ADJUSTED BAYES AND EMPIRICAL BAYES ESTIMATION IN FINITE POPULATION-SAMPLING

The article considers estimation of several strata means where each stratum contains a finite number of elements. Let gamma-i denote the i-th stratum mean (i = 1, ..., m). Our object is to find an estimator-gamma-triple-over-dot = (gamma-triple-over-dot-1, .., gamma-triple-over-dot-m) of gamma = (gamma-1, .., gamma-m) which minimizes the Bayes [GRAPHICS] In the above, 'E' represents the expectation taken over both the prior and the sample. The main purpose of this procedure is to make the empirical cumulative distribution function (c.d.f.) of estimates of the parameter ensemble close to the c.d.f. of the unobserved parameters. Throughout the paper, we call the conditions (i) and (ii) the "closeness requirements" and the resulting optimal estimator-gamma-triple-over-dot the "adjusted" Bayes estimator of gamma. We show that the standard Bayes estimator of gamma does not satisfy the second closeness requirement and thus it shrinks "too far towards the prior mean". We propose the "adjusted" Bayes estimator of gamma which corrects this "overshrinking". Adjusted empirical Bayes estimators when all the prior parameters are unknown are also proposed and their optimality properties are studied. The absence of normality assumption in motivating the estimators as well as deriving their properties is interesting.