are nonoverlapping and the pair (q0 , q1 ) is least favorable, so that the maximin test of P0 against P1 rejects when q1 (x)/q0 (x) is sufficiently large. Note. Suppose a < b, and let r(x) = Then p1 (x) , p0 (x) r∗ (x) = ⎧ ⎨ ka kr(x) r (x) = ⎩ kb ∗ q1 (x) , q0 (x) when when when and k= 1 − 1 . 1 − 0 r(x) ≤ a, a < r(x) < b, b ≤ r(x). (8.12) The maximin test thus replaces the original probability ratio with a censored version. Proof. The proof will be given under the simplifying assumption that p0 (x) and p1 (x) are positive for all x in the sample space. 8.3. Comparing Two Approximate Hypotheses 327 (i): For q1 to be a probability density, a must satisfy the equation P1 [r(X) > a] + aP0 [r(X) ≤ a] = 1 . 1 − 1 (8.13) If (8.13) holds, it is easily checked that q1 ∈ P1 (Problem 8.12). To prove existence and uniqueness of a solution a of (8.13), let γ(c) = P1 [r(X) > c] + cP0 [r(X) ≤ c]. Then γ(0) = 1 and Furthermore (Problem 8.14) γ(c + ∆) − γ(c) = γ(c) → ∞ as c → ∞. (8.14)  ∆ p0 (x) dµ(x)  (8.15) r(x)≤c [c + ∆ − r(x)]p0 (x) dµ(x). + c c0 and this proves uniqueness. The proof for b is exactly analogous (Problem 8.13). (ii): As 1 → 0, the solution a of (8.13) tends to c0 . Analogously, as 1 → 0, b → ∞ (Problem 8.13). (iii): This will follow from the following facts: (a) When X is distributed according to a distribution in P0 , the statistic r∗ (X) is stochastically largest when the distribution of X is Q0 . (b) When X is distributed according to a distribution in P1 , r∗ (X) is stochastically smallest for Q1 . (c) r∗ (X) is stochastically larger when the distribution of X is Q1 than when it is Q0 . These statements are summarized in the inequalities Q0 [r∗ (X) < t] ≥ Q0 [r∗ (X) < t] ≥ Q1 [r∗ (X) < t] ≥ Q1 [r∗ (X) < t] Qi (8.18) for all t and all ∈ Pi . From (8.12), it is seen that (8.18) is obvious when t ≤ ka or t > kb. Suppose therefore that ak < t ≤ bk, and denote the event r∗ (X) < t by E. Then Q0 (E) ≥ (1 − 0 )P0 (E) by (8.10). But r∗ (x) < t < kb implies r(X) < b and hence Q0 (E) = (1 − )P0 (E). Thus Q0 (E) ≥ Q0 (E), and analogously Q1 (E) ≤ Q1 (E). Finally, the middle inequality of (8.18) follows from Corollary 3.2.1. 328 8. The Minimax Principle If the ’s are sufficiently small so that Q0 = Q1 , it follows from (a)–(c) that P0 and P1 are nonoverlapping. That (Q0 , Q1 ) is least favorable and the associated test ϕ is maximin now follows from Theorem 8.1.1, since the most powerful test ϕ for testing Q0 against Q1 is a nondecreasing function of q1 (X)/q0 (X). This shows that Eϕ(X) takes on its sup over P0 at Q0 and its inf over P1 at Q1 , and this completes the proof. Generalizations of this theorem are given by Huber and Strassen (1973, 1974). See also Rieder (1977) and Bednarski (1984). An optimum permutation test, with generalizations to the case of unknown location and scale parameters, is discussed by Lambert (1985). When the data consist of n identically, independently distributed random variables X1 , . . . , Xn , the neighborhoods (8.10) may not be appropriate, since they do not preserve the assumption of independence. If Pi has density pi (x1 , . . . , xn ) = fi (x1 ) . . . fi (xn ) (i = 0, 1), (8.19) a more appropriate model approximating (8.19) may then assign to X = (X1 , . . . , Xn ) the family Pi∗ of distributions according to which the Xj are independently distributed, each with distribution (1 − i )Fi (xj ) + i Gi (xj ), (8.20) where Fi has density fi and where as before the Gi are arbitrary. Corollary 8.3.1 Suppose q0 and q1 defined by (8.11) with x = xj satisfy (8.18) and hence are a least favorable pair for testing P0 against P1 on the basis of the single observation Xj . Then the pair of distributions with densities qi (x1 ) . . . qi (xn ) (i = 0, 1) is least favorable for testing P0∗ against P1∗ , so that the maximin test is given by ⎧  n  ⎨ 1  q1 (xj ) > γ if = c. ϕ(x1 , . . . , xn ) = (8.21) ⎩ q0 (xj ) < j=1 0 Proof. By assumption, the random variables Yj = q1 (Xj )/q0 (Xj ) are stochastically increasing as one moves successively from Q0 ∈ P0 to Q0 to Q1 to Q1 ∈ P1 . The same is then true of any function ψ(Y1 , . . . , Yn ) which is nondecreasing in each of its arguments by Lemma 3.4.1, and hence of ϕ defined by (8.21). The proof now follows from Theorem 8.3.1. Instead of the problem of testing P0 against P1 , consider now the situation of Lemma 8.2.1 where H : θ ≤ θ0 is to be tested against θ ≥ θ1 (θ0 < θ1 ) on the basis of n independent observations Xj , each distributed according to a distribution Fθ (xj ) whose density fθ (xj ) is assumed to have monotone likelihood ratio in xj . A robust version of this problem is obtained by replacing Fθ with (1 − )Fθ (xj ) + G(xj ), j = 1, . . . , n, (8.22) P0∗∗ where  is given and for each θ the distribution G is arbitrary. Let and P1∗∗ be the classes of distributions (8.22) with θ ≤ θ0 and θ ≥ θ1 respectively; and let P0∗ and P1∗ be defined as in Corollary 8.3.1 with fθi in place of fi . Then the 8.4. Maximin Tests and Invariance 329 maximin test (8.21) of P0∗ against P1∗ retains this property for testing P0∗∗ against P1∗∗ . This is proved in the same way as Corollary 8.3.1, using the additional fact that if Fθ is stochastically larger than Fθ , then (1 − )Fθ + G is stochastically larger than (1 − )Fθ + G. 8.4 Maximin Tests and Invariance When the problem of testing ΩH against ΩK remains invariant under a certain group of transformations, it seems reasonable to expect the existence of an invariant pair of least favorable distributions (or at least of sequences of distributions which in some sense are least favorable and invariant in the limit), and hence also of a maximin test which is invariant. This suggests the possibility of bypassing the somewhat cumbersome approach of the preceding sections. If it could be proved that for an invariant problem there always exists an invariant test that maximizes the minimum power over ΩK , attention could be restricted to invariant tests; in particular, a UMP invariant test would then automatically have the desired maximin property (although it would not necessarily be admissible). These speculations turn out to be correct for an important class of problems, although unfortunately not in general. To find out under what conditions they hold, it is convenient first to separate out the statistical aspects of the problem from the group-theoretic ones by means of the following lemma. Lemma 8.4.1 Let P = {Pθ , θ ∈ Ω} be a dominated family of distributions on (X , A), and let G be a group of transformations of (X , A), such that the induced group Ḡ leaves the two subsets ΩH and ΩK of Ω invariant. Suppose that for any critical function ϕ there exists an (almost) invariant critical function ψ satisfying inf Eḡθ ϕ(X) ≤ Eθ ψ(X) ≤ sup Eḡθ ϕ(X) Ḡ (8.23) Ḡ for all θ ∈ Ω. Then if there exists a level-α test ϕ0 maximizing inf Ωk Eθ ϕ(X), there also exists an (almost) invariant test with this property. Proof. Let inf ΩK Eθ ϕ0 (X) = β, and let ψ0 be an (almost) invariant test such that (8.23) holds with ϕ = ϕ0 , ψ = ψ0 . Then Eθ ψ0 (X) ≤ sup Eḡθ ϕ0 (X) ≤ α for all θ ∈ ΩH Ḡ and Eθ ψ0 (X) ≥ inf Eḡθ ϕ0 (X) ≥ β for all Ḡ θ ∈ ΩK , as was to be proved. To determine conditions under which there exists an invariant or almost invariant test ψ satisfying (8.23), consider first the simplest case that G is a finite group, G = {g1 , . . . , gN } say. If ψ is then defined by ψ(x) = N 1  ϕ(gi x), N i=1 (8.24) 330 8. The Minimax Principle it is clear that ψ is again a critical function, and that it is invariant under G. It also satisfies (8.23), since Eθ ϕ(gX) = Eḡθ ϕ(X) so that Eθ ψ(X) is the average of a number of terms of which the first and last member of (8.23) are the minimum and maximum respectively. An illustration of the finite case is furnished by Example 8.2.3. Here the problem remains invariant under the n! permutations of the variables (X1 , . . . , Xn ). Lemma 8.4.1 is applicable and shows that there exists an invariant test maximizing inf ΩK Eθ ϕ(X). Thus in particular the UMP invariant test obtained in Example 6.3.5 has this maximin property and therefore constitutes a solution of the problem. It also follows that, under the setting of Theorem 6.3.1, the UMPI test given by (6.9) is maximin. The definition (8.24) suggests the possibility of obtaining ψ(x) also in other cases by averaging the values of ϕ(gx) with respect to a suitable probability distribution over the group G. To see what conditions would be required of this distribution, let B be a σ-field of subsets of G and ν a probability distribution over (G, B). Disregarding measurability problems for the moment, let ψ be defined by  ψ(x) = ϕ(gx) dν(g). (8.25) Then 0 ≤ ψ ≤ 1, and (8.23) is seen to hold by applying Fubini’s theorem (Theorem 2.2.4) to the integral of ψ with respect to the distribution Pθ . For any g0 ∈ G,   ψ(g0 x) = ϕ(gg0 x) dν(g) = ϕ(hx) dν ∗ (h) , where h = gg0 and where ν ∗ is the measure defined by ν ∗ (B) = ν(Bg0−1 ) for all B ∈ B, into which ν is transformed by the transformation h = gg0 . Thus ψ will have the desired invariance property, ψ(g0 x) = ψ(x) for all g0 ∈ G, if ν is right invariant, that is, if it satisfies ν(Bg) = ν(B) for all B ∈ B, g ∈ G. (8.26) Such a condition was previously used in (6.16). The measurability assumptions required for the above argument are: (i) For any A ∈ A, the set of pairs (x, g) with gx ∈ A is measurable (A × B). This insures that the function ψ defined by (8.25) is again measurable. (ii) For any B ∈ B, g ∈ G, the set Bg belongs to B. Example 8.4.1 If G is a finite group with elements g1 , . . . , gN , let B be the class of all subsets of G and ν the probability measure assigning probability 1/N to each of the N elements. The condition (8.26) is then satisfied, and the definition (8.25) of ψ in this case reduces to (8.24). Example 8.4.2 Consider the group G of orthogonal n × n matrices Γ, with the group product Γ1 Γ2 defined as the corresponding matrix product. Each matrix can be interpreted as the point in n2 -dimensional Euclidean space whose coordinates are the n2 elements of the matrix. The group then defines a subset of this 8.5. The Hunt–Stein Theorem 331 space; the Borel subsets of G will be taken as the σ-field B. To prove the existence of a right invariant probability measure over (G, B), we shall define a random orthogonal matrix whose probability distribution satisfies (8.26) and is therefore the required measure. With any nonsingular matrix x = (xij ), associate the orthogonal matrix y = f (x) obtained by applying the following Gram–Schmidt orthogonalization process to the n row vectors xi = (xi1 , . . . , xin ) of x : y1 is the unit vector in the direction of x1 ; y2 the unit vector in the plane spanned by x1 and x2 which is orthogonal to y1 and forms an acute angle with x2 ; and so on. Let y = (yij ) be the matrix whose ith row is yi . Suppose now that the variables Xij (i, j = 1, . . . , n) are independently distributed as N (0, 1), let X denote the random matrix (Xij ), and let Y = f (X). To show that the distribution of the random orthogonal matrix Y satisfies (8.26), consider any fixed orthogonal matrix Γ and any fixed set B ∈ B. Then P {Y ∈ BΓ} = P {Y Γ ∈ B} and from the definition of f it is seen that Y Γ = f (XΓ ). Since the n2 elements of the matrix XΓ have the same joint distribution as those of the matrix X, the matrices f (XΓ ) and f (X) also have the same distribution, as was to be proved. Examples 8.4.1 and 8.4.2 are sufficient for the applications to be made here. General conditions for the existence of an invariant probability measure, of which these examples are simple special cases, are given in the theory of Haar measure. [This is treated, for example, in the books by Halmos (1974), Loomis (1953), and Nachbin (1965). For a discussion in a statistical setting, see Eaton (1983, 1989), Farrell (1985a), and Wijsman (1990), and for a more elementary treatment Berger (1985a).] 8.5 The Hunt–Stein Theorem Invariant measures exist (and are essentially unique) for a large class of groups, but unfortunately they are frequently not finite and hence cannot be taken to be probability measures. The situation is similar and related to that of the nonexistence of a least favorable pair of distributions in Theorem 8.1.1. There it is usually possible to overcome the difficulty by considering instead a sequence of distributions which has the desired property in the limit. Analogously we shall now generalize the construction of ψ as an average with respect to a right-invariant probability distribution, by considering a sequence of distributions over G which are approximately right-invariant for n sufficiently large. Let P = {Pθ , θ ∈ Ω} be a family of distributions over a Euclidean space (X , A) dominated by a σ-finite measure µ, and let G be a group of transformations of (X , A) such that the induced group Ḡ leaves Ω invariant. Theorem 8.5.1 (Hunt–Stein.) Let B be a σ-field of subsets of G such that for any A ∈ A the set of pairs (x, g) with gx ∈ A is in A × B and for any B ∈ B and g ∈ G the set Bg is in B. Suppose that there exists a sequence of probability distributions νn over (G, B) which is asymptotically right-invariant in the sense that for any g ∈ G, B ∈ B, lim |νn (Bg) − νn (B)| = 0. n→∞ (8.27) 332 8. The Minimax Principle Then given any critical function ϕ, there exists a critical function ψ which is almost invariant and satisfies (8.23). Proof. Let  ψn (x) = ϕ(gx) dνn (g), which as before is measurable and between 0 and 1. By the weak compactness theorem (Theorem A.5.1 of the Appendix) there exists a subsequence {ψni } and a measurable function ψ between 0 and 1 satisfying   ψni p dµ = ψp dµ lim i−∞ for all µ-integrable functions p, so that in particular lim Eθ ψni (X) = Eθ ψ(X) i→∞ for all θ ∈ Ω. By Fubini’s theorem,   Eθ ψni (X) = [Eθ ϕ(gX)] dνni (g) = Eḡθ ϕ(X) dνni (g) , so that inf Eḡθ ϕ(X) ≤ Eθ ψni (X) ≤ sup Eḡθ ϕ(X), Ḡ Ḡ and ψ satisfies (8.23). In order to prove that ψ is almost invariant we shall show below that for all x and g, ψni (gx) − ψni (x) → 0. (8.28) Let IA (x) denote the indicator function of a set A ∈ A. Using the fact that IgA (gx) = IA (x), we see that (8.28) implies   ψni (x)IA (x) dPθ (x) ψ(x) dPθ (x) = lim i→∞ A  = lim ψni (gX)IgA (gx) dPθ (x) i→∞   = ψ(x)IgA (x) dPḡθ (x) = ψ(gx) dPθ (x) , A and hence ψ(gx) = ψ(x) (a.e. P), as was to be proved. To prove (8.28), consider any fixed x and any integer m, and let G be partitioned into the mutually exclusive sets   1 Bk = h ∈ G : ak < ϕ(hx) ≤ ak + , k = 0, . . . , m, m where ak = (k − 1)/m. In particular, B0 is the set {h ∈ G : ϕ(hx) = 0}. It is seen from the definition of the sets Bk that  m m  m    1 ak νni (Bk ) ≤ ϕ(hx) dνni (h) ≤ ak + νni (Bk ) m Bk k=0 k=0 ≤ k=0 m  k=0 ak νni (Bk ) + 1 , m 8.5. The Hunt–Stein Theorem 333 and analogously that % %m  m % %  1 % −1 % ϕ(hgx) dνni (h) − ak νni (Bk g )% ≤ , % % % m −1 Bk g k=0 k=0 from which it follows that ψni (gx) − ψni (x) :≤  |ak | · |νni (Bk g −1 ) − νni (Bk )| + 2 . m By (8.27) the first term of the right-hand side tends to zero as i tends to infinity, and this completes the proof. When there exist a right-invariant measure ν over G and a sequence of subsets Gn of G with Gn ⊆ Gn+1 , ∪Gn = G, and ν(Gn ) = cn < ∞, it is suggestive to take for the probability measures νn of Theorem 8.5.1 the measures ν/cn truncated on Gn . This leads to the desired result in the example below. On the other hand, there are cases in which there exists such a sequence of subsets of Gn but no invariant test satisfying (8.23) and hence no sequence νn satisfying (8.27). Example 8.5.1 Let x = (x1 , . . . , xn ), A be the class of Borel sets in n-space, and G the group of translations (x1 + g, . . . , xn + g), −∞ < g < ∞. The elements of G can be represented by the real numbers, and the group product gg  is then the sum g + g  . If B is the class of Borel sets on the real line, the measurability assumptions of Theorem 8.5.1 are satisfied. Let ν be Lebesgue measure, which is clearly invariant under G, and define νn to be the uniform distribution on the interval I(−n, n) = {g : −n ≤ g ≤ n}. Then for all B ∈ B, g ∈ G, |νn (B) − νn (Bg)| = |g| 1 , |ν[B ∩ I(−n, n)] − ν[B ∩ I(−n − g, n − g)]| ≤ 2n 2n so that (8.27) is satisfied. This argument also covers the group of scale transformations (ax1 , . . . , axn ), 0 < a < ∞, which can be transformed into the translation group by taking logarithms. When applying the Hunt–Stein theorem to