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# Statement of a problem № m40604

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An alternative approach to linearization variance estimators. Demnati and Rao (2004) derive a unified theory for linearization variance estimation using weights. Let θ be the population quantity of interest, and define the estimator ˆθ to be a function of the vector of sampling weights and the population values: θ= g (w, y1, y2. . . yk ), Where w= (w1. . . wN)T with wi the sampling weight of unit i (wi =0 if i is not in the sample), and yj is the vector of population values for the jth response variable. Then a linearization variance estimator can be found by taking the partial derivatives of the function with respect to the weights. Let Evaluated at the sampling weights wi. Then we can estimate V (ˆθ) by For example, considering the ratio estimator of the population total, The partial derivative of ˆθ = g (w, x, y) with respect to wi is For an SRS, finding the estimated variance of ṫz gives (4.11). Consider the post stratified estimator in Exercise 17. - Write the estimator as ṫpost =g (w, y, x1. . . xL), where xli =1 if observation i is in post stratum l and 0 otherwise. - Find an estimator of V (ṫpost) using the Demnati–Rao (2004) approach.

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