STA 304H1F-1003H Fall 2019
Assignment 2-Question 4 Question 1. (5 marks)
Consider a stratified design composed of H strata of size Nh, h = 1, . . . , H. We want to estimate the population mean μy of the characteristic y. Let μx,h, h = 1, . . . , H be the means in the strata (in the population) of an auxiliary characteristic x. The μx,h are supposedly known and we propose to estimate μy using the following estimator:
μD = yst + μx − xst
where yst and xst are the basic estimate of the population means μy and μx for y and x,
respectively.
(a) (1 mark) Give an expression of μx in terms of μx,h, h = 1,…,H.
(b) (1 mark) Show that μD is unbiased estimates for μy.
(c) (1 mark) Give the variance of μD
(d) (1 mark) Let n = Hh=1 nh be the sample size. What is the optimal allocation of the nh in order to minimise the variance of μD? We consider that the init cost of the survey does not depend on the stratum.
(e) (1 mark) In which favourable case is μD preferable to yst?
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