On the Asymptotic Behaviour of Empirical Bayes Tests for the Continuous One-Parameter Exponential Family

Abstract

Several authors have proposed empirical Bayes tests (EBT) for the continuous one-parameter exponential family for the case that the prior distribution is completely unspecified. They investigated the convergence rate of the (unconditional) Bayes risk, and gave upper bounds for this convergence rate. In this paper it is proposed to study the convergence of the conditional Bayes risk. A method is presented which makes it possible to derive the exact convergence rate of the conditional risk and its limit distribution. Several results are given. Also the question is considered whether monotonizing an empirical Bayes test influences its asymptotic properties.