On empirical Bayes procedures for selecting good populations in a positive exponential family

This paper deals with the problem of selecting good populations with respect to a control from k(⩾2) populations. The random variable associated with population πi is assumed to be positive-valued and has density of form f(xi|θi)=c(θi)exp(-xi/θi)h(xi) with unknown parameter θi, for each i=1,…,k. A nonparametric empirical Bayes approach is used to construct the selection procedures. The performance of the procedures is investigated by studying their regret Bayes risks rn. Under the simple assumption on the prior Gi of ∫θ3Gi(θ)<∞, we show that nrn→m, where m is a constant that can be calculated explicitly. The result indicates that the rate O(n-1) can be easily obtained for nonparametric procedures under weak conditions.