obtain invariant minimax tests, it is frequently convenient to carry out the calculation in steps, as was done in Theorem 6.6.1. Suppose that the problem remains invariant under two groups D and E, and denote by y = s(x) a maximal invariant with respect to D and by E ∗ the group defined in Theorem 6.2.2, which E induces in y-space. If D and E ∗ satisfy the conditions of the Hunt–Stein theorem, it follows first that there exists a maximin test depending only on y = s(x), and then that there exists a maximin test depending only on a maximal invariant z = t(y) under E ∗ . Example 8.5.2 Consider a univariate linear hypothesis in the canonical form in which Y1 , . . . , Yn are independently distributed as N (ηi , σ 2 ), where it is given that ηs+1 = · · · = ηn = 0, and where the hypothesis to be tested is η1 = · · · = ηr = 0. It was shown in Section 7.1 that this problem remains invariant under certain groups of transformations and that with respect to these groups there exists a UMP invariant test. The groups involved are the group of orthogonal transformations, translation groups of the kind considered in Example 8.5.1, and 334 8. The Minimax Principle a group of scale changes. Since each of these satisfies the assumptions of the Hunt–Stein theorem, and since they leave invariant the problem of maximizing the minimum power over the set of alternatives r  ηi2 ≥ ψ12 σ2 i=1 (ψ1 > 0), (8.29) it follows that the UMP invariant test of Chapter 7 is also the solution of this maximin problem. It is also seen slightly more generally that the test which is UMP invariant under the same groups for testing r  ηi2 ≤ ψ02 2 σ i=1 (Problem 7.4) maximizes the minimum power over the alternatives (8.29) for ψ0 < ψ1 . Example 8.5.3 (Stein) Let G be the group of all nonsingular linear transformations of p-space. That for p > 1 this does not satisfy the conditions of Theorem 8.5.1 is shown by the following problem, which is invariant under G but for which the UMP invariant test does not maximize the minimum power. Generalizing Example 6.2.1, let X = (X1 , . . . , Xp ), Y = (Y1 , . . . , Yp ) be independently distributed according to p-variate normal distributions with zero means and nonsingular covariance matrices E(Xi Xj ) = σij and E(Yi Yj ) = ∆σij , and let H : ∆ ≤ ∆0 be tested against ∆ ≥ ∆1 (∆0 < ∆1 ), the σij being unknown. This problem remains invariant if the two vectors are subjected to any common nonsingular transformation, and since with probability 1 this group is transitive over the sample space, the UMP invariant test is trivially ϕ(x, y) ≡ α. The maximin power against the alternatives ∆ ≥ ∆1 that can be achieved by invariant tests is therefore α. On the other hand, the test with rejection region Y12 /X12 > C has a strictly increasing power function β(∆), whose minimum over the set of alternatives ∆ ≥ ∆1 is β(∆1 ) > β(∆0 ) = α. It is a remarkable feature of Theorem 8.5.1 that its assumptions concern only the group G and not the distributions Pθ .3 When these assumptions hold for a certain G it follows from (8.23) as in the proof of Lemma 8.4.1 that for any testing problem which remains invariant under G and possesses a UMP invariant test, this test maximizes the minimum power over any invariant class of alternatives. Suppose conversely that a UMP invariant test under G has been shown in a particular problem not to maximize the minimum power, as was the case for the group of linear transformations in Example 8.5.3. Then the assumptions of Theorem 8.5.1 cannot be satisfied. However, this does not rule out the possibility that for another problem remaining invariant under G, the UMP invariant test may maximize the minimum power. Whether or not it does is no longer a property of the group alone but will in general depend also on the particular distributions. 3 These assumptions are essentially equivalent to the condition that the group G is amenable. Amenability and its relationship to the Hunt–Stein theorem are discussed by Bondar and Milnes (1982) and (with a different terminology) by Stone and von Randow (1968). 8.5. The Hunt–Stein Theorem 335 Consider in particular the problem of testing H : ξ1 = · · · = ξp = 0 on the basis of a sample (Xα1 , . . . , Xαp ), α = 1, . . . , n, from a p-variate normal distribution with mean E(Xαi ) = ξi and common covariance matrix (σij ) = (aij )−1 . This problem remains invariant under a number of groups, including that of all nonsingular linear transformations of p-space, and a UMP invariant test exists. An invariant class of alternatives under these groups is   aij ξi ξj ≥ ψ12 . (8.30) σ2 Here, Theorem 8.5.1 is not applicable, and the question of whether the T 2 -test of H : ψ = 0 maximizes the minimum power over the alternatives  aij ξi ξj = ψ12 (8.31) [and hence a fortiori over the alternatives (8.30)] presents formidable difficulties. The minimax property was proved for the case p = 2, n = 3 by Giri, Kiefer, and Stein (1963), for the case p = 2, n = 4 by Linnik, Pliss, and Salaevskii (1968), and for p = 2 and all n ≥ 3 by Salaevskii (1971). The proof is effected by first reducing the problem through invariance under the group G1 of Example 6.6.11, to which Theorem 8.5.1 is applicable, and then applying Theorem 8.1.1 to the reduced problem. It is a consequence of this approach that it also establishes the admissibility of T 2 as a test of H against the alternatives (8.31). In view of the inadmissibility results for point estimation when p ≥ 3 (see TPE2, Sections 5.4-5.5, it seems unlikely that T 2 is admissible for p ≥ 3, and hence that the same method can be used to prove the minimax property in this situation. The problem becomes much easier when the minimax property is considered against local or distant alternatives rather than against (8.31). Precise definitions and proofs of the fact that T 2 possesses these properties for all p and n are provided by Giri and Kiefer (1964) and in the references given in Section 7.9. The theory of this and the preceding section can be extended to confidence sets if the accuracy of a confidence set at level 1 − α is assessed by its volume or some other appropriate measure of its size. Suppose that the distribution of X depends on the parameters θ to be estimated and on nuisance parameters ϑ, and that µ is a σ-finite measure over the parameter set ω = {θ : (θ, ϑ) ∈ Ω}, with ω assumed to be independent of ϑ. Then the confidence sets S(X) for θ are minimax with respect to µ at level 1 − α if they minimize sup Eθ,ϑ µ[S(X)] among all confidence sets at the given level. The problem of minimizing Eµ[S(X)] is related to that of minimizing the probability of covering false values (the criterion for accuracy used so far) by the relation (Problem 8.34)  Eθ0 ,ϑ µ[S(X)] = Pθ0 ,ϑ [θ ∈ S(X)] dµ(θ), (8.32) θ=θ0 which holds provided µ assigns measure zero to the set {θ = θ0 }. (For the special case that θ is real-valued and µ Lebesgue measure, see Problem 5.26.) Suppose now that the problem of estimating θ is invariant under a group G in the sense of Section 6.11 and that it satisfies the invariance condition µ[S(gx)] = µ[S(x)]. (8.33) 336 8. The Minimax Principle If uniformly most accurate equivariant confidence sets exist, they minimize (8.32) among all equivariant confidence sets at the given level, and one may hope that under the assumptions of the Hunt–Stein theorem, they will also be minimax with respect to µ among the class of all (not necessarily equivariant) confidence sets at the given level. Such a result does hold and can be used to show for example that the most accurate equivariant confidence sets of Examples 6.11.2 and 6.11.3 minimize their maximum expected Lebesgue measure. A more general class of examples is provided by the confidence intervals derived from the UMP invariant tests of univariate linear hypotheses such as the confidence spheres for θi = µ + αi or for αi given in Section 7.4. Minimax confidence sets S(x) are not necessarily admissible; that is, there may exist sets S  (x) having the same confidence level but such that Eθ,ϑ µ[S  (X)] ≤ Eθ,ϑ µ[S(X)] for all θ, ϑ with strict inequality holding for at least some (θ, ϑ). Example 8.5.4 Let Xi (i = 1, . . . , s) be independently normally distributed with mean E(Xi ) = θi and variance 1, and let G be the group generated by translations Xi +ci (i = 1, . . . , s) and orthogonal transformations of (X1 , . . . , Xs ). (G is the Euclidean group of rigid motions in s-space.) In Example 6.12.2, it was argued that the confidence sets  C0 = {(θ1 , . . . , θs ) : (θi − Xi )2 ≤ c} (8.34) are uniformly most accurate equivariant. The volume µ[S(X)] of any confidence set S(X) remains invariant under the transformations g ∈ G, and it follows from the results of Problems 8.26 and 8.4 and Examples 8.5.1 and 8.5.2 that the confidence sets (8.34) minimize the maximum expected volume. However, very surprisingly, they are not admissible unless s = 1 or 2. In the case s ≥ 3, Stein (1962) suggested the region (8.34) can be improved by recentered regions of the form C1 = {(θ1 , . . . , θs ) : (θi − b̂Xi )2 ≤ c} , (8.35)  2 where b̂ = max(0, 1 − (s − 2)/ i Xi ). In fact, Brown (1966) proved that, for s ≥ 3, Pθ {θ ∈ C1 } > Pθ {θ ∈ C0 } for all θ. This result, which will not be proved here, is closely related to the inadmissibility of X1 , . . . , Xs as a point estimator of (θ1 , . . . , θs ) for a wide variety of loss functions. The work on point estimation, which is discussed in TPE2, Sections 5.4-5.6, for squared error loss, provides easier access to these ideas than the present setting. Further entries into the literature on admissibility are Stein (1981), Hwang and Casella (1982), and Tseng and Brown (1997); additional references are provided in TPE2, p.423. The inadmissibility of the confidence sets (8.34) is particularly surprising in that the associated UMP invariant tests of the hypotheses H : θi = θi0 (i = 1, . . . , s) are admissible (Problems 8.24, 8.25). 8.6. Most Stringent Tests 337 8.6 Most Stringent Tests One of the practical difficulties in the consideration of tests that maximize the minimum power over a class ΩK of alternatives is the determination of an appropriate ΩK . If no information is available on which to base the choice of this set, and if a natural definition is not imposed by invariance arguments, a frequently reasonable definition can be given in terms of the power that can be achieved against the various alternatives. The envelope power function βα∗ was defined in Problem 6.25 by βα∗ (θ) = sup βϕ (θ), where βϕ denotes the power of a test ϕ and where the supremum is taken over all level-α tests of H. Thus βα∗ (θ) is the maximum power that can be attained at level α against the alternative θ. (That it can be attained follows under mild restrictions from Theorem A.5.1 of the Appendix.) If ∗ S∆ = {θ : βα∗ (θ) = ∆}, ∗ ∗ then of two alternatives θ1 ∈ S∆ , θ2 ∈ S∆ , θ1 can be considered closer to H, 1 2 equidistant, or further away than θ2 as ∆1 is <, =, or > ∆2 . The idea of measuring the distance of an alternative from H in terms of the available information has been encountered before. If for example X1 , . . . , Xn is a sample from N (ξ, σ 2 ), the problem of testing H : ξ ≤ 0 was discussed (Section 5.2) both when the alternatives ξ are measured in absolute units and when they are measured in σ-units. The latter possibility corresponds to the present proposal, since it follows from invariance considerations (Problem 6.25) that βα∗ (ξ, σ) is constant on the lines ξ/σ = constant. Fixing a value of ∆ and taking as ΩK the class of alternatives θ for which βα∗ (θ) ≥ ∆, one can determine the test that maximizes the minimum power over ΩK . Another possibility, which eliminates the need of selecting a value of ∆, is to consider for any test ϕ the difference βα∗ (θ) − βϕ (θ). This difference measures the amount by which the actual power βϕ (θ) falls short of the maximum power attainable. A test that minimizes sup [βα∗ (θ) − βϕ (θ)] (8.36) Ω−ω is said to be most stringent. Thus a test is most stringent if it minimizes its maximum shortcoming. ∗ Let ϕ∆ be a test that maximizes the minimum power over S∆ , and hence ∗ ∗ minimizes the maximum difference between βα (θ) and βϕ (θ) over S∆ . If ϕ∆ happens to be independent of ∆, it is most stringent. This remark makes it possible to apply the results of the preceding sections to the determination of most stringent tests. Suppose that the problem of testing H : θ ∈ ω against the alternatives θ ∈ Ω − ω remains invariant under a group G, that there exists a UMP almost invariant test ϕ0 with respect to G, and that the assumptions ∗ of Theorem 8.5.1 hold. Since βα∗ (θ) and hence the set S∆ is invariant under Ḡ ∗ (Problem 6.25), it follows that ϕ0 maximizes the minimum power over S∆ for each ∆, and ϕ0 is therefore most stringent. As an example of this method consider the problem of testing H : p1 , . . . , pn ≤ 1 against the alternative K : pi > 12 for all i, where pi is the probability of success 2 338 8. The Minimax Principle in the ith trial of a sequence of n independent trials. If Xi is 1 or 0 as the ith trial is a success or failure, then the problem remains invariant under permutations of  the X’s, and the UMP invariant test rejects (Example 6.3.5) when Xi > C. It now follows from the remarks above that this test is also most stringent. Another illustration is furnished by the general univariate linear hypothesis. Here it follows from the discussion inExample 8.5.2 that the standard test for testing H : η1 = · · · = ηr = 0 or H  : ri=1 ηi2 /σ 2 ≤ ψ02 is most stringent. When the invariance approach is not applicable, the explicit determination of most stringent tests typically is difficult. The following is a class of problems for which they are easily obtained by a direct approach. Let the distributions of X constitute a one-parameter exponential family, the density of which is given by (3.19), and consider the hypothesis H : θ = θ0 . Then according as θ > θ0 or θ < θ0 , the envelope power βα∗ (θ) is the power of the UMP one-sided test for testing H against θ > θ0 or θ < θ0 . Suppose that there exists a two-sided test ϕ0 given by (4.3), such that sup [βα∗ (θ) − βϕ0 (θ)] = sup [βα∗ (θ) − βϕ0 (θ)], θ<θ0 (8.37) θ>θ0 and that the supremum is attained on both sides, say at points θ1 < θ0 < θ2 . If βϕ0 (θi ) = βi , i = 1, 2, an application of the fundamental lemma [Theorem 3.6.1(iii)] to the three points θ1 , θ2 , θ0 shows that among all tests ϕ with βϕ (θ1 ) ≥ β1 and βϕ (θ2 ) ≥ β2 , only ϕ0 satisfies βϕ (θ0 ) ≤ α. For any other level-α test, therefore, either βϕ (θ1 ) < β1 or βϕ (θ2 ) < β2 , and it follows that ϕ0 is the unique most stringent test. The existence of a test satisfying (8.37) can be proved by a continuity consideration [with respect to variation of the constants Ci and γi which define the boundary of the test (4.3)] from the fact that for the UMP one-sided test against the alternatives θ > θ0 the right-hand side of (8.37) is zero and the left-hand side positive, while the situation is reversed for the other one-sided test. 8.7 Problems Section 8.1 Problem 8.1 Existence of maximin tests.4 Let (X , A) be a Euclidean sample space, and let the distributions Pθ , θ ∈ Ω, be dominated by a σ-finite measure over (X , A). For any mutually exclusive subsets ΩH , ΩK of Ω there exists a level-α test maximizing (8.2). [Let β = sup[inf Ωk Eθ ϕ(X)], where the supremum is taken over all level-α tests of H : θ ∈ ΩH . Let ϕn be a sequence of level-α tests such that inf ΩK Eθ ϕn (X) tends to β. If ϕni is a subsequence and ϕ a test (guaranteed by Theorem 8.5.1 of the Appendix) such that Eθ ϕni (X) tends to Eθ ϕ(X) for all θ ∈ Ω, then ϕ is a level-α test and inf Ωk Eθ ϕ(X) = β.] 4 The existence of maximin tests is established in considerable generality in Cvitanic and Karatzas (2001). 8.7. Problems 339 Problem 8.2 Locally most powerful tests. 