On Equivariance and the Compound Decision Problem

Abstract

This paper obtains some extensions of Gilliland and Hannan's results on equivariance and the compound decision problem. Consider a compound decision problem with restricted component risk and component distributions in a norm compact set of mutually absolutely continuous probability measures. Then the method of proof of a theorem of Gilliland and Hannan translates the results of Mashayekhi on symmetrization of product measures into uniform convergence to zero of the excess of the simple envelop over the equivariant envelope. Our envelope results strengthen, among other things, the results of Datta who obtained admissible asymptotically optimal solutions to the compound estimation problem for a large subclass of the real one parameter exponential family under squared error loss. Sufficient conditions for asymptotic optimality of "delete bootstrap" rules are given and, for squared error loss estimation of continuous functions and for finite action space problems with continuous loss functions, the problem of treating the asymptotic excess compound risk of Bayes compound rules is reduced to the question of $L_1$-consistency of certain mixtures. Examples of estimates satisfying the above consistency condition are provided.