We obtain a limit of a hierarchical Bayes estimator of a finite population mean when the sample size is large. The limit is in the sense of ordinary calculus, where the sample observations are treated as fixed quantities. Our result suggests a simple way to correct the hierarchical Bayes estimator to achieve design-consistency, a well-known property in the traditional randomization approach to finite population sampling. We also suggest three different measures of uncertainty of our proposed estimator.
"On the design-consistency property of hierarchical Bayes estimators in finite population sampling." Ann. Statist. 35 (2) 724 - 737, April 2007. https://doi.org/10.1214/009053606000001262