5 Let d be a measure of the distance of an alternative θ from a given hypothesis H. A level-α test ϕ0 is said to be locally most powerful (LMP) if, given any other level-α test ϕ, there exists ∆ such that βϕ0 (θ) ≥ βϕ (θ) for all θ with 0 < d(θ) < ∆. (8.38) Suppose that θ is real-valued and that the power function of every test is continuously differentiable at θ0 . (i) If there exists a unique level-α test ϕ0 of H : θ = θ0 , maximizing βϕ (θ0 ), then ϕ0 is the unique LMP level-α test of H against θ > θ0 for d(θ) = θ−θ0 . (ii) To see that (i) is not correct without the uniqueness assumption, let X take on the values 0 and 1 with probabilities Pθ (0) = 12 − θ3 , Pθ (1) = 12 + θ3 , − 12 < θ3 < 12 , and consider testing H : θ = 0 against K : θ > 0. Then every test ϕ of size α maximizes βϕ (0), but not every such test is LMP. [Kallenberg et al. (1984).] (iii) The following6 is another counterexample to (i) without uniqueness, in which in fact no LMP test exists. Let X take on the values 0, 1, 2 with probabilities #  x $ Pθ (x) = α +  θ + θ2 sin for x = 1, 2, θ Pθ (0) = 1 − pθ (1) − pθ (2), where −1 ≤ θ ≤ 1 and  is a sufficiently small number. Then a test ϕ at level α maximizes β  (0) provided ϕ(1) + ϕ(2) = 1 , but no LMP test exists. (iv) A unique LMP test maximizes the minimum power locally provided its power function is bounded away from α for every set of alternatives which is bounded away from H. (v) Let X1 , . . . , Xn be a sample from a Cauchy distribution with unknown location parameter θ, so that the joint density of the X’s is π −n n i=1 [1 + (xi − θ)2 ]−1 . The LMP test for testing θ = 0 against θ > 0 at level α < 12 is not unbiased and hence does not maximize the minimum power locally. [(iii): The unique most powerful test against θ is    1 > 2 ϕ(1) = = 1 if sin sin , ϕ(2) θ < θ and each of these inequalities holds at values of θ arbitrarily close to 0. (v): There exists M so large that any point with xi ≥ M for all i = 1, . . . , n lies in the acceptance region of the LMP test. Hence the power of the test tends to zero as θ tends to infinity.] 5 Locally optimal tests for multiparameter hypotheses are given in Gupta and Vermeire (1986). 6 Due to John Pratt. 340 8. The Minimax Principle Problem 8.3 A level-α test ϕ0 is locally unbiased (loc. unb.) if there exists ∆0 > 0 such that βϕ0 (θ) ≥ α for all θ with 0 < d(θ) < ∆0 ; it is LMP loc. unb. if it is loc. unb. and if, given any other loc. unb. level-α test ϕ, there exists ∆ such that (8.38) holds. Suppose that θ is real-valued and that d(θ) = |θ − θ0 |, and that the power function of every test is twice continuously differentiable at θ = θ0 . (i) If there exists a unique test ϕ0 of H : θ = θ0 against K : θ = θ0 which among all loc. unb. tests maximizes β  (θ0 ), then ϕ0 is the unique LMP loc. unb. level-α test of H against K. (ii) The test of part (i) maximizes the minimum power locally provided its power function is bounded away from α for every set of alternatives that is bounded away from H. [(ii): A necessary condition for a test to be locally minimax is that it is loc. unb.] Problem 8.4 Locally uniformly most powerful tests. If the sample space is finite and independent of θ, the test ϕ0 of Problem 8.2(i) is not only LMP but also locally uniformly most powerful (LUMP) in the sense that there exists a value ∆ > 0 such that ϕ0 maximizes βϕ (θ) for all θ with 0 < θ − θ0 < ∆. [See the argument following (6.21) of Section 6.9.] Problem 8.5 The following two examples show that the assumption of a finite sample space is needed in Problem 8.4. (i) Let X1 , . . . , Xn be i.i.d. according to a normal distribution N (σ, σ 2 ) and test H : σ = σ0 against K : σ > σ0 . (ii) Let X and Y be independent Poisson variables with E(X) = λ and E(Y ) = λ + 1, and test H : λ = λ0 against K : λ > λ0 . In each case, determine the LMP test and show that it is not LUMP. [Compare the LMP test with the most powerful test against a simple alternative.] Section 8.2 Problem 8.6 Let the distribution of X depend on the parameters (θ, ϑ) = (θ1 , . . . , θr , ϑ1 , . . . , ϑs ). A test of H : θ = θ0 is locally strictly unbiased if for each ϕ, (a) βϕ (θ0 , ϕ) = α, (b) there exists a θ-neighborhood of θ0 in which βϕ (θ, ϑ) > α for θ = θ0 . (i) Suppose that the first and second derivatives % % % % ∂ ∂2 βϕi (ϑ) = βϕ (θ, ϑ)%% and βϕij (ϑ) = βϕ (θ, ϑ)%% ∂θi ∂θ ∂θ i j θ0 θ0 exist for all critical functions ϕ and all ϑ. Then a necessary and sufficient condition for ϕ to be locally strictly unbiased is that βϕ = 0 for all i and ϑ, and that the matrix (βϕij (ϑ)) is positive definite for all ϑ. (ii) A test of H is said to be of type E (type D is s = 0 so that there are no nuisance parameters) if it is locally strictly unbiased and among all tests 8.7. Problems 341 with this property maximizes the determinant |(βϕij )|.7 (This determinant under the stated conditions turns out to be equal to the Gaussian curvature of the power surface at θ0 .) Then the test ϕ0 given by (7.7) for testing the general linear univariate hypothesis (7.3) is of type E. [(ii): With θ = (η1 , . . . , ηr ) and ϑ = (ηr+1 , . . . , ns , σ), the test ϕ0 , by Problem 7.5, has the property of maximizing the surface integral  [βϕ (η, σ 2 ) − α] dA S among all similar (and hence all locally unbiased) tests where S = {(η1 , . . . , ηr ) :  r 2 2 2 i=1 ηi = ρ σ }. Letting ρ tend to zero and utilizing the conditions   βϕi (ϑ) = 0, ηi ηj dA = 0 for i = j, ηi2 dA = k(ρσ), S r S ii 2 i=1 βϕ (η, σ ) among all ij matrix, |(βϕ )| ≤ βϕii , one finds that ϕ0 maximizes locally unbiased tests. Since for any positive definite it follows that for any locally strictly unbiased test ϕ,  ii  Σβϕii r  Σβϕii r 0 βϕ ≤ ≤ = [βϕ110 ]r = |(βϕij0 )|.] |(βϕij )| ≤ r r Problem 8.7 Let Z1 , . . . , Zn be identically independently distributed according to a continuous distribution D, of which it is assumed only that it is symmetric about some (unknown) point. For testing the hypothesis H : D(0) = 12 , the sign test maximizes the minimum power against the alternatives K : D(0) ≤ q(q < 12 ). [A pair of least favorable distributions assign probability 1 respectively to the distributions F ∈ H, G ∈ K with densities [|x|] |[x]| 1 − 2q q q f (x) = , g(x) = (1 − 2q) 2(1 − q) 1 − q 1−q where for all x (positive, negative, or zero) [x] denotes the largest integer ≤ x.] Problem 8.8 Let fθ (x) = θg(x) + (1 − θ)h(x) with 0 ≤ θ ≤ 1. Then fθ (x) satisfies the assumptions of Lemma 8.2.1 provided g(x)/h(x) is a nondecreasing function of x. Problem 8.9 Let x = (x1 , . . . , xn ), and let gθ (x, ξ) be a family of probability densities depending on θ = (θ1 , . . . , θr ) and the real parameter ξ, and jointly measurable in x and ξ. For each θ, let hθ (ξ) be a probability density with respect to a σ-finite measure ν such that pθ (x) = gθ (x, ξ)hθ (ξ) dν(ξ) exists. We shall say that a function f of two arguments u = (u1 , . . . , ur ), v = (v1 , . . . , vs ) is nondecreasing in (u, v) if f (u , v)/f (u, v) ≤ f (u , v  )/f (u, v  ) for all (u, v) satisfying ui ≤ ui , vj ≤ vj (i = 1, . . . , r; j = 1, . . . , s). Then pθ (x) is nondecreasing in (x, θ) provided the product gθ (x, ξ)hθ (ξ) is (a) nondecreasing in (x, θ) for each fixed ξ; 7 An interesting example of a type-D test is provided by Cohen and Sackrowitz (1975), who show that the χ 2 -test of Chapter 14.3 has this property. Type D and E tests were introduced by Isaacson (1951). 342 8. The Minimax Principle (b) nondecreasing in (θ, ξ) for each fixed x; (c) nondecreasing in (x, ξ) for each fixed θ. [Interpreting gθ (x, ξ) as the conditional density of x given ξ, and hθ (ξ) as the a priori density of ξ, let ρ(ξ) denote the a posteriori density of ξ given x, and let ρ (ξ) be defined analogously with θ in place of θ. That pθ (x) is nondecreasing in its two arguments is equivalent to   gθ (x , ξ) gθ (x , ξ)  ρ(ξ) dν(ξ) ≤ ρ (ξ) dν(ξ). gθ (x, ξ) gθ (x, ξ) By (a) it is enough to prove that  gθ (x , ξ)  [ρ (ξ) − ρ(ξ)] dν(ξ) ≥ 0. D= gθ (x, ξ) Let S− = {ξ : ρ (ξ)/ρ(ξ) < 1} and S+ = {ξ : ρ(ξ)/ρ(ξ) ≥ 1}. By (b) the set S− lies entirely to the left of S+ . It follows from (c) that there exists a ≤ b such that   D=a [ρ (ξ) − ρ(ξ)] dν(ξ) + b [ρ (ξ) − ρ(ξ)] dν(ξ), S− and hence that D = (b − a) S+  S+ [ρ (ξ) − ρ(ξ)] dν(ξ) ≥ 0.] Problem 8.10 (i) Let X have binomial distribution b(p, n), and consider testing H : p = p0 at level α against the alternatives ΩK : p/q ≤ 12 p0 /q0 or ≥ 2p0 /q0 . For α = .05 determine the smallest sample size for which there exists a test with power ≥ .8 against ΩK if p0 = .1, .2, .3, .4, .5. (ii) Let X1 , . . . , Xn be independently distributed as N (ξ, σ 2 ). For testing σ = 1 at level α = .05, determine the smallest sample size for which there exists a test with power ≥ .9 against the alternatives σ 2 ≤ 12 and σ 2 ≥ 2. [See Problem 4.5.] Problem 8.11 Double-exponential distribution. Let X1 , . . . , Xn be a sample from the double-exponential distribution with density 12 e−|x−θ| . The LMP test for testing θ ≤ 0 against θ > 0 is the sign test, provided the level is of the form   m 1  n α= n , 2 k k=0 so that the level-α sign test is nonrandomized. [Let Rk (k = 0, . . . , n) be the subset of the sample space in which k of the X’s are positive and n − k are negative. Let 0 ≤ k < l < n, and let Sk , Sl be subsets of Rk , Rl such that P0 (Sk ) = P0 (Sl ) = 0. Then it follows from a consideration of Pθ (Sk ) and P0 (Sl ) for small θ that there exists ∆ such that Pθ (Sk ) < Pθ (Sl ) for 0 < θ < ∆. Suppose now that the rejection region of a nonrandomized test of θ = 0 against θ > 0 does not consist of the upper tail of a sign test. Then it can be converted into a sign test of the same size by a finite number of steps, each of which consists in replacing an Sk by an Sl with k < l, and each of which therefore increases the power for θ sufficiently small.] 8.7. Problems 343 Section 8.3 Problem 8.12 If (8.13) holds, show that q1 defined by (8.11) belongs to P1 . Problem 8.13 Show that there exists a unique constant b for which q0 defined by (8.11) is a probability density with respect to µ, that the resulting q0 belongs to P0 , and that b → ∞ as 0 → 0. Problem 8.14 Prove the formula (8.15). Problem 8.15 Show that if P0 = P1 and 0 , 1 are sufficiently small, then Q0 = Q1 . Problem 8.16 Evaluate the test (8.21) explicitly for the case that Pi is the normal distribution with mean ξi and known variance σ 2 , and when 0 = 1 . Problem 8.17 Determine whether (8.21) remains the maximin test if in the model (8.20) Gi is replaced by Gij . Problem 8.18 Write out a formal proof of the maximin property outlined in the last paragraph of Section 8.3. Section 8.4 Problem 8.19 Let X1 , . . . , Xn be independently normally distributed with means E(Xi ) = µi and variance 1. The test of H : µ1 = · · ·  = µn = 0 that maximizes the minimum power over ω  : µi ≥ d rejects when Xi ≥ C. [If the least favorable distribution assigns probability 1 to a single point, invariance under permutations suggests that this point will be µ1 = · · · = µn = d/n]. In the preceding problem determine the maximin test if Problem 8.20 8 (i)  ω  is replaced by ai µi ≥ d, where the a’s are given positive constants. (ii) Solve part (i) with V ar(Xi ) = 1 replaced by V ar(Xi ) = σi2 (known). [(i): Determine the point (µ∗1 , . . . , µ∗n ) in ω  for which the MP test of H against K : (µ∗1 , . . . , µ∗n ) has the smallest power, and show that the MP test of H against K is a maximin solution.] Problem 8.21 Let X1 , . . . , Xn be independent normal variables with variance 1 and means ξ1 , . . . , ξn , and consider the problem of testing H : ξ1 = · · · = ξn = 0 against the alternatives K = {K1 , . . . , Kn }, where Ki : ξj = 0 for j = i, ξi = ξ (known and positive). Show that the problem remains invariant under permutation  of the X’s and that there exists a UMP invariant test φ0 which rejects when e−ξxj > C, by the following two methods. (i) The order statistics X(1) < · · · < X(n) constitute a maximal invariant. 8 Due to Fritz Scholz. 344 8. The Minimax Principle (ii) Let f0 and fi denote the densities underH and Ki respectively. Then the level-α test φ0 of H vs. K  : f = (1/n) fi is UMP invariant for testing H vs. K. [(ii): If φ0 is not UMP invariant for H vs. K, there exists an invariant test φ1 whose (constant) power against K exceeds that of φ0 . Then φ1 is also more powerful against K  .] Problem 8.22 The UMP invariant test φ0 of Problem 8.21 (i) maximizes the minimum power over K; (ii) is admissible. (iii) For testing the hypothesis H of Problem 8.21 against the alternatives K  = {K1 , . . . , Kn , K1 , . . . , Kn }, where under Ki : ξj = 0 for all j = i, ξi = −ξ, determine the UMP test under a suitable group G , and show that it is both maximin and invariant. [ii): Suppose φ is uniformly at least as powerful as φ0 , and more powerful for at least one Ki , and let   φ (xi1 , . . . , xin ) φ∗ (x1 , . . . , xn ) = , n! where the summation extends over all permutations. Then φ∗ is invariant, and its power is independent of i and exceeds that of φ0 .] Problem 8.23 For testing H : f0 against K : {f1 , . . . , fs }, suppose there exists a finite group G = {g1 , . . . , gN } which leaves H and K invariant and which is transitive in the sense that given fj , fj  (1 ≤ j, j  ) there exists g ∈ G such that ḡfj = fj  . In generalization of Problems 8.21, 8.22, determine a UMP invariant test, and show that it is both maximin against K and admissible. Problem 8.24 To generalize the results of the preceding problem to the testing of H : f vs. K : {fθ , θ ∈ ω}, assume: (i) There exists a group G that leaves H and K invariant. (ii) Ḡ is transitive over ω. (iii) There exists a probability distribution Q over G which is right-invariant in the sense of Section 8.4. Determine a UMP invariant test, and show that it is both maximin against K and admissible. Problem 8.25 Let X1 , . . . , Xn be independent normal with means θ1 , . . . , θn and variance 1. (i) Apply the results of the preceding problem to the testing of H : θ1 = · · · = θn = 0 against K : θi2 = r2 , for any fixed r > 0. (ii) Showthat the results  of2 (i) 2remain valid if H and K are replaced by H : θi2 ≤ r02 , K  : θi ≥ r1 (r0 < r1 ). 8.7. Problems 345 Problem 8.26 Suppose in Problem 8.25(i) the variance σ 2 is unknown and that the data consist of X1 , . . . , Xn together with an independent variable S 2 random 2 2 2 2 2 for which S /σ has a χ -distribution. If K is replaced by θi /σ = r2 , then  (i) the confidence sets (θi − Xi )2 /S 2 ≤ C are uniformly most accurate equivariant under the group generated by the n-dimensional generalization of the group G0 of Example 6.11.2, and the scale changes Xi = cXi , S 2 = c2 S 2 . (ii) The confidence sets of (i) are minimax with respect to the measure µ given by 1 µ[C(X, S 2 )] = 2 [ volume of C(X, S 2 )]. σ  2 θi .] [Use polar coordinates with θ2 = Section 8.5 Problem 8.27 Let X = (X1 , . . . , Xp ) and Y = (Y1 , . . . , Yp ) be independently distributed according to p-variate normal distributions with zero means and covariance matrices E(Xi Xj ) = σij and E(Yi Yj ) = ∆σij . (i) The problem of testing H : ∆ ≤ ∆0 remains invariant under the group G of transformations X ∗ = XA, Y ∗ = Y A, where A = (aij ) is any nonsingular p × p matrix with aij = 0 for i > j, and there exists a UMP invariant test under G with rejection region Y12 /X12 > C. (ii) The test with rejection region Y12 /X12 > C maximizes the minimum power for testing ∆ ≤ ∆0 against ∆ ≥ ∆1 (∆0 < ∆1 ). [(ii): That the Hunt–Stein theorem is applicable to G can be proved in steps by considering the group Gq of transformations Xq = α1 X1 + · · · + αq Xq , Xi = Xi for i = 1, . . . , q − 1, q + 1, . . . , p, successively for q = 1, . . . , p − 1. Here αq = 0, since the matrix A is nonsingular if and only if aii = 0 for all i. The group product (γ1 , . . . , γq ) of two such transformations (α1 , . . . , αq ) and (β1 , . . . , βq ) is given by γ1 = αq + β1 , γ2 = a2 βq + β2 , . . . , γq−1 = αq−1 βq + βq−1 , γq = αq , βq , which shows Gq to be isomorphic to a group of scale changes (multiplication of all components by βq ) and translations [addition of (β1 , . . . , βq−1 , 0)]. The result now follows from the Hunt–Stein theorem and Example 8.5.1, since the assumptions of the Hunt– Stein theorem, except for the easily verifiable measurability conditions, concern only the abstract structure (G, B), and not the specific realization of the elements of G as transformations of some space.] Problem 8.28 Suppose that the problem of testing θ ∈ ΩH against θ ∈ ΩK remains invariant under G, that there exists a UMP almost invariant test ϕ0 with respect to G, and that the assumptions of Theorem 8.5.1 hold. Then ϕ0 maximizes inf ΩK [w(θ)Eθ ϕ(X) + u(θ)] for any weight functions w(θ) ≥ 0, u(θ) that are invariant under Ḡ. Problem 8.29 Suppose X has the multivariate normal distribution in Rk with unknown mean vector h and known positive definite covariance matrix C −1 . 