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Unformatted text preview: STAT 854/454 Sampling Theory and Practice Assignment 4 Due: Wednesday, March 11, in class Format: standard A4 size paper, single-sided, with a cover page 1. Let t y = ∑ N i =1 y i be the population total. Consider linear estimators of t y in the form of ˆ t L = ∑ i ∈ s w i y i , where s is the set of sampled units and w i is a constant to be used as a weight for the i th element whenever it is selected for the sample ( i = 1 , 2 , ··· ,N ). (a) Find E ( ˆ t L ) and V ( ˆ t L ) under a general sampling design using the π i and π ij . (b) Argue that the Horvitz-Thompson estimator is the only unbiased estimator in the class of linear estimators in the form of ˆ t L . (c) ( STAT 854 ONLY ) Show that for any given sampling design there exists an unbiased estimator for the population total t y = ∑ N i =1 y i if and only if π i = P ( i ∈ s ) > 0 for i = 1 , 2 , ··· ,N . 2. Midzuno’s method for unequal probability sampling: Select the first element with prob- ability q...
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This note was uploaded on 11/23/2010 for the course STAT 454 taught by Professor Da during the Fall '09 term at Waterloo.
- Fall '09