Empirical Bayes Selections under Multicriteria

Abstract

Suppose there are normal populations each with unknown mean and unknown variance. We consider two problems in this paper. The first, for some given values of s and t, we are interested in selecting the one whose mean is between s and s+t'(t'>0) and simultaneously its mean is closest to t, otherwise, none is selected. The second, for given values of s, a square of c, and t'(t'> 0), we are interested in selecting the one whose mean deviates from s within t' and those variance is no large than a square of c and simultaneously its mean is closest to s. The problem is formulated in a Bayes set up. Empirical Bayes rules are derived and they have been shown to be asymptotically optimal.