346 8. The Minimax Principle Consider testing h = 0 versus |C 1/2 h| ≥ b for some b > 0, where | · | denotes the Euclidean norm. (i) Show the test that rejects when |C 1/2 X|2 > ck,1−α is maximin, where ck,1−α denotes the 1 − α quantile of the Chi-squared distribution with k degrees of freedom. (ii) Show that the maximin power of the above test is given P {χ2k (b2 ) > ck,1−α }, where χ2k (b2 ) denotes a random variable that has the noncentral Chi-squared distribution with k degrees of freedom and noncentrality parameter b2 . Problem 8.30 Suppose X1 , . . . , Xk are independent, with Xi ∼ N (θi , 1). Consider testing the null hypothesis θ1 = · · · = θk = 0 against max |θi | ≥ δ, for some δ > 0. Find a maximin level α test as explicitly as possible.  Compare this test with the maximin test if the alternative parameter space were i θi2 ≥ δ 2 . Argue they are quite similar for small δ. Specifically, consider the power of each test against (δ, 0, . . . , 0) and show that it is equal to α + Cα δ 2 + o(δ 2 ) as δ → 0, and the constant Cα is the same for both tests. Section 8.6 Problem 8.31 Existence of most stringent tests. Under the assumptions of Problem 8.1 there exists a most stringent test for testing θ ∈ ΩH against θ ∈ Ω − ΩH . Problem 8.32 Let {Ω∆ } be a class of mutually exclusive sets of alternatives such that the envelope power function is constant over each Ω∆ and that ∪Ω∆ = Ω − ΩH , and let ϕ∆ maximize the minimum power over Ω∆ . If ϕ∆ = ϕ is independent of ∆, then ϕ is most stringent for testing θ ∈ ΩH . Problem 8.33 Let (Z1 , . . . , ZN ) = (X1 , . . . , Xm , Y1 , . . . , Yn ) be distributed according to the joint density (5.55), and consider the problem of testing H : η = ξ against the alternatives that the X’s and Y ’s are independently normally distributed with common variance σ 2 and means η = ξ. Then the permutation test with rejection region |Ȳ − X̄| > C[T (Z)], the two-sided version of the test (5.54), is most stringent. [Apply Problem 8.32 with each of the sets Ω∆ consisting of two points (ξ1 , η1 , σ), (ξ2 , η2 , σ) such that n δ, m+n n ξ2 = ζ + δ, m+n ξ1 = ζ − m δ; m+n m η2 = ζ − δ m+n η1 = ζ + for some ζ and δ.] Problem 8.34 Show that the UMP invariant test of Problem 8.21 is most stringent. 8.8. Notes 347 8.8 Notes The concepts and results of Section 8.1 are essentially contained in the minimax theory developed by Wald for general decision problems. An exposition of this theory and some of its applications is given in Wald’s book (1950). For more recent assessments of the important role of the minimax approach, see Brown (1994, 2000). The ideas of Section 8.3, and in particular Theorem 8.3.1, are due to Huber (1965) and form the core of his theory of robust tests [Huber (1981, Chapter 10)]. The material of sections 8.4 and 8.5, including Lemma 8.4.1, Theorem 8.5.1, and Example 8.5.2, constitutes the main part of an unpublished paper of Hunt and Stein (1946). 9 Multiple Testing and Simultaneous Inference 9.1 Introduction and the FWER When testing more than one parameter, say H: θ1 = · · · = θs = 0 (9.1) against the alternatives that one or more of the θ’s are positive, it is typically not enough simply to accept or reject H. In case of acceptance, nothing more is required: the finding is that none of the parameter values are significant. However, when H is rejected, one will in most cases want to know just which of the parameters θ are significant. And when H is tested against the two-sided alternatives that one or more of the θ’s are different from 0, one would in case of rejection usually want to know the signs of the significant θ’s.1 Example 9.1.1 (Normal one-sample problem) Suppose that X1 , . . . , Xn is a sample from N (ξ, σ 2 ) and consider the hypothesis H: ξ ≤ ξ0 , σ ≤ σ0 . In case of rejection one would want to know whether it is the mean or the variance that is rejected, or perhaps both. Example 9.1.2 (Comparing several treatments with a control) When tes ing several treatments against a control, the overall null hypothesis states that none of the treatments is an improvement over, or differs from, the control. In case of rejection one will wish to know just which of the treatments show a significant difference. 1 We 9.3. shall here disregard this latter issue, but see Comment 2 at the end of Section 9.1. Introduction and the FWER 349 Example 9.1.3 (Testing equality of several treatments) Instead of comparing several treatments with a control, one may wish to compare a number of possible alternative situations with each other. If the quality of the ith of s alternatives is measured by a parameter θi , the hypothesis is H: θ1 = · · · = θs . (9.2) Since most multiple testing problems, like those in Examples 9.1.2 and 9.1.3, are concerned with multiple comparisons, the whole subject of multiple testing is frequently, and somewhat inaccurately, called multiple comparisons. When comparing several medical, agricultural, or industrial treatments, the numbers of treatments is typically fairly small, say, in the single digits. Larger numbers occur in some educational studies, where for example it may be desired to compare performance in the 50 of the U.S. states. A fairly recent application of multiple comparison theory occurs in microarrays where thousands or even tens of thousands of genes are tested simultaneously. Each microarray corresponds to one unit (plant, animal or person) and in these experiments the sample size (the number of such units) is typically of a much smaller order of magnitude (in the tens) than the number of comparisons being tested. Let us now consider the general problem of simultaneously testing a finite numbers of hypotheses Hi (i = 1, . . . , s). We shall assume that tests for the individual hypotheses are available and the problem is how to combine them into a simultaneous test procedure. The easiest approach is to disregard the multiplicity and simply test each hypothesis at level α. However, with such a procedure the probability of one or more false rejections rapidly increases with s. When the number of true hypotheses is large, we shall be nearly certain to reject some of them. To get a numerical idea of this phenomenon, the following Table shows (to 2 decimals) the probability of one or more false rejections when all of the hypotheses H1 , . . . , Hs are true, when the test statistics used for testing H1 , . . . , Hs are independent, and when the level at which each of the s hypotheses is tested is α = .05. s P(at least one false rejection) 1 .05 2 .10 5 .23 10 .40 50 .92 In this sense the claim that the procedure controls the probability of false rejections at level .05 is clearly very misleading. We shall therefore in the present chapter replace the usual condition for testing a single hypothesis, that the probability of a false rejection not exceed α, by the requirement, when testing several hypotheses, that the probability of one or more false rejections, not exceed a given level. This probability is called the family-wise error rate (FWER). Here the term “family” refers to the collection of hypotheses H1 , . . . , Hs that is being considered for joint testing. In a laboratory testing blood samples, this might be all the tests performed in a day, or those performed in a day by a given tester. Alternatively, the tests given in the morning and afternoon might be considered as separate families, and so on. Which tests are to be treated jointly as a family depends on the situation. 350 9. Multiple Testing and Simultaneous Inference Once the family has been defined, we shall require that F W ER ≤ α (9.3) for all possible constellations of true and false hypotheses. This is sometimes called strong error control to distinguish it from the much weaker (and typically not very meaningful) condition of weak control which requires (9.3) to hold only when all the hypotheses of the family are true. Methods that control the FWER are often described by the p-values of the individual tests, which were introduced in Section 3.2. We now present two simple methods that control the FWER which can be stated easily in terms of p-values. Each hypothesis Hi can be viewed as a subset, ωi , of Ω. Assume that p̂i is a p-value for testing Hi ; specifically, we assume P {p̂i ≤ u} ≤ u (9.4) for any u ∈ (0, 1) and any P ∈ ωi . Note that it is not required that the distribution of p̂i be uniform on (0, 1) whenever Hi is true. (For example, if Hi corresponds to testing θi ≤ 0 but the true θi is < 0, exact uniformity is too strong. Also, even if the null hypothesis is simple, the p-value may have a discrete distribution.) Theorem 9.1.1 (Bonferroni Procedure) If, for i = 1, . . . , s, hypothesis Hi is rejected when p̂i ≤ α/s, then the FWER for the simultaneous testing of H1 , . . . , Hs satisfies (9.3). Proof. Suppose hypotheses Hi with i ∈ I are true and the remainder false, with |I| denoting the cardinality of I. From the Bonferroni inequality it follows that  F W ER = P {reject any Hi with i ∈ I} ≤ P {reject Hi } i∈I =  i∈I P {p̂i ≤ α α }≤ ≤ |I|α/s ≤ α . s s i∈I While such Bonferroni based procedures satisfactorily control the FWER, their ability to detect cases in which Hi is false will typically be very low since Hi is tested at level α/s which - particularly if s is large - is orders smaller than the conventional α levels. For this reason procedures are prized for which the levels of the individual tests are increased over α/s without an increase in the FWER. It turns out that such a procedure due to Holm (1979) is available under the present minimal assumptions. The Holm procedure can conveniently be stated in terms of the p-values p̂1 , . . . , p̂s of the s individual tests. Let the ordered p-values be denoted by p̂(1) ≤ . . . ≤ p̂(s) , and the associated hypotheses by H(1) , . . . , H(s) . Then the Holm procedure is defined stepwise as follows: Step 1. If p̂(1) ≥ α/s, accept H1 , . . . , Hs and stop. If p̂(1) < α/s reject H(1) and test the remaining s − 1 hypotheses at level α/(s − 1). Step 2. If p̂(1) < α/s but p̂(2) ≥ α/(s − 1), accept H(2) , . . . , H(s) and stop. If p̂(1) < α/s and p̂(2) < α/(s − 1), reject H(2) in addition to H(1) and test the remaining s − 2 hypotheses at level α/(s − 2). 9.1. Introduction and the FWER 351 And so on. Theorem 9.1.2 The Holm procedure satisfies (9.3). Proof. Suppose Hi with i ∈ I is the set of true hypotheses, so P ∈ ωi if and only if i ∈ I. Let j be the smallest (random) index satisfying p̂(j) = min p̂i . i∈I Note that j ≤ s − |I| + 1. Now, the Holm procedure commits a false rejection if p̂(1) ≤ α/s, p̂(2) ≤ α/(s − 1), . . . , p̂(j) ≤ α/(s − j + 1) , which certainly implies that min p̂i = p̂(j) ≤ α/(s − j + 1) ≤ α/|I| . i∈I Therefore, by the Bonferroni inequality, the probability of a false rejection is bounded above by  P {min p̂i ≤ α/|I|} ≤ P {p̂i ≤ α/|I|} ≤ α . i∈I i∈I The Bonferroni method is an example of a single-step procedure, meaning any hypothesis is rejected if its corresponding p-value is less than a common cutoff value (which in the Bonferroni case is α/s). The Holm procedure is a special case of a class of stepdown procedures, which we now briefly describe. Roughly speaking, stepdown procedures begin by determining whether the test that looks most significant should be rejected. If each individual test is summarized by a p-value, this can be described as follows. Let α1 ≤ α2 ≤ · · · ≤ αs (9.5) be constants. If p̂(1) ≥ α1 , accept all hypotheses. Otherwise, for r = 1, . . . , s, reject hypotheses H(1) , . . . , H(r) if p̂(1) < α1 , . . . , p̂(r) < αr . (9.6) That is, a stepdown procedure starts with the most significant p-value and continues rejecting hypotheses as long as their corresponding p-values are small. The Holm procedure uses αi = α/(s − i + 1). (Alternatively, if the rejection region of each test corresponds to large value of a test statistic, a stepdown procedure begins by determining whether or not the hypothesis corresponding to the largest test statistic should be rejected; see Procedure 9.1.1 below.) On the other hand, stepup procedures begin by looking at the least significant p-value (or the smallest value of a test statistic when the individual tests reject for large values). For a given set of constants (9.5), reject all hypotheses if p̂(s) < αs . Otherwise, for r = s, . . . , 1, reject hypotheses H(1) , . . . , H(r) if p̂(s) ≥ αs , . . . , p̂(r+1) ≥ αr+1 but p̂(r) < αr . (9.7) Safeguards against false rejections are of course not the only concern of multiple testing procedures. Corresponding to the power of a single test one must also consider the ability of a multiple test procedure to detect departures from the 352 9. Multiple Testing and Simultaneous Inference hypotheses when they do occur. For certain parametric models, optimality results for some stepwise procedures will be developed in the next section. For now, we show that it is possible to improve upon the Holm method by incorporating the dependence structure of the individual tests. To see how, suppose that a test of the individual hypothesis Hj is based on a test statistic Tn,j , with large values indicating evidence against Hj . (The use of the subscript n in the test statistics will be for asymptotic purposes later on.) If P is the true probability distribution generating the data, let I = I(P ) ⊂ {1, . . . , s} denote the indices of the set of true hypotheses; that is, i ∈ I if and only P ∈ ωi . For K ⊂ {1, . . . , s}, let HK denote the intersection > hypothesis that all Hi with i ∈ K are true; that is, HK is equivalent to P ∈ i∈K ωi . In order to improve upon the Holm method, the basic idea is to use critical values that more accurately approximate the distribution of maxj∈K Tn,j when testing HK , at least when K is in fact true. Let Tn,r1 ≥ Tn,r2 ≥ · · · ≥ Tn,rs (9.8) denote the observed ordered test statistics, and let H(1) , H(2) , . . . , H(s) be the corresponding hypotheses. A stepdown procedure begins with the most significant test statistic. First, test the joint null hypothesis H{1,...,s} that all hypotheses are true. This hypothesis is rejected if Tn,r1 is large. If it is not large, accept all hypotheses; otherwise, reject the hypothesis corresponding to the largest test statistic. Once a hypothesis is rejected, remove it and test the remaining hypotheses by rejecting for large values of the maximum of the remaining test statistics, and so on. To be specific, consider the following generic procedure, based on critical values ĉn,K (1 − α), where ĉn,K (1 − α) is designed for testing the intersection hypothesis HK at nominal level α. Although we are not specifying the constants at this point, we note that they could be nonrandom or data-dependent. Procedure 9.1.1 (Generic Stepdown Method) 1. Let K1 = {1, . . . , s}. If Tn,r1 ≤ ĉn,K1 (1 − α), then accept all hypotheses and stop; otherwise, reject H(1) and continue. 2. Let K2 be the indices of the hypotheses not previously rejected. If Tn,r2 ≤ ĉn,K2 (1 − α), then accept all remaining hypotheses and stop; otherwise, reject H(2) and continue. .. . j. Let Kj be the indices of the hypotheses not previously rejected. If Tn,rj ≤ ĉn,Kj (1 − α), then accept all remaining hypotheses and stop; otherwise, reject H(j) and continue. .. . s. If Tn,s ≤ ĉn,Ks (1 − α), then accept H(s) ; otherwise, reject H(s) . The problem now is how to construct the ĉn,K (1 − α) so that the FWER is controlled. The following result reduces the multiple testing problem of controlling the FWER to that of constructing single tests that control the probability of a Type 1 error. 