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Biometrika 1977 64(1):97-103; doi:10.1093/biomet/64.1.97
© 1977 by Biometrika Trust
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Two-stage sampling with exchangeable prior distributions

D. R. BELLHOUSE, M. E. THOMPSON and V. P. GODAMBE

Social Science Computer Laboratory, University of Western Ontario
Department of Statistics, University of Waterloo Ontario

Consider a two-stage population of K primary units, consisting respectively of M1,...,MK secondary units. Denote the characteristic values by yij. Assume a class of priors under which yij for fixedd i are the first Mi elements of the ith row of a rectangular array such that (i) the row vectors are exchangeable, and (ii) elements within rows are independently exchangeable. It is shown that among two-stage designs with specified fixed sample sizes at each stage, a design having inclusion probabilities proportional to Mi at the iimt stage and equal within primaries at the second stage is optimal for estimation of the population total. The unbiased estimator having smallest expected variance is {Sigma}Mi times the sample mean.

Key Words: Completeness • Finite population • Multistage sampling • Order statistics • Sample mean • Symmetric prior


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