On the Convergence Rates of Empirical Bayes Rules for Two-Action Problems: Discrete Case

Abstract

The purpose of this paper is to investigate the convergence rates of a sequence of empirical Bayes decision rules for the two-action decision problems where the distributions of the observations belong to a discrete exponential family. It is found that the sequence of the empirical Bayes decision rules under study is asymptotically optimal, and the order of associated convergence rates is $O(\exp(-cn))$, for some positive constant $c$, where $n$ is the number of accumulated past experience (observations) at hand. Two examples are provided to illustrate the performance of the proposed empirical Bayes decision rules. A comparison is also made between the proposed empirical Bayes rules and some earlier existing empirical Bayes rules.