9.1. Introduction and the FWER 353 Theorem 9.1.3 Let P denote the true distribution generating the data. Consider Procedure 9.1.1 based on critical values ĉn,K (1−α) which satisfy the monotonicity requirement: for any K ⊃ I(P ), ĉn,K (1 − α) ≥ ĉn,I(P ) (1 − α) . (9.9) F W ERP ≤ P {max(Tn,j : j ∈ I(P )) > ĉn,I(P ) (1 − α)} . (9.10) (i) Then, (ii) Also suppose that if ĉn,K (1 − α) is used to test the intersection hypothesis HK , then it is level α when K = I(P ); that is, P {max(Tn,j : j ∈ I(P )) > ĉn,I(P ) (1 − α)} ≤ α . (9.11) Then FWERP ≤ α. Proof. Consider the event that a true hypothesis is rejected, so that for some i ∈ I(P ), hypothesis Hi is rejected. Let ĵ be the smallest index j in the method where this occurs, so that max{Tn,j : j ∈ I(P )} > ĉn,Kĵ (1 − α) . (9.12) Since Kĵ ⊃ I(P ), assumption (9.9) implies ĉn,Kĵ (1 − α) ≥ ĉn,I(P ) (1 − α) (9.13) and so (i) follows. Part (ii) follows immediately from (i). Example 9.1.4 (Multivariate Normal Mean) Suppose (X1 , . . . , Xs ) is multivariate normal with unknown mean µ = (µ1 , . . . , µs ) and known covariance matrix Σ having (i, j) component σi,j . Consider testing Hj : µj ≤ 0 versus √ √ µj > 0. Let Tn,j = Xj / σj,j , since the test that rejects for large Xj / σj,j is U M P for testing Hj . To apply Theorem 9.1.3, let ĉn,K (1−α) be the 1−α quantile of the distribution of max(Xj : j ∈ K) when µ = 0. Since max(Xj : j ∈ I) ≤ max(Xj : j ∈ K) whenever I ⊂ K, the monotonicity requirement (9.9) is satisfied. Moreover, the resulting test procedure rejects at least as many hypotheses as the Holm procedure (Problem 9.5) In the special case when σi,i = σ 2 is independent of i and σi,j as the product structure σi,j = λi λj , then Appendix 3 (p.374) of Hochberg and Tamhane (1987) reduces the problem of determining the distribution of the maximum of a multivariate normal vector to a univariate integral. In general, one can resort to simulation to approximate the critical values; see Example 11.2.13. Example 9.1.5 (One-way Layout) Suppose for i = 1, . . . , s and j = 1, . . . , ni , Xi,j = µi + i,j , where the i,j are i.i.d. N (0, σ 2 ); the vector µ = (µ1 , . . . , µs ) and σ 2 are unknown. Consider testing Hi : µi = 0 against µi = 0. 1/2 Let tn,i = ni X̄i· /S, where X̄i· = n−1 i ni  j=1 Xi,j , S2 = ni s   (Xi,j − X̄i· )2 /ν , i=1 j=1 354 9. Multiple Testing and Simultaneous Inference  and ν = i (ni −1). Under Hi , tn,i has a t-distribution with ν degrees of freedom. Let Tn,i = |tn,i |, and let ĉn,K (1 − α) denote the 1 − α quantile of the distribution of max(Tn,i : i ∈ K) when µ = 0 and σ = 1. Since max(Tn,i : i ∈ I) ≤ max(Tn,i : i ∈ K) , the monotonicity requirement (9.9) follows. Note that the joint distribution of (tn,1 , . . . , tn,s ) follows an s-variate multivariate t-distribution with ν degrees of freedom; see Hochberg and Tamhane (1987, p.374-5). When the number of tests is in the tens or hundreds of thousands, control of the FWER at conventional levels becomes so stringent that individual departures from the hypothesis have little chance of being detected, and it is unreasonable to control the probability of even one false rejection. A radical weakening of the FWER was proposed by Benjamini and Hochberg (1995), who suggested the following. For a given multiple testing decision rule, let N be the total number of rejections and let F be the number of false rejections, i.e., the number of rejections among the N rejections corresponding to true null hypotheses. Define Q to be F/N (and defined to be 0 if N = 0). Thus Q is the proportion of rejected hypotheses that are rejected erroneously. When none of the hypotheses are rejected, both numerator and denominator of that proportion are 0, and Q is then defined to be 0. The false discovery rate (FDR) is F DR = E(Q). (9.14) When all hypotheses are true, F DR = F W ER. In general, F DR ≤ F W ER (Problem 9.9), and typically this inequality is strict, so that the FDR is more liberal (in the sense of permitting more rejections) than the FWER. The FDR is a fairly recent idea, and its properties and behavior are the subject of very active research. We shall here only mention some recent papers on this topic: Finner and Roters (2001), Benjamini and Yekutielli (2001) and Sarkar (2002). 9.2 Maximin Procedures In the present section we shall obtain optimal procedures for a class of problems of the kind illustrated in Examples 9.1.1 and 9.1.2. Consider the general problem of testing simultaneously s hypotheses Hi: θi ≤ 0 against the alternatives θi > 0, (i = 1, . . . , s) and suppose that we would reject the individual hypotheses Hi if a test statistic Ti were sufficiently large. The joint c.d.f. of (T1 , . . . , Ts ) will be denoted by Fθ , θ = (θ1 , . . . , θs ), and we shall assume that the marginal distribution of Ti depends only on θi . The parameter and sample space will be assumed to be finite or infinite open rectangles θi < θi < θi and ti < ti < ti respectively. For ease of notation we shall suppose that θi = ti = −∞ and θ i = ti = ∞ for all i . We shall assume further that, for any B, Pθi {Ti ≤ B} → 1 as θi → −∞ and Pθi {Ti ≥ B} → 1 as θi → +∞ . A crucial assumption will be that the distributions Fθ are stochastically increasing in the following sense, which generalizes the univariate definition in 9.2. Maximin Procedures 355 Section 3.4 to s dimensions. A set ω in IRs is said to be monotone increasing if t = (t1 , . . . , ts ) ∈ ω and ti ≤ ti for all i implies t ∈ ω , and the distributions Fθ will be called stochastically increasing if θi ≤ θi for all i implies   dFθ ≤ (9.15) dFθ ω ω for every monotone increasing set ω. The condition will be assumed not only for the distributions of (T1 , . . . , Ts ) but also for (±T1 , . . . , ±Ts ). Thus, for example, for (−T1 , . . . , −Ts ) it means that for any decreasing region the inequality (9.15) will be reversed. A class of models for which (9.15) holds is given in Problem 9.10. For the sake of simplicity, we shall suppose that when θ1 = . . . = θs , the variables (T1 , . . . , Ts ) are exchangeable, i.e., that the joint distribution is invariant under permutations of the components. In addition, we assume that the joint distribution of (T1 , . . . , Ts ) has a density with respect to Lebesgue measure.2 In order for the critical constants to be uniquely defined, we further assume that the joint density is positive on its (assumed rectangular) region of support, but this can be weakened. Under these assumptions we shall restrict attention to decision rules satisfying the following monotonicity condition. A decision procedure E for the simultaneous testing of H1 , . . . , Hs based on T = (T1 , . . . , Ts ) states for each possible observation vector t the subset It of {1, . . . , s} of values i for which the hypothesis Hi is rejected. A decision rule E is said to be monotone increasing if ti ≤ ti for i ∈ It and ti < ti for i ∈ / It implies that It = It . The ordered T -values will be denoted by T(1) ≤ T(2) ≤ · · · ≤ T(s) and the corresponding hypotheses by H(1) , . . . , H(s) . Consider the following monotone decision procedure D, which can be viewed as an application of Procedure 9.1.1. The Stepdown Procedure D: Step 1. If T(s) < C1 , accept H1 , . . . , Hs . If T(s) ≥ C1 but T(s−1) < C2 , reject H(s) and accept H(1) , . . . , H(s−1) . Step 2. If T(s) ≥ C1 , and T(s−1) ≥ C2 , but T(s−2) < C3 reject H(s) and H(s−1) and accept H(1) , . . . , H(s−2) . And so on. The C’s are determined by P0, . . . , 0 {max(T1 , . . . , Tj ) ≥ Cs−j+1 } = α , ? @A B (9.16) j and therefore the C’s are nonincreasing. Lemma 9.2.1 Under the above assumptions, the procedure D with critical constants given by (9.16) controls the FWER in the strong sense. 2 This assumption is used only so that the critical constants of the optimal procedures lead to control at exact level α. 356 9. Multiple Testing and Simultaneous Inference Proof. Apply Theorem 9.1.3 with ĉn,K (1 − α) = Cs−|K|+1 , where |K| is the cardinality of K. Then, by the monotonicity of the Cs, condition (9.9) holds. We must verify (9.11) for every Pθ . Suppose θ is such that exactly p hypotheses are true. By exchangeability, we can assume H1 , . . . , Hp are true and Hp+1 , . . . , Hs are false. A false rejection occurs if and only if at least one of H1 , . . . , Hp is rejected. Since D is monotone, the probability of this event is largest when θ1 = · · · = θp = 0 and θp+1 → ∞, · · · , θs → ∞ , and, by (9.16), the sup of this probability is equal to P0, . . . , 0 {Ti ≥ Cs−p+1 for some i = 1, . . . , p} = α . ? @A B p The procedure D defined above is an example of a stepdown procedure in that it starts with the most significant (or, in this case, the largest) test statistic and continues rejecting hypotheses as long as their corresponding test statistics are large. In contrast, stepup procedures begin with the least significant test statistic. Consider the following monotone stepup procedure U . The Stepup Procedure U : Step 1. If T(1) > C1∗ reject H1 , . . . , Hs . If T(1) ≤ C1∗ but T(2) > C2∗ , accept H(1) and reject H(2) , . . . , H(s) . Step 2. If T(1) ≤ C1∗ , and T(2) ≤ C2∗ but T(3) > C3∗ , accept H(1) and H(2) and reject H(3) , . . . , H(s) . And so on. The C ∗ ’s are determined by P0, . . . , 0 {Lj } = 1 − α , ? @A B (9.17) j where ∗ Lj = {Tπ(1) ≤ C1∗ , . . . , Tπ(j) ≤ Cj∗ for some permutation of {1, . . . , j}} . The following lemma proves control of the FWER and is left as an exercise (Problem 9.11). Lemma 9.2.2 Under the above assumptions, the stepup procedure U with critical constants given by (9.17) controls the FWER in the strong sense. Subject to controlling the FWER we want to maximize what corresponds to the power of a single test, i.e., the probability of rejecting hypotheses that are in fact false. Let βi (θ) = Pθ {reject at least i hypotheses} and, for any  > 0, let Ai () denote the set in the parameter space for which at least i of the θ’s are > . Then we shall be interested in maximizing inf θ∈Ai () βi (θ) for i = 1, 2, . . . , s. (9.18) This is in the same spirit as the maximin criterion of Chapter 8. However, it is the false hypotheses we should like to reject, and so we also consider maximizing inf θ∈Ai () Pθ {reject at least i false hypotheses} . (9.19) 9.2. Maximin Procedures 357 We note the following obvious fact. Lemma 9.2.3 Under (9.15), for any monotone increasing procedure E, the functions βi (θ1 , . . . , θs ) are nondecreasing in each of the variables θ1 , . . . , θs . For the sake of simplicity we shall now consider the maximin problem first for the case s = 2. Corresponding to any decision rule E, let e0,0 denote the part of the sample space where both hypotheses are accepted, e0,1 where H1 is accepted and H2 is rejected, e1,0 where H1 is rejected and H2 is accepted, and e1,1 where both H1 and H2 are rejected. The following is an optimality result for the stepdown procedure D. It will be convenient in the following theorem to restate the procedure D in the case s = 2. Theorem 9.2.1 Assume the conditions described at the beginning of this section. (i) A monotone increasing decision procedure with FWER ≤ α will maximize (9.18) for i = 1 if and only if it rejects at least one hypothesis when max(T1 , T2 ) ≥ C1 , (9.20) in which case Hi is rejected if Ti > C1 ; in the contrary case, both hypotheses are accepted. The constant C1 is determined by P0,0 {max(T1 , T2 ) ≥ C1 } = α (9.21) The minimum value of β1 (θ) over A1 () is P {Ti ≥ C1 }. (ii) A monotone increasing decision rule with FWER ≤ α and satisfying (9.20) will maximize (9.18) for i = 2 if and only if it takes the following decisions: d0,0 : accept H1 and H2 when max(T1 , T2 ) < C1 d1,0 : reject H1 and accept H2 when T1 ≥ C1 and T2 < C2 d0,1 : accept H1 and reject H2 when T1 < C2 and T2 ≥ C1 d1,1 : reject both H1 and H2 when both T1 and T2 are ≥ C2 (and when 9.20 holds). Here C2 is determined by P0 {Ti ≥ C2 } = α, (9.22) and hence C2 < C1 . The minimum probability over A2 () of rejecting both hypotheses is P, {at least one Ti is ≥ C1 and both are ≥ C2 } . (iii) The result (i) holds if the criterion (9.18) is replaced by (9.19) with i = 1, and P {Ti ≥ C1 } is also the maximum value of criterion (9.19). Proof. To prove (i), note that the claimed optimal solution has minimum power when θ = (, −∞) and D has P {T1 ≥ C1 } for the claimed optimal value of β1 (θ). Now, suppose that E is any other monotone decision rule with FWER ≤ α. Assume there exists (t1 , t2 ) ∈ / d0,0 , i.e., rejecting at least one hypothesis, but (t1 , t2 ) ∈ e0,0 . Then, there exists at least one component of (t1 , t2 ) that is ≥ C1 , say t1 ≥ C1 . It follows that P,−∞ {e0,0 } ≥ P,−∞ {T1 < t1 , T2 < t2 } = P {T1 < t1 } > P {T1 < C1 } and hence P,−∞ {ec0,0 } < P,−∞ {T1 ≥ C1 } = P {T1 ≥ C1 } . 358 9. Multiple Testing and Simultaneous Inference Thus, E has a smaller value of criterion (9.18) than does the claimed optimal D. Therefore, e0,0 cannot have points outside of d0,0 , i.e., e0,0 must be a proper subset of d0,0 . But then, since both procedures are monotone, ec0,0 is bigger than dc0,0 on a set of positive Lebesgue measure and so P0,0 {ec0,0 } > P0,0 {dc0,0 } = α . It follows that for the maximin procedure, the region dc0,0 must be given by (9.20). To prove (ii), the goal now is to show that, among all monotone nondecreasing procedures which control the FWER and satisfy (9.20), D maximizes inf β2 (θ) = inf Pθ {d1,1 } . A2 () A2 () To prove this, consider any other monotone procedure E which controls the FWER and satisfying e0,0 = d0,0 , and suppose that e1,1 contains a point (t1 , t2 ) with ti < C2 for some i, say t1 < C2 . Then, since E is monotone, it contains the quadrant {T1 ≥ t1 , T2 ≥ t2 }, and hence P0,∞ {e1,1 } ≥ P0,∞ {T1 ≥ t1 , T2 ≥ t2 } = P0 {T1 ≥ t1 } > P0 {T1 ≥ C2 } = α , which contradicts strong control. It follows that e1,1 is a proper subset of d1,1 , and Pθ {e1,1 } < Pθ {d1,1 } for all θ . Since the inf over A2 () of both sides is attained at (, ), inf Pθ {e1,1 } < inf Pθ {d1,1 } , A2 () A2 () as was to be proved. To prove (iii), observe that, for any θ, Pθ {rejecting at least one false Hi } ≤ Pθ {rejecting at least one Hi } , and so inf θ∈A1 () Pθ {rejecting at least one false Hi } ≤ inf θ∈A1 () Pθ {rejecting at least one Hi } But, the right side is P {T1 > C1 }, and so it suffices to show that D satisfies inf θ∈A1 () Pθ {D rejects at least one false Hi } = P {T1 > C1 } . But, this last result is easily checked. Finally, once d0,0 and d1,1 are determined, so are d0,1 and d1,0 by monotonicity, and this completes the proof. Theorem 9.2.1 provides the maximin test which first maximizes inf β1 (θ) and then inf β2 (θ). In the next result, the order in which these aspects are maximized is reversed, which results in the stepup procedure U being optimal. Theorem 9.2.2 Assume the conditions described at the beginning of this section. (i) A monotone decision rule with FWER ≤ α will maximize (9.18) for i = 2 if and only if it rejects both hypotheses, i.e., takes decision u1,1 , when min(T1 , T2 ) ≥ C1∗ (9.23) 9.2. Maximin Procedures 359 and accepts Hi if Ti < C1∗ , where C1∗ = C2 is determined by (9.22). Its minimum power β2 (θ) over A2 () is P {min(T1 , T2 ) ≥ C1∗ } . (9.24) (ii) The monotone procedure with FWER ≤ α and satisfying (9.23) maximizes (9.18) for i = 1 if and only it takes the following decisions: u0,1 = {T1 < C1∗ , T2 ≥ C2∗ } u1,0 = {T1 ≥ C2∗ , T2 < C1∗ } u0,0 = {T1 < C1∗ , T2 < C2∗ } 2 uc1,1 , where C2∗ is determined by P0,0 {uc0,0 } = α . (9.25) Its minimum power β1 (θ) over A1 () is P {Ti ≥ C2∗ } . (9.26) (iii) The result (ii) holds if criterion (9.18) with i = 1 is replaced by (9.19) with i = 1. Note that C1∗ = C2 < C1 < C2∗ . (9.27) Also, the best minimum power β1 (θ) over A1 () for the procedure of Theorem 9.2.1 exceeds that for Theorem 9.2.2, while the situation is reversed for the best minimum power of β2 (θ) over A2 (). This is, of course, as it must be since the first of these two procedures maximized the minimum value of β1 (θ) over A1 () while the second maximized the minimum value of β2 (θ) over A2 (). Proof. (i) Suppose that E is any other monotone procedure with FWER ≤ α. Assume there exists (t1 , t2 ) ∈ e1,1 such that ti < C1∗ for some i, say t1 < C1∗ . Then, P0,∞ {e1,1 } ≥ P0,∞ {T1 ≥ t1 , T2 ≥ t2 } = P0 {T1 ≥ t1 } > P0 {T1 ≥ C1∗ } = α , which would violate the FWER condition. Therefore, e1,1 ⊂ u1,1 . But then inf β2 (θ) A2 () is smaller for E than for U , as was to be proved. (ii) Note that the claimed solution inf A1 () β(θ) is given by inf θ∈A1 () Pθ {uc0,0 } = P,−∞ {uc0,0 } = P {T1 ≥ C1∗ } . We now seek to determine u0,0 , as in Theorem 9.2.1, but with the added constraint that u0,0 ⊂ uc1,1 . To prove optimality for the claimed solution, suppose that E is another monotone procedure controlling FWER at α, and satisfying e1,1 = u1,1 with u1,1 given by (9.23). Assume (t1 , t2 ) ∈ e0,0 but ∈ / u0,0 , so that Ti > C2∗ for some i, say i = 1. Then, P,−∞ {e0,0 } ≥ P,−∞ {T1 ≤ t1 , T2 ≤ t2 } = P {T1 ≤ t1 } > P {T1 > C2∗ } . 