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Unformatted text preview: STAT 854/454 Short Solutions to Assignment 4 1. Let t y = N i =1 y i be the population total. Consider linear estimators of t y in the form of 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 ( t L ) and V ( t L ) under a general sampling design using the i and ij . Soln: Re-write t L as t L = N i =1 a i w i y i , where a i = 1 if i s and a i = 0 otherwise. Can show that E ( t L ) = N i =1 i w i y i and V ( t L ) = N i =1 i (1- i ) w 2 i y 2 i +2 N- 1 i =1 N j = i +1 ( ij- i j ) w i w j y i y j . (b) Argue that the Horvitz-Thompson estimator is the only unbiased estimator in the class of linear estimators in the form of t L . Soln: To make E ( t L ) = N i =1 i w i y i = t y = N i =1 y i for an arbitrary y variable, we must have i w i = 1, i.e.= 1, i....
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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