Analytical expressions for Rao–Blackwell estimators in adaptive cluster sampling

Abstract

Abstract In this paper, we consider adaptive cluster sampling with the initial sample selected either by simple random sampling without replacement or by unequal probability sampling with replacement. We derive analytical expressions for the Rao–Blackwell versions of the following two estimators originally proposed by Thompson (1990, J. Amer. Statist. Assoc. 85, 1050–1059): the modified Horvitz–Thompson estimator and the modified Hansen–Hurwitz one. Furthermore, we derive analytical expressions for the variances of the two Rao–Blackwell estimators above mentioned, as well as analytical expressions for unbiased Rao–Blackwell estimators of those variances. In the without-replacement case, the derivations are based on the multivariate hypergeometric distribution, whereas in the with-replacement case, they are based on the multinomial distribution.