360 9. Multiple Testing and Simultaneous Inference Hence, P,−∞ {ec0,0 } < P {T1 > C2∗ } , so that E cannot be optimal. It follows that e0,0 ⊂ u0,0 . But if e0,0 is a proper subset of u0,0 , the set ec0,0 in which E rejects at least one hypothesis contains uc0,0 and so P0,0 {ec0,0 } > P0,0 {uc0,0 } = α , and E does not control the FWER at α. Finally, the proof of (iii) is analogous to the proof of (iii) in Theorem 9.2.1. Theorems 9.2.1 and 9.2.2 have natural extensions to the case of s hypotheses where the aim is to maximize the s quantities (9.18). As in the case s = 2, these maximizations lead to different procedures, and one must choose their order of importance. The two most natural choices are the following: (a) Begin by maximizing inf β1 (θ), which will lead to an optimal choice for d0,0,...,0 , the decision to accept all hypotheses. With d0,...,0 fixed, the partition of dc0,...,0 into the subsets in which the remaining decisions should be taken is begun by maximizing the minimum of β2 (θ) over the part of the parameter space in which at least 2 hypotheses are false, and so on. (b) Alternatively, we may start at the other end by maximizing inf βs (θ), and from there proceed downward. We shall here only state the result for case (a). For its proof and the statement and proof for case (b), see Lehmann, Romano, and Shaffer (2003). Theorem 9.2.3 Under the assumptions made at the beginning of this section, among all monotone procedures E with FWER ≤ α, the stepdown procedure D with critical constants given by (9.16), has the following properties: (i) it maximizes inf β1 (θ) over A1 () (ii) it maximizes inf β2 (θ) over A2 () subject to the additional condition es,2 ⊂ ds,1 , where es,i and ds,i denote the events that the procedures E and D reject at least i of the hypotheses H1 , . . . , Hs . (iii) Quite generally, it maximizes both (9.18) and (9.19) among all monotone procedures E with FWER ≤ α and satisfying es,i ⊂ ds,i−1 . We shall now provide a canonical form for certain stepdown procedures, and particularly for the maximin procedure D of Theorem 9.2.3, that provides additional insights. Let p̂1 , . . . , p̂s be the p-values of the statistics T1 , . . . , Ts , and denote the ordered p-values by p̂(1) ≤ · · · ≤ p̂(s) . If F denotes the common marginal distribution of Ti under θi = 0, we have that p̂i = 1 − F (Ti ) (9.28) p̂(1) = 1 − F (T(s) ) . (9.29) and hence that In terms of the p̂’s, the steps of the stepdown procedure T(s) ≥ C1 , T(s−1) ≥ C2 , . . . (9.30) 9.2. Maximin Procedures 361 are equivalent respectively to p̂(1) ≤ α1 , p̂(2) ≤ α2 , . . . (9.31) for suitable α’s. In particular, T(s) ≥ C1 is equivalent to p̂(1) ≤ α1 . Thus, by (9.29), T(s) < C1 is equivalent to F (T(s) ) < 1 − α1 , so that C1 = F −1 (1 − α1 ) . On the other hand, if Gs denotes the distribution of T(s) when all the θi are 0, it follows from (9.16) that C1 = G−1 s (1 − α) and hence that 1 − α1 = F [G−1 s (1 − α)] , (9.32) which gives α1 as a function of α. It is of interest to determine the ranges of the step levels α1 , . . . , αs . Since Gs (t) ≤ F (t) for all t, it follows from (9.32) that 1 − α1 ≥ 1 − α for all F , or α1 ≤ α for all F , (9.33) with equality when F = G, i.e., when T1 = · · · Ts . To find a lower bound for α1 , put u = G−1 (1 − α) in (9.32) so that 1 − α1 = F (u) with 1 − α = Gs (u) and note that for all u 1 − Gs (u) = P {at least one Ti ≥ u} ≤  (9.34) P {Ti ≥ u} = s[1 − F (u)] . Thus, F (u) ≤ 1 − α 1 [1 − G(u)] = 1 − s s and hence α . (9.35) s We shall now show that the lower bound (9.35) is sharp by giving an example of a joint distribution of (T1 , . . . , Ts ) for which it is attained. α1 ≥ Example 9.2.1 (A Least Favorable Distribution) Let U be uniformly distributed on (0, 1) and suppose that when H1 , . . . , Hs are all true, 1 s−1 (mod 1), . . . , Ys = U + (mod 1) . s s Since (Y1 , . . . , Ys ) does not satisfy our assumption of exchangeability, replace it by the exchangeable set of variables (X1 , . . . , Xs ) = (Yπ(1) , . . . , Yπ(s) ), where (π(1), . . . , π(s)) is a random permutation of (1, . . . , s) (and independent of U ). Let Ti = 1 − Xi and suppose that Hi is rejected when Ti is large. To show that α , (9.36) F [G−1 s (1 − α)] = 1 − s note that the T ’s are uniformly distributed on (0, 1) so that (9.36) becomes α Gs (1 − ) = 1 − α . s Now α α α 1 − Gs (1 − ) = P {at least one Ti ≥ 1 − } = P {at least one Xi ≤ } . s s s Y1 = U, Y2 = U + 362 9. Multiple Testing and Simultaneous Inference But the events {Xi ≤ α/s} are mutually exclusive, and therefore  α α α P {Xi ≤ } = s · = α , }= s s s i=1 s P {at least one Xi ≤ which implies (9.36). We shall now briefly sketch the corresponding development for α2 , defined by the fact that p̂(2) ≤ α2 is equivalent to T(s−1) ≥ C2 , where C2 is determined by (9.16) so that Gs−1 (C2 ) = 1 − α . Note that Gs−1 is not the distribution of T(s−1) , i.e., of the 2nd largest of s T ’s, but of the largest of T1 , . . . , Ts−1 (i.e., the largest of s − 1 T ’s). In exact analogy with the derivation of (9.32) it now follows that 1 − α2 = F [G−1 s−1 (1 − α)] . (9.37) The maximum value of α2 , as in the case of α1 , is equal to α and is attained when T1 = · · · = Ts−1 . The argument giving the lower bound shows that α2 ≥ α/(s − 1). To show that this value is attained, we must find an example for which α Gs−1 (1 − )=1−α . s−1 Example 9.2.1 will serve this purpose since in that case α α ) = P {at least one of T1 , . . . , Ts−1 ≥ 1 − } 1 − Gs−1 (1 − s−1 s−1 = s−1  i=1 P {Xi ≤ α α } = (s − 1) · =α s−1 s−1 for any α satisfying α/(s − 1) < 1/s, i.e., α < (s − 1)/s. Continuing in this way we arrive at the following result. Theorem 9.2.4 (i) The step levels αi defined by the procedure D with critical constants given by (9.16) and the equivalence of (9.30) and (9.31) are given by 1 − αi = F [Gs−i+1 (1 − α)] , where Gj is the distribution of max(T1 , . . . , Tj ). (ii) The range of αi is α ≤ αi ≤ α . s−i+1 (9.38) (9.39) Furthermore, the upper bound α is attained when T1 = · · · = Ts , i.e., when there really is no multiplicity. The lower bound α/(s − i + 1) is attained when the distribution of T1 , . . . , Ts−i+1 is that of Example 9.2.1. Not all points in the s-dimensional rectangle (9.39) are possible for (α1 , . . . , αs ). In particular, since for all t Gi (t) ≥ Gj (t) when i < j , 9.3. The Hypothesis of Homogeneity 363 it follows that α1 ≤ α2 ≤ · · · ≤ αs . (9.40) The values of αi given by (9.38) can be determined when the joint distribution of (T1 , . . . , Ts ) (and hence the distributions Gs ) is known. Consider, however, the situation in which the common marginal distribution F of the statistics Ti needed to carry out the tests of the individual hypotheses Hi at a given level is known, but the joint distribution of the T ’s is unknown. Then, we are unable to determine the step levels (9.38). It follows, however, from (9.39) that the procedure (9.31) with αi = α/(s − i + 1) for i = 1, . . . , s (9.41) will control the FWER for all joint distributions of (T1 , . . . , Ts ), since these levels are conservative in all cases. This is just the Holm procedure of Theorem 9.1.2. Also, none of the levels αi can be larger than α/(s − i + 1) without violating the FWER condition for some distribution. To see this, note that if levels αi are used in Example 9.2.1, it follows from the discussion of this example that when i of the hypotheses are true, the probability of at least one false rejection is (s − i + 1)αi . Thus, if αi exceeds α/(s − i + 1), the FWER condition will be violated. Of course, if the class of joint distributions of the T ’s is restricted, the range of αi may be smaller than (9.39). For example, suppose that the T ’s are independent. Then, putting u = G−1 s (1 − α) as before, we see from (9.34) that 1 − α1 = F (u) and 1 − α = F s (u) so that α1 = 1 − (1 − α)1/s , and more generally that αi = 1 − (1 − α)1/(s−i+1) . In this case, the range reduces to a single point. More interesting is the case of positive quadrant dependence when Gs (u) ≥ F s (u) and hence 1 − α ≥ (1 − α1 )1/s and 1 − (1 − α)s ≤ α1 ≤ α . (9.42) The bounds are sharp since the upper bound is attained when T1 = · · · = Ts and the lower bound is attained in the case of independence. 9.3 The Hypothesis of Homogeneity The previous section dealt with situations in which each of the parameters varies independently, so that any subset of the hypotheses H1 , . . . , Hs can be true with 364 9. Multiple Testing and Simultaneous Inference the remaining ones being false. This condition is not satisfied, for example, when the set of hypotheses is s Hi,j : θi = θj , i x and F2 (y) ≤ 0 if y < x. Now, Cov[F1 (S), F2 (S)] = E{[F1 (S) − F1 (x)] · F2 (S)} . If S ≥ x, F1 (S) − F1 (x) ≥ 0 and F2 (S) ≥ 0, and so the quantity inside the expectation is ≥ 0. Similarly, if S < x, F1 (S) − F1 (x) ≤ 0 and F2 (S) ≤ 0 and so the quantity inside the expectation is ≥ 0. Lemma 9.3.5 Assume Y1 , . . . , Yr , S are independent, where S is a nonnegative random variable. Then, Ti = Yi /S satisfy (9.58). Proof. Let Gi denote the distribution of Yi . Fix t1 , . . . , tr . By conditioning on S,  P {T1 ≤ t1 , . . . , Tr ≤ tr } = E[ Gi (ti S)] . i Apply Lemma 9.3.4 with Fi (s) = Gi (ti s) to get the last quantity is an upper bound for   E[Gi (ti S)] = P {Ti ≤ ti } . i i For this situation, we have the following result. Theorem 9.3.2 If the test statistics for testing the r hypotheses (9.49) are positively quadrant dependent in the sense of (9.58), then sup α(µ1 , . . . , µs ) ≤ 1 − r  (1 − αvi ) , (9.60) i=1 where, as before, α1 = 0. Proof. That the right side of (9.60) is an upper bound for α(µ1 , . . . , µs ) follows from the proof of Lemma 9.3.1 and the assumption of positive quadrant dependence. Note, however, that we can no longer assert that the upper bound is sharp. For the F and Studentized range tests, the sharp upper bound will depend on the total sample size n. Theorem 9.3.2 guarantees that the procedures using the α-levels derived under the assumption of independence, continue to control the FWER even in the case 9.3. The Hypothesis of Homogeneity 373 of positive dependence. The proof of Lemma 9.3.2 shows that αs = αs−1 = α continues to be necessary for admissibility even in the positively dependent case. However, the maximization results for α2 , . . . , αs can then no longer be asserted. They nevertheless have the great advantage that they define procedures that do not require detailed knowledge of the joint distribution of the various test statistics. Even in the simplified version with known variance the multiple testing problem considered in the present section is clearly much more difficult than the testing of a single hypothesis; the procedures presented above still ignore many important aspects of the problem. 1. Choice of test statistic. The most obvious feature that has not been dealt with is the choice of test statistics. Unfortunately it does not appear that the invariance considerations which were so helpful in the case of a single hypothesis play a similar role here. 2. Order relation of significant means. Whenever two means µi and µj are judged to differ, we should like to state not only that µi = µj , but that if X̄i < X̄j then also µi < µj . Such additional statements introduce the possibility of additional errors (stating µi < µj when in fact µi > µj ), and it is not obvious that when these are included, the probability of at least one error is still bounded by α. [For recent work on directional errors, see Finner (1999) and Shaffer (1990, 2002).] 3. Nominal versus true levels. The levels α2 , . . . , αs , sometimes called nominal levels, are the levels at which the hypotheses µi = µj , µi = µj = µk , . . . are tested. They are however not the true probabilities of falsely rejecting the homogeneity of these sets, but only the upper bounds of these probabilities with respect to variation of the remaining µ’s. The true probabilities tend to be much smaller (particularly when s is large), since they take into account that homogeneity of a set S0 is rejected only if it is also rejected for all sets S containing S0 . 4. Interpretability. As pointed out at the beginning of the section, the totality of acceptance and rejection statements resulting from a multiple comparison procedure typically does not lead to a simple partition of means. This is illustrated by the possibility that the hypothesis of homogeneity is rejected for a set S but for none of its subsets. As another example, consider the case s = 3, where it may happen that the hypotheses µi = µj and µj = µk are accepted but µi = µk is rejected. The number of such “inconsistencies” and the corresponding difficulty of interpreting the results may be formidable. Measures of the complexity of the totality of statements as a third criterion (besides level and power) are discussed by Shaffer (1981). The inconsistencies and resulting difficulties of interpretation suggest the consideration of an alternative formulation of the problem which avoids this  difficulty. Instead of testing the 2s hypotheses Hi,j : µi = µj , estimate the (unknown) partition of the µ’s defined by (9.48). Possible approaches to such procedures are discussed for example in Hochberg and Tamhane (1987, Chapter 10, Section 6) and by Dayton (2003). 374 9. Multiple Testing and Simultaneous Inference 5. Procedures (i) and (ii) can be inverted to provide simultaneous confidence intervals for all differences µj − µi . The T -method (discussed in Problems 9.29–9.32) was designed to give simultaneous intervals for all differences µj − µi ; it can be extended to cover   also all contrasts in the µ’s, that is, all linear functions ci µi with ci = 0, but against more complex contrasts the intervals tend to be longer than those of Scheffés S-method, which was intended for the simultaneous consideration of all contrasts. [For a comparison of the two methods, see for example Scheffé (1959, Section 3.7) and Arnold (1981, Chapter 12).] It is a disadvantage of the remaining (truly stagewise or sequential) procedures of this section that the problem of corresponding confidence sets is considerably more complicated. For a discussion of such confidence methods, see Holm (1999) and the references cited there. 6. To control the rate of false rejections, we have restricted attention to procedures controlling the FWER, the probability of at least one error. Instead, one might wish to control the false discovery rate as defined at the end of Section 9.1; see Benjamini and Hochberg (1995). Alternatively, an optimality theory based on the number of false rejections is given in Spjøtvoll (1972). Another possibility is the control the k-FWER, the probability of making k or more false rejections, as well as the probability that the false discovery proportion exceeds some threshold; see Korn et al. (2004), Romano and Shaikh (2004) and Lehmann and Romano (2005). 7. The optimal choice of the αk discussed in this section can be further improved, at the cost of considerable additional complication, by permitting the α’s to depend on the outcomes of the other tests. This possibility is discussed, for example, in Marcus, Peritz, and Gabriel (1976); see also Holm (1979) and Shaffer (1984). The procedures discussed in this section were concerned with testing the equality of means. In more complex situations, further problems arise. Consider, for example, the two-way layout of 7.5 with     µi,j = µ + αi + βj + γi,j ( αi = βj = γi,j = γi,j = 0) . i j If we are interested in multiple testing of the α’s, β’s, and γ’s, the first question that arises is whether we want to treat these three cases (α’s, β’s, γ’s) as a single family, as two families (the main effects forming one family, the interactions the other), or as three families in which each of the three sets is handled separately. The most appropriate designation of what constitutes a family depends very much on context. Consider, for example, the National Assessment of Educational Progress which makes it possible to compare the progress made by any two states. For a federal report, the set of all 50 possible hypotheses would constitute an 2 appropriate family. However, a particular state would be interested primarily in the comparison of its performance with those of the other 49 states, thus leading to a family of size 49. A comparison which is not significant in the federal report might then turn out to be significant in the state report. Some of the issues concerning the most suitable definition of family are discussed in Tukey (1991) 9.4. Scheffé’s S-Method: A Special Case 375 and in the books by Hochberg and Tamhane (1987), and Westfall and Young (1993). We shall in the next two sections consider simultaneous inferences for various families of linear functions of means in normal linear models. However, since we are assuming fully articulated parametric models, we shall consider the slightly more demanding problem of obtaining simultaneous confidence intervals rather than restricting attention to hypothesis testing. As the simplest example, suppose that X1 , . . . , Xs are normal variables with means µ1 , . . . , µs and unit variance. We can then apply to the hypotheses Hi : µi = µi,0 the approach of Section 9.1 and test these hypotheses by means of a stepdown procedure. The resulting acceptance regions can then be converted in the usual way into confidence sets. It is shown in Holm (1999) that these sets are rather complicated and not rectangular, so that they do not consist of intervals for the individual µi ’s. (They can, of course, be enclosed in a larger rectangle, but the intervals obtained by such a process tend to be unnecessarily large.) 