Statistical Theory Assignment

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Homework Sheet 3 Basic Questions 1. Let X be a random sample from a binomial distribution with n = 100 and p unknown. (a) Show that the maximum likelihood estimate for p is unbiassed. (b) The variance of X is 100p(1 − p). Find the bias of the maximum likelihood estimate for this variance. 2. Show that X+1 n+2 is a biased estimator of the binomial parameter θ. Is this estimator asymptotically unbiased? 3. Let Ymin be the smallest order statistic in a random sample of size n drawn from the uniform pdf, fY (y; θ) = 1/θ, 0 ≤ y ≤ θ. Find an unbiased estimator for θ based on Ymin . 4. A random sample of size 2, Y1 and Y2 , is drawn from the pdf fY (y; θ) = 2yθ2 , 0 < y < 1/θ. What must c equal if the statistic c(Y1 + 2Y2 ) is to be an unbiased estimator for 1/θ? 5. Let X1 , . . . , Xn be a random sample from a N (0, σ 2 ) distribution. We want to the standard deviation σ. Find the constant c so that Pestimate n Y = c i=1 |Xi | is an unbiased estimator for σ and determine its efficiency (efficiency is defined as 1/var(Y )). 1 ...
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