Asymptotic Optimality of Bayes Compound Estimators in Compact Exponential Families

Abstract

The problem of finding admissible, asymptotically optimal compound rules is pursued in the infinite state case. The components involve the estimation of an arbitrary continuous transform of the natural parameter of a real exponential family with compact parameter space. We show that all Bayes estimators are admissible. Our main result is that any Bayes compound estimator versus a mixture of i.i.d. priors on the compound parameter is asymptotically optimal if the mixing hyperprior has full support. The asymptotic optimality results are generalized to weighted squared error loss with continuous weight function and applications to some nonexponential situations are also considered. Several examples of such hyperpriors are given and for some of them practically useful forms of the corresponding Bayes estimators are obtained.