9.4 Scheffé’s S-Method: A Special Case If X1 , . . . , Xr are independent normal with common variance σ 2 and expectations E(Xi ) = α+βti , confidence sets for (α, β) were obtained in Section 7.6. A related problem is that of determining confidence bands for the whole regression line ξ = α + βt, that is, functions L (t; X), M  (t; X) such that P {L (t; X) ≤ α + βt ≤ M  (t; X) for all t} = γ. (9.61) The problem of obtaining simultaneous confidence intervals for a continuum of parametric functions arises also in other contexts. In the present section, a general problem of this kind will be considered for linear models. Confidence bands for an unknown distribution function were treated in Section 6.13. Suppose first that X1 , . . . , Xr are independent normal with variance σ 2 = 1 and with means E(Xi ) = ξi , and  that simultaneous confidence intervals  are required for all linear functions ui ξi . No generality is lost by dividing ui ξi and its lower and upper bound by u2i , so that attention can be restricted to confidence sets  (9.62) S(x) = {ξ : L(u; x) ≤ ui ξi ≤ M (u; x) for all u ∈ U } , where x, u denote both the vectors with coordinates xi , ui and ther × 1 column matrices with these elements, and where U is the set of all u with u2i = 1. The sets S(x) are to satisfy Pξ {ξ ∈ S(X)} = γ for all ξ = (ξ1 , . . . , ξr ). (9.63) Since u = (u1 , . . . , ur ) ∈ U if and only if −u = (−u  1 , . . . , −ur ) ∈ U , the simultaneous inequalities (9.62) imply L(−u; x) ≤ − ui ξi ≤ M (−u; x), and hence  −M (−u; x) ≤ ui ξi ≤ −L(−u; x) and max(L(u; x), −M (−u; x)) ≤  ui ξi ≤ min(M (u; x), −L(−u; x)). 376 9. Multiple Testing and Simultaneous Inference Nothing is therefore lost by assuming that L and M satisfy L(u; x) = −M (−u; x). (9.64) The problem of determining suitable confidence bounds L(u; x) and M (u; x) is invariant under the group G1 of orthogonal transformations (Q an orthogonal r × r matrix). G1 : gx = Qx, ḡξ = Qξ  Writing ui ξi = u ξ, we have g ∗ S(x) = {Qξ : L(u; x) ≤ u ξ ≤ M (u; x) for all u ∈ U } = {ξ : L(u; x) ≤ u (Q−1 ξ) ≤ M (u; x) for all u ∈ U } = {ξ : L(Q−1 u; x) ≤ u ξ ≤ M (Q−1 u; x) for all u ∈ U }, where the last equality uses the fact that U is invariant under orthogonal transformations of u. Since S(gx) = {ξ : L(u; Qx) ≤ u ξ ≤ M (u; Qx) for all u ∈ U }, the confidence sets S(x) are equivariant under G1 if and only if L(u; Qx) = L(Q−1 u; x), M (u; Qx) = M (Q−1 u; x), or equivalently if L(Qu; Qx) = L(u; x), for all M (Qu; Qx) = M (u; x) (9.65) x, Q and u ∈ U, that is, if L and M are invariant under common orthogonal transformations of u and x. A function L of u and x is invariant under these transformations if and only if it depends on u and x only through u x, x x, and u u [Problem 9.23(i)] and hence (since u u = 1) if there exists h such that L(u; x) = h(u x, x x). (9.66) A second group of transformations leaving the problem invariant is the group of translations G2 : gx = x + a, ḡξ = ξ + a where x + a = (x1 + a1 , . . . , xr + ar ). An argument paralleling that leading to (9.65) shows that L(u; x) is equivariant under G2 if and only if [Problem 9.23(ii)]  L(u; x + a) = L(u; x) + ai ui for all x, a, and u. (9.67) The function h of (9.66) must therefore satisfy h[u (x + a), (x + a) (x + a)] = h(u x, x x) + a u for all and hence, putting x = 0, h(u a, a a) = a u + h(0, 0). a, x and u ∈ U, 9.4. Scheffé’s S-Method: A Special Case 377 A necessary condition (which clearly is also sufficient) for S(x) to be equivariant under both G1 and G2 is therefore the existence of constants c and d such that . -    ui ξi ≤ ui xi + d for all u ∈ U S(x) = ξ : ui xi − c ≤ From (9.64) it follows that c = d, so that the only equivariant families S(x) are given by % - % . % % S(x) = ξ : % ui (xi − ξi )% ≤ c for all u ∈ U (9.68) The constant c is determined by (9.63), which now reduces to % . -% % % ui Xi % ≤ c for all u ∈ U = γ. P0 %   (9.69)  Xi2 , since u2i = 1, and hence By the Schwarz inequality ( ui Xi )2 ≤ % %  % % (9.70) Xi2 ≤ c2 . ui Xi % ≤ c for all u ∈ U if and only if % The constant c in (9.68) is therefore given by P (χ2r ≤ c2 ) = γ. (9.71) In (9.68), it is of course possible to drop the restriction u ∈ U by writing (9.68) in the equivalent form   % , % % % (9.72) u2i for all u . S(x) = ξ : % ui (xi − ξi )% ≤ c So far attention has been restricted to the confidence bands (9.62). However, confidence sets do not have to be intervals, and it may be of interest to consider more general simultaneous confidence sets  S(x) : ui ξi ∈ A(u, x) for all u ∈ U. (9.73) For these sets, the equivariance conditions (9.65) and (9.67) become respectively (Problem 9.24) A(Qu, Qx) = A(u, x) for all x, Q and u ∈ U (9.74) A(u, x + a) = A(u, x) + u a for all u, x, and a. (9.75) and The first of these is equivalent to the condition that the set A(u, x) depends on u ∈ U and x only through u x and x x. On the other hand putting x = 0 in (9.75) gives A(u, a) = A(u, 0) + u a. It follows from (9.74) that A(u, 0) is a fixed set A1 independent of u, so that A(u, x) = A1 + u x. (9.76) The most general equivariant sets (under G1 and G2 ) are therefore of the form  (9.77) ui (xi − ξi ) ∈ A for all u ∈ U, where A = −Ai . 378 9. Multiple Testing and Simultaneous Inference We shall now suppose that r > 1 and then show that among all A which define confidence sets (9.77) with confidence coefficient ≥ γ, the sets (9.68) are smallest3 in the very strong sense that if A0 = [−c0 , c0 ] denotes the set (9.68) with confidence coefficient γ, then A0 is a subset of A. To see this, note that if Yi = Xi − ξi , the sets A are those satisfying   (9.78) ui Yi ∈ A for all u ∈ U ≥ γ. P  Now the set of values taken on by ui yi for a fixed y = (y1 , . . . , yr ) as u ranges over U is the interval (Problem 9.24)  , ,  I(y) = − yi2 , + yi2 . Let c∗ be the largest value of c for which the interval [−c, c] is contained in A. Then the probability (9.78) is equal to P {I(Y ) ⊂ A} = P {I(Y ) ⊂ [−c∗ , c∗ ]}. Since P {I(Y ) ⊂ A} ≥ γ, it follows that c∗ ≥ c0 , and this completes the proof. It is of interest to compare the simultaneous confidence intervals (9.68) for all ui ξi , u ∈ U , with the joint confidence spheres for (ξ1 , . . . , ξr ) given by (6.43). These two sets of confidence statements are equivalent in the following sense. Theorem 9.4.1 The parameter vector (ξ1 , . . . , ξr ) satisfies and only if it satisfies (9.68).  (Xi − ξi )2 ≤ c2 if Proof. The result follows immediately from (9.70) with Xi replaced by Xi −ξi . Another comparison of interest is that of the simultaneous confidence intervals (9.72) for all u with the corresponding interval  % ,  % % %   S (x) = ξ : % u2i ui (xi − ξi )% ≤ c (9.79)   2 ui has a standard normal distrifor a single given u. Since ui (Xi − ξi )/  bution, the constant c is determined by P (χ21 ≤ c2 ) = γ instead of by (9.71). If r > 1, the constant c2 = c2r is clearly larger than c2 = c21 . The lengthening of the confidence intervals by the factor cr /c1 in going from (9.79) to (9.72) is the price one must pay for asserting confidence γ for all ui ξi instead of a single one. In (9.79), it is assumed that the vector u defines the linear combination of interest and is given before any observations  are available. However, it often happens that an interesting linear combination ûi ξi to be estimated is suggested by the data. The intervals , % % % % ûi (xi − ξi )% ≤ c û2i (9.80) %  with c given by (9.71) then provide confidence limits for ûi ξi at confidence level γ, since they are included in the set of intervals (9.72). [The notation ûi 3 A more general definition of smallness is due to Wijsman (1979). It has been pointed out by Professor Wijsman that his concept is equivalent to that of tautness defined by Wynn and Bloomfield (1971). 9.4. Scheffé’s S-Method: A Special Case 379 in (9.80) indicates that the u’s were suggested by the data rather than fixed in advance.] Example 9.4.1 (Two groups) Suppose the data exhibit a natural split into a lower and upper group, say ξi1 , . . . , ξik , and ξj1 , . . . , ξjr−k , with averages ξ¯− and ξ¯+ , and that confidence limits are required for ξ¯+ − ξ¯− . Letting X̄ − = (Xi1 + · · · + Xik )/k and X̄ + = (Xj1 + · · · + Xjr−k )/(r − k) denote the associated averages of the X’s we see that 1 1 1 1 1 1 + ≤ ξ¯+ − ξ¯− ≤ X̄ + − X̄ − + c + (9.81) X̄ + − X̄ − − c k r−k k r−k with c given by (9.71) provide the desired limits. Similarly c c c c ≤ ξ¯+ ≤ X̄ + + √ (9.82) X̄ + − √ X̄ − − √ ≤ ξ¯− ≤ X̄ − + √ , r−k r−k k k provide simultaneous confidence intervals for the two group means separately, with c again given by (9.71). For a discussion of related examples and issues see Peritz (1965).  Instead of estimating a data-based function  ûi ξi , one may be interested in testing it. At level α = 1 − γ, the hypothesis ûi ξi = 0 is rejected when the confidence intervals (9.80) do not cover the origin, i.e., when , % % % % ûi xi % ≥ c û2i . % Equivariance with respect to the group G1 of orthogonal transformations assumed at the beginning of this section is appropriate only when all linear combinations ui ξi with u ∈ U are of equal importance. Suppose instead that interest focuses on the individual means, so that simultaneous confidence intervals are required for ξ1 , . . . , ξr . This problem remains invariant under the translation group G2 . However, it is no longer invariant under G1 , but only under the much smaller subgroup G0 generated by the n! permutations and the 2n changes of sign of the X’s. The only simultaneous intervals that are equivariant under G0 and G2 are given by [Problem 9.25(i)] S(x) = {ξ : xi − ∆ ≤ ξi ≤ xi + ∆ for all i} (9.83) where ∆ is determined by P [S(X)] = P (max |Yi | ≤ ∆) = γ (9.84) with Y1 , . . . , Yr being independent N (0, 1). These maximum-modulus intervals for the ξ’s can be extended to all linear  combinations ui ξi of the ξ’s by noting that the right side of (9.83) is equal to the set [Problem 9.25(ii)] % - % .  % % ξ:% (9.85) ui (Xi − ξi )% ≤ ∆ |ui | for all u , which therefore also has probability γ, but which is not equivariant under G1 . A comparison of the intervals (9.85) with the Scheffé intervals (9.72) shows [Problem  9.25(iii)] that the intervals (9.85) are shorter when uj ξj = ξi (i.e. when uj = 1 for j = i, and uj = 0 otherwise), but that they are longer for example when u 1 = · · · = ur . 380 9. Multiple Testing and Simultaneous Inference 9.5 Scheffé’s S-Method for General Linear Models The results obtained  in the preceding section for the simultaneous estimation of all linear functions ui ξi when the common variance of the variables Xi is known easily extend to the general linear model of Section 7.1. In the canonical form (7.2), the observations are n independent normal random variables with common unknown variance σ 2 and with means E(Yi ) = ηi for i = 1, . . . , s and E(Yi ) = 0 for i = s + 1, . . . , n. Simultaneous confidence intervals are required for all linear r functions ui ni with u ∈ U , where U is the set of all u = (u1 , . . . , ur ) r i=1 2 with u = 1. Invariance under the translation group Yi = Yi + ai , i=1 i i = r + 1, . . . , s, leaves Y1 , . . . , Yr ; Ys+1 , . . . , Yn as maximal invariants,  and suf2 ficiency justifies restricting attention to Y = (Y1 , . . . , Yr ) and S 2 = n j=s+1 Yj . The confidence intervals corresponding to (9.62) are therefore of the form L(u; y, S) ≤ r  ui ηi ≤ M (u; y, S) for all u ∈ U, (9.86) i=1 and in analogy to (9.64) may be assumed to satisfy L(u; y, S) = −M (−u; y, S). (9.87) By the argument leading to (9.66), it is seen in the present case that equivariance of L(u; y, S) under G1 requires that L(u; y, S) = h(u y, y  y, S), and equivariance under G2 requires that L be of the form L(u; y, S) = r  ui yi − c(S). i=1 Since σ 2 is unknown, the problem is now also invariant under the group of scale changes G3 : yi = byi (i = 1, . . . , r), S  = bS (b > 0). Equivariance of the confidence intervals under G3 leads to the condition [Problem 9.26(i)] L(u; by, bS) = bL(u; y, S) and hence to b  ui yi − c(bS) = b  for all b > 0,  ui yi − c(S) , or c(bS) = bc(S). Putting S = 1 shows that c(S) is proportional to S. Thus   L(u; y, S) = ui yi − cS, M (u; y, S) = ui yi + dS, and by (9.87), c = d, so that the equivariant simultaneous intervals are given by    ui yi − cS ≤ u i ηi ≤ ui yi + cS for all u ∈ U. (9.88) Since (9.88) is equivalent to  (yi − ηi )2 ≤ c2 , S2 9.5. Scheffé’s S-Method for General Linear Models the constant c is determined from the F -distribution by   2  Yi /r n − s 2. n−s 2 ≤ c c = γ. = P0 Fr,n−s ≤ P0 2 S /(n − s) r r 381 (9.89) As in (9.72), the restriction u ∈ U can be dropped; this only requires replacing c  u2i = c Var ui Yi /σ 2 . in (9.88) and (9.89) by c As in the case of known variance, instead of restricting attention to the confidence bands (9.88), one may wish to permit more general simultaneous confidence sets  ui ηi ∈ A(u; y, S). (9.90) The most general equivariant confidence sets are then of the form [Problem 9.26(ii)]  ui (yi − ηi ) ∈ A for all u ∈ U, (9.91) S and for a given confidence coefficient, the set A is minimized by A0 = [−c, c], so that (9.91) reduces to (9.88). For applications, it is convenient to express the intervals (9.88) in terms of the original variables Xi and ξi . Suppose as in Section 7.1 that X1 , . . . , Xn are independently distributed as N (ξi , σ 2 ), where ξ = (ξ1 , . . . , ξn ) is assumed to lie in a given s-dimensional linear subspace Ω (s < n). Let V be an r-dimensional subspace of Ω (r ≤ s), let ξˆi be the least squares estimates of the ξ’s under  2 (Xi − ξˆi )2 . Then the inequalities Ω , and let S = ; ;     < < < Var  v ξˆ < Var  v ξˆ    i i i i = = vi ξˆi − cS ≤ vi ξi ≤ vi ξˆi + cS σ2 σ2 for all v ∈ V, (9.92)  with c given by (9.89), provide simultaneous confidence intervals for vi ξi for all v ∈ V with confidence coefficient γ. This result is an immediate consequence of (9.88) and (9.89) together with the following three facts, which will be proved below:   s n ˆ (i) If si=1 ui ηi = n j=1 vj ξj , then i=1 ui Yi = j=1 vj ξ j ; (ii) n i=s+1 Yi2 = n j=1 (Xj − ξˆj )2 , To state (iii), note that the η’s are obtained as linear functions of the ξ’s through the relationship (η1 , . . . , ηr , ηr+1 , . . . , ηs , 0, . . . , 0) = C(ξ1 , . . . , ξn ) (9.93) where C is defined by (7.1) and the prime indicates a transpose. This is seen by taking the expectation of both sides of (7.1). For each vector u = (u1 , . . . , ur ),   (u) (9.93) expresses ui ηi as a linear function vj ξj of the ξ’s. (u) (u) (iii) As u ranges over r-space, v (u) = (v1 , . . . , vn ) ranges over V . 382 9. Multiple Testing and Simultaneous Inference Proof of (i) Recall from Section 7.2 that n s n    (Xj − ξj )2 = (Yi − ηi )2 + Yj2 . j=1 i=1 j=s+1 Since the right side is minimized by ηi = Yi and the left side by ξj = ξˆj , this shows that (Y1 · · · Ys 0 · · · 0) = C(ξˆ1 · · · ξˆj ) , and the result now follows from comparison with (9.93). Proof of (ii) This is just equation (7.13).    (u) (u) Proof of (iii) Since ηi = n u i ηi = vj ξj with vj = j=1 cij ξj , we have r (u) (u) (u) = (v1 , . . . , vn ) are linear combinations, with i=1 ui cij . Thus, the vectors v weights u1 , . . . , ur , of the first r row vectors of C. Since the space spanned by these row vectors is V , the result  follows. The set of linear functions vi ξi , v ∈ V , for which the interval (9.92) does not cover the origin—that is, for which v satisfies ;   < < Var  v ξˆ % % i i = % % vi ξˆi % > cS % σ2 (9.94) —is declared significantly different from 0 by the intervals (9.92).  