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Handout4-W09 - -y j π j 2 = 1 2 X i ∈ s X j ∈ s π i...

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STAT 454/854 Winter 2009 Handout #4 Unequal Probability Sampling and the HT Estimator Let π i and π ij be respectively the first order and the second order inclusion probabilities under a general sampling design. The Horvitz-Thompson (HT) estimator for the population total t y = N i =1 y i is defined as ˆ t HT = X i s y i π i = X i s d i y i , where d i = 1 i are often referred to as the basic design weights. (1) The HT estimator is unbiased, i.e. E ( ˆ t HT ) = t y . (2) The theoretical variance of ˆ t HT is given by V ( ˆ t HT ) = N X i =1 1 - π i π i y 2 i + N X i =1 N X j 6 = i ( π ij - π i π j ) y i π i y j π j = N X i =1 N X j =1 ( π ij - π i π j ) y i π i y j π j . Here we used π ii = π i in the last expression. (3) An unbiased variance estimator is given by v ( ˆ t HT ) = X i s 1 - π i π 2 i y 2 i + X i s X j 6 = i π ij - π i π j π ij y i π i y j π j = X i s X j s π ij - π i π j π ij y i π i y j π j . (4) The Yates-Grundy-Sen variance formula when the sample size n is fixed. V ( ˆ t HT ) = N - 1 X i =1 N X j = i +1 ( π i π j - π ij ) y i π i - y j π j ! 2 = 1 2 N X i =1 N X j =1 ( π i π j - π ij ) y i π i - y j π j ! 2 v ( ˆ t HT ) = X i s X j>i π i π j - π ij π ij
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Unformatted text preview: -y j π j ! 2 = 1 2 X i ∈ s X j ∈ s π i π j-π ij π ij y i π i-y j π j ! 2 (5) Techniques useful for proofs: i) Let a i = 1 if i ∈ s and 0 otherwise, i = 1 , 2 , ··· ,N . Then E ( a i ) = π i , E ( a i a j ) = π ij for i 6 = j , V ( a i ) = π i (1-π i ), and Cov ( a i ,a j ) = π ij-π i π j . ii) To find out the expectation (or variance) of A = ∑ i ∈ s Z i and B = ∑ i ∈ s ∑ j ∈ s Z i Z j , we re-write A and B as A = N X i =1 a i Z i and B = N X i =1 N X j =1 a i a j Z i Z j . iii) When the sample size n is fixed, we have N X i =1 π i = n ; N X j 6 = i π ij = ( n-1) π i and N X i =1 N X j 6 = i π ij = n ( n-1) ....
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