10_unequalprob

10_unequalprob - Sampling with Unequal Probabilities...

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Unformatted text preview: Sampling with Unequal Probabilities Lecture 10 STAT 651: Survey Sampling Methods, Kaizar – p.1/8 Lecture 10: Sampling with Unequal Probabilities Reading: Lohr Chapter 6, sections 1 - 6 Connections to Standard Sampling Plans Horvitz-Thompson Estimation STAT 651: Survey Sampling Methods, Kaizar – p.2/8 Connections to Standard Sampling Plans STAT 651: Survey Sampling Methods, Kaizar – p.3/8 Example: Sample Clusters Proportional to Size STAT 651: Survey Sampling Methods, Kaizar – p.4/8 Horvitz-Thompson Estimation ˆ tHT = wi yi i∈S ˆ yHT = ¯ wi yi i∈S wi i∈S STAT 651: Survey Sampling Methods, Kaizar – p.5/8 Variance of HT Estimators One-Stage Cluster Sample: N ¯ V [tHT ] = i=1 1 = 2 1 − πi 2 ti + πi N N N i=1 k =i,k =1 N (πi πk − πik ) i=1 k =i,k =1 ˆ [tHT ] = 1 V¯ 2 (πi πk − πik ) i∈S k =i,k ∈S πik − πi πk ti tk πi πk ti ti − πi πk ti ti − πi πk 2 2 STAT 651: Survey Sampling Methods, Kaizar – p.6/8 Variance of HT Estimators Two-Stage Cluster Sample: N ¯ V [tHT ] = i=1 1 = 2 ˆ [tHT ] = 1 V¯ 2 1 − πi 2 ti + πi N N N i=1 k =i,k =1 N (πi πk − πik ) i=1 k =i,k =1 (πi πk − πik ) i∈S k =i,k ∈S πik − πi πk ti tk + πi πk ti ti − πi πk i=1 N 2 ti ti − πi πk N 2 + i=1 + i∈S ˆ V ti πi ˆ V ti πi ˆˆ V ti πi STAT 651: Survey Sampling Methods, Kaizar – p.7/8 Example: Random Digit Dialing STAT 651: Survey Sampling Methods, Kaizar – p.8/8 ...
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