Thus (9.94) is a rejection region at level α = 1 − γ of the hypothesis H : vi ξi = 0 for all v ∈ V in the sense that H is rejected if and only if at least one v ∈ V satisfies (9.94). If ω denotes the (s − r)-dimensional space of vectors v ∈ Ω which are orthogonal to V , then H states that ξ ∈ ω , and the rejection region (9.94) is in fact equivalent to the F -test of H : ξ ∈ ω of Section 7.1. In canonical form, this was seen in the sentence following (9.88). To implement the intervals (9.92) in specific situations in which the correspond ing intervals for a single given function vi ξi are known, it is only necessary to designate the space V and to obtain its dimension r, the constant c then being determined by (9.89). Example 9.5.1 (All contrasts) Let Xij (j = 1, . . . , ni ; i = 1, . . . , s) be independently distributed as N (ξi , σ 2 ), and is suppose V is the space of all vectors v = (v1 , . . . , vn ) satisfying  vi = 0. (9.95)  Any function vi ξi with v ∈ V is called a contrast among the ξi . The set of contrasts includes in particular the differences ξ¯+ − ξ¯− discussed in Example 9.4.1. The space Ω is the set of all vectors (ξ1 , . . . , ξ1 ; ξ2 , . . . , ξ2 ; ξs , . . . , ξs ) and has dimension s, while V is the subspace of vectors Ω that are orthogonal to (1, . . . , 1) and hence has dimension r = s − 1. It was seen in Section 7.3 that 9.5. Scheffé’s S-Method for General Linear Models ξˆi = Xi· , and if the vectors of V are denoted by w1 w1 w2 w2 ws ws ,..., ; ,..., ; ,..., n1 n1 n2 n2 ns ns 383  , the simultaneous confidence intervals (9.92) become (Problem 9.28) 6 6    wi2 wi2 ≤ wi ξi ≤ wi Xi· + cS (9.96) wi Xi· − cS ni ni  for all (w1 , . . . , ws ) satisfying wi = 0,   (Xij − Xi· )2 . with S 2 = In the present case the space ω is the set of vectors with all coordinates equal, so that the associated hypothesis is H : ξ1 = · · · = ξs . The rejection region (9.94) is thus equivalent to that given by (7.19). Instead of testing the overall homogeneity hypothesis H, we may be interested in testing one or more subhypotheses suggested by the data. In the situation corresponding to that of Example 9.4.1 (but with replications), for instance, interest may focus on the hypotheses H1 : ξi1 = · · · = ξik and H2 : ξj1 = · · · = ξjs−k . A level α simultaneous test of H1 and H2 is given by the rejection region (1) (2) (1) (2) ni (Xi· − X·· )2 /(k − 1) ni (Xi· − X·· )2 /(s − k − 1) > C, > C, S 2 /(n − s) S 2 /(n − s)   (1) (2) averaging extends where (1) , (2) , X·· , X·· indicate that the summation or   over the sets (i1 , . . . , ik ) and (j1 , . . . , js−k ) respectively, S 2 = (Xij − Xi· )2 , α = 1 − γ, and the constant C is given by (9.89) with r = s and is therefore the same as in (7.19), rather than being determined by the Fk−1,n−s and Fs−k−1,n−s distributions. The reason for this larger critical value is, of course, the fact the H1 and H2 were suggested by the data. The present procedure is an example of Gabriel’s simultaneous test procedure mentioned in Section 9.3. Example 9.5.2 (Two-way layout) As a second example, consider first the additive model in the two-way classification of Section 7.4 or 7.5, and then the more general interaction model of Section 7.5. Suppose Xij are independent N (ξij , σ 2 ) (i = 1, . . . , a; j  = 1, . . . , b), with ξij given by (7.20), and let V be the space of all linear functions wi αi = wi (ξi· − ξ·· ). As was seen in Section 7.4, s = a + b − 1. To determine r, note that V can   also be represented as i=1 wi ξi· with wi = 0 [Problem 9.27(i)], which shows that r = a − 1. The least-squares estimators ξˆi were found in Section 7.4 to be  ξˆij = Xi· + X·j − X·· , so that ξˆi· = Xi· and S 2 = (Xij − Xi· − X·j + X·· )2 . The simultaneous confidence intervals (9.92) therefore can be written as 1 1    wi2 wi2 wi Xi· − cS wi Xi· + cS ≤ wi ξi· ≤ b b a  wi = 0. for all w with i=1 If there are m observations in each cell, and the model is  additive   as before, the (Xijk − Xi·· − only changes required are to replace Xi· by Xi·· , S 2 by  X·j· + X··· )2 , and the expression under the square root by wi2 /bm. 384 9. Multiple Testing and Simultaneous Inference Let us now drop the assumption of additivity and consider the general linear model ξijk = µ + αi + βj + γij , with µ and the α’s, β’s, and γ’s defined as in Section 7.5. The dimension s of Ω is then ab, and the least squares estimators of the parameters were seen in Section 7.5 to be µ̂ = X··· , α̂i = Xi·· − X··· , β̂j = X·j· − X··· , γ̂ ij = Xij· − Xi·· − X·j· + X···    The simultaneous intervals for all wi αi , or for all w wi = 0, are i ξi·· with 2 therefore unchanged except for the replacement of S = (X ijk − Xi·· − X·j· +  X··· )2 by S 2 = (Xijk − Xij· )2 and of n − s = n − a − b + 1 by n − s = n − ab = (m − 1)ab in (9.89). Analogously, onecan obtain simultaneous confidence intervalsfor the totality of linear functions wij γ wij ξij· for the ij , or equivalently the set of functions  totality of w’s satisfying i wij = j wij = 0 [Problem 9.27(ii), (iii)]. Example 9.5.3 (Regression line) As a last example consider the problem of obtaining confidence bands for a regression line, mentioned at the beginning of the section. The problem was treated for a single value t0 in Section 5.6 (with a different notation) and in Section 7.6. The simultaneous confidence intervals in the present case become  1/2 (t − t̄)2 1 α̂ + β̂t − cS ≤ α + βt (9.97) + n (ti − t̄)2  1/2 (t − t̄)2 1 , + ≤ α̂ + β̂t + cS n (ti − t̄)2 where α̂ and β̂ are given by (7.23),    (ti − t̄)2 (Xi − α̂ − β̂ti )2 = (Xi − X̄)2 − β̂ 2 S2 = and c is determined by (9.89) with r = s = 2. This is the Working–Hotelling confidence band for a regression line. At the beginning of the section, the Scheffé intervals were derived as the only confidence bands that are equivariant under the indicated groups. If the requirement of equivariance (particular under orthogonal transformations) is dropped, other bounds exist which are narrower for certain sets of vectors u at the cost of being wider for others [Problems 9.26(iii) and 9.32]. A general method that gives special emphasis to a given subset is described by Richmond (1982). Some optimality results not requiring equivariance but instead permitting bands which are narrower for some values of t at the expense of being wider for others are provided, among others, by Bohrer (1973), Cima and Hochberg (1976), Richmond (1982), Naiman (1984a,b), and Piegorsch (1985a, b). If bounds are required only for a subset, it may be possible that intervals exist at the prescribed confidence level, which are uniformly narrower than the Scheffé intervals. This is the case for example for the intervals (9.97) when t is restricted to a given finite interval. For a discussion of this and related problems, and references to the literature, see for example Wynn and Bloomfield (1971) and Wynn (1984). 9.6. Problems 385 9.6 Problems Section 9.1 Problem 9.1 Show the Bonferroni procedure, while generally conservative, can have FWER = α by exhibiting a joint distribution for (p̂1 , . . . , p̂s ) and satisfying (9.4) such that P {mini p̂i ≤ α/s} = α. Problem 9.2 (i) Under the assumptions of Theorem 9.1.1, suppose also that the p-values are mutually independent. Then, the procedure which rejects any Hi for which p̂i < c(α, s) = 1 − (1 − α)1/s controls the FWER. (i) Compare α/s with c(α, s) and show lim s→∞ − log(1 − α) c(α, s) = . (α/s) α For α = .05, this limiting value to 3 decimals is 1.026, so the increase in cutoff value is not substantial. Problem 9.3 Show that, under the assumptions of Theorem 9.1.2, it is not possible to increase any of the critical values αi = α/(s − i + 1) in the Holm procedure (9.6) without violating the FWER. Problem 9.4 Under the assumptions of Theorem 9.1.2 and independence of the p-values, the critical values α/(s − i + 1) can be increased to 1 − (1 − α)1/(s−i+1) . For any i, calculate the limiting value of the ratio of these critical values, as s → ∞. Problem 9.5 In Example 9.1.4, verify that the stepdown procedure based on √ the maximum of Xj / σj,j improves upon the Holm procedure. By Theorem 9.1.3, the procedure has FWER ≤ α. Compare the two procedures in the case σi,i = 1, σi,j = ρ if i = j; consider ρ = 0 and ρ → ±1. Problem 9.6 Suppose Hi is specifies the unknown probability P belongs to a subset of the parameter space ωi , for > i = 1, . . . , s. For any K ⊂ {1, . . . , k}, let HK be the intersection hypothesis P ∈ j∈K ωj . Suppose φK is level α for testing HK . Consider the multiple testing procedures that rejects Hi if φK rejects HK whenever i ∈ K. Show, the FWER ≤ α. [This method of constructing tests that control the FWER is called the closure method of Marcus, Peritz and Gabriel (1976).] Problem 9.7 As in Procedure 9.1.1, suppose that a test of the individual hypothesis Hj is based on a > test statistic Tn,j , with large values indicating evidence against the Hj . Assume sj=1 ωj is not empty. For any subset K of {1, . . . , s}, let cn,K (α, P ) denote an α-quantile of the distribution of maxj∈K Tn,j under P . 386 9. Multiple Testing and Simultaneous Inference Concretely, cn,K (α, P ) = inf{x : P {max Tn,j ≤ x} ≥ α} . j∈K (9.98) For testing the intersection hypothesis HK , it is only required to approximate a > critical value for P ∈ j∈K ωj . Because there may be many such P , we define 2 cn,K (1 − α) = sup{cn,K (1 − α, P ) : P ∈ ωj } . (9.99) j∈K (i) In Procedure 9.1.1, show that the choice ĉn,K (1 − α) = cn,K (1 − α) controls the FWER, as long as (9.9) holds. (ii) Further assume that for every subset K ⊂ {1, . . . , k}, there exists a distribution PK which satisfies cn,K (1 − α, P ) ≤ cn,K (1 − α, PK ) (9.100) for all P such that I(P ) ⊃ K. Such a P> K may be referred to being least favorable among distributions P such that P ∈ j∈K ωj . (For example, if Hj corresponds to a parameter θj ≤ 0, then intuition suggests a least favorable configuration should correspond to θj = 0.) In addition, assume the subset pivotality condition of Westfall and Young (1993); that is, assume there exists a P0 with I(P0 ) = {1, . . . , s} such that the joint distribution of {Tn,j : j ∈ I(PK )} under PK is the same as the distribution of {Tn,j : j ∈ I(PK )} under P0 . This condition says the (joint) distribution of the test statistics used for testing the hypotheses Hj , j ∈ I(PK ) is unaffected by the truth or falsehood of the remaining hypotheses (and therefore we assume all hypotheses are true by calculating the distribution of the maximum under P0 ). Show we can use ĉn,K (1 − α, P0 ) for ĉn,K (1 − α). (iii) Further assume the distribution of (Tn,1 , . . . , Tn,s ) under P0 is invariant under permutations (or exchangeable). Then, the critical values ĉn,K (1 − α) can be chosen to depend only on |K|. Problem 9.8 Rather than finding multiple tests that control the FWER, consider the k-FWER, the probability of rejecting k or more false hypotheses. For a given k, if there are s hypotheses, consider the procedure that rejects any hypothesis whose p-value is ≤ kα/s. Show that the resulting procedure controls the k-FWER. [Additional stepdown procedures that control the number of false rejections, as well as the probability that the proportion of false rejections exceeds a given bound, are obtained in Lehmann and Romano (2005).] Problem 9.9 In general, show that F DR ≤ F W ER, and equality holds when all hypotheses are true. Therefore, control of the FWER at level α implies control of the FDR. Section 9.2 Problem 9.10 . Suppose (X1 , . . . , Xk )T has a multivariate c.d.f. F (·). For θ ∈ IRk , let Fθ (x) = F (x − θ) define a multivariate location family. Show that (9.15) is satisfied for this family. (In particular, it holds if F is any multivariate normal distribution.) 9.6. Problems 387 Problem 9.11 Prove Lemma 9.2.2. Problem 9.12 We have suppressed the dependence of the critical constants C1 , . . . , Cs in the definition of the stepdown procedure D, and now more accurately call them Cs,1 , . . . , Cs,s . Argue that, for fixed s, Cs,j is nonincreasing in j and only depends on s − j. Problem 9.13 Under the assumptions of Theorem 9.2.1, suppose there exists another monotone rule E that strongly controls the FWER, and such that Pθ {dc0,0 } ≤ Pθ {ec0,0 } c for all θ ∈ ω0,0 , (9.101) with strict inequality for some θ ∈ Argue that the ≤ in (9.101) is an equality, and hence e0,0 d0,0 has Lebesgue measure 0, where AB denotes the symmetric difference between sets A and B. A similar result for the region d1,1 can be made as well. c ω0,0 . Problem 9.14 In general, the optimality results of Section 9.2 require the procedures to be monotone. To see why this is required, consider 9.2.2 (i). Show the procedure E to be inadmissible. Hint: One can always add large negative values of T1 and T2 to the region u1,1 without violating the FWER. Problem 9.15 Prove part (i) of Theorem 9.2.3. Problem 9.16 In general, show Cs = C1∗ . In the case s = 2, show (9.27). Section 9.3 Problem 9.17 Show that r+1  i=1 Yi − Y1 + · · · + Yr+1 r+1 2 − r  i=1 Yi − Y1 + · · · + Yr r 2 ≥ 0. Problem 9.18 (i) For the validity of Lemma 9.3.1 it is only required that the probability of rejecting homogeneity of any set containing {µi1 , . . . , µiv1 } as a proper subset tends to 1 as the distance between the different groups (9.48) all → ∞, with the analogous condition holding for H2 , . . . , Hr . (ii) The condition of part (i) is satisfied for example if homogeneity of a set S  is rejected for large values of |Xi· − X·· |, where the sum extends over the subscripts i for which µi ∈ S. Problem 9.19 In Lemma 9.3.2, show that αs−1 admissibility. = α is necessary for Problem 9.20 Prove Lemma 9.3.3 when s is odd. Problem 9.21 Show that the Tukey levels (vi) satisfy (9.54) when s is even but not when s is odd. 388 9. Multiple Testing and Simultaneous Inference Problem 9.22 The Tukey T -method leads to the simultaneous confidence intervals CS for all i, j. (9.102) |(Xj· − Xi· ) − (µj − µi )| ≤ sn(n − 1) [The probability of (9.102) is independent of the µ’s and hence equal to 1 − αs .] Section 9.4 Problem 9.23 (i) A function L satisfies the first equation of (9.65) for all u, x, and orthogonal transformations Q if and only if it depends on u and x only through u x, x x, and u u. (ii) A function L is equivariant under G2 if and only if it satisfies (9.67). Problem 9.24 (i) For the confidence sets (9.73), equivariance under G1 and G2 reduces to (9.74) and (9.75) respectively.  (ii) For fixed ui yi ∈ A hold for all (u1 , . . . , ur ) 1 , . . . , yr ), the statements  (y 2 with  ui =1 if and only if A contains the interval I(y) = [− Yi2 , + Yi2 ]. (iii) Show that the statement following (9.77) ceases to hold when r = 1. Problem 9.25 Let Xi (i = 1, . . . , r) be independent N (ξi , 1). (i) The only simultaneous confidence intervals equivariant under G0 are those given by (9.83). (ii) The inequalities (9.83) and (9.85) are equivalent.  (iii) Compared with the uj ξj Scheffé intervals (9.72), the intervals (9.85) for are shorter when uj ξj = ξi and longer when u1 = · · · = ur .  ui yi is maximized subject to |yi | ≤ ∆ for all [(ii): For a fixed u = (u1 , . . . , ur ), i, by yi = ∆ when ui > 0 and yi = −∆ when ui < 0.] Section 9.5  Problem 9.26 (i) The confidence intervals L(u; y, S) = ui yi − c(S) are equivariant under G3 if and only if L(u; by, bS) = bL(u; y, S) for all b > 0. (ii) The most general confidence sets (9.90) which are equivariant under G1 , G2 , and G3 are of the form (9.91).  Problem (i) In Example 9.5.2, the set of linear functions wi αi =  9.27 w i (ξi· − ξ·· ) for all w can also be represented as the set of functions   wi ξi· for all w satisfying wi = 0.   (ii) The set of linear functions wij γij = wij (ξij· − ξi·· − ξ ·j· + ξ··· )  for all w is equivalent to the set w ij ξij· for all w satisfying i wij =  w = 0. ij j 9.6. Problems 389 (iii) Determine the simultaneous confidence intervals (9.92) for the set of linear functions of part (ii). Problem 9.28 (i) In Example 9.5.1, the simultaneous confidence intervals (9.92) reduce to (9.96). (ii) What change is needed in the confidence intervals of Example 9.5.1 if the v’s are not required to satisfy (9.95), i.e.,  if simultaneous confidence intervals are desired for all linear functions vi ξi instead of all contrasts? Make a table showing the effect of this change for s = 2, 3, 4, 5; ni = n = 3, 5, 10. Problem 9.29 Tukey’s T -Method. Let Xi (i = 1, . . . , r) be independent N (ξi , 1), and consider simultaneous confidence intervals L[(i, j); x] ≤ ξj −