Two-stage sampling with exchangeable prior distributions

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 ΣMi times the sample mean.