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Sampling solved exercises 2 sampling with unequal probabilites

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3 Sampling with Unequal Probabilities 3.1 Calculation of inclusion probabilities If we have an auxiliary variable xk > 0, k ∈ U , ‘sufficiently’ proportional to the variable yk , it is often interesting to select the units with unequal probabilities proportional to xk . To do this, we first calculate the inclusion probabilities according to xk (3.1) πk = n  . x ∈U If Expression (3.1) gives πk > 1, the corresponding units are selected in the sample (with an inclusion probability equal to 1), and we then recalculate the πk according to (3.1) on the remaining units. 3.2 Estimation and variance The Horvitz-Thompson estimator of the total is  yk Yπ = , πk k∈S and its variance is: var(Yπ ) =   yk y ∆k , πk π k∈U ∈U where ∆k = πk − πk π , and πk is the second-order inclusion probability. If k = , then πkk = πk . To obtain a positive estimate of the variance (see page 4), a sufficient constraint is to have ∆k ≤ 0 for all k =  in U. This constraint is called the Sen-Yates-Grundy constraint. There exist several algorithms that allow for the selection of units with unequal probabilities. Two books give a brief overview of such methods: Brewer 60 3 Sampling with Unequal Probabilities and Hanif (1983) and Gabler (1990). The most well-known methods are systematic sampling (Madow, 1948), sampling with replacement (Hansen and Hurwitz, 1943), the method of Sunter (1977) and Sunter (1986). As well, the method of Brewer (197 ...
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