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as094-soln - STAT 854/454 Short Solutions to Assignment 4 1...

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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 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 . Soln: Re-write ˆ t L as ˆ 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 ( ˆ t L ) = N i =1 π i w i y i and V ( ˆ t L ) = N i =1 π i (1 - π i ) w 2 i y 2 i +2 N - 1 i =1 N j = i +1 ( π ij - π i π 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 ˆ t L . Soln: To make E ( ˆ t L ) = N i =1 π i w i y i = t y = N i =1 y i for an arbitrary y variable, we must have π i w i = 1, i.e. w i = 1 i .
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