Estimation of the Mixing Distribution for a Normal Mean with Applications to the Compound Decision Problem

Abstract

A procedure for estimating the mixing distribution for a normal mean is presented. The estimator is shown to be consistent regardless of the mixing distribution, and it is suggested that the estimation techniques may be applied in a similar manner to estimate mixing distributions for a wide class of location parameter families. Implied marginal density and derivative estimates for the normal case are shown to be consistent, converging at near-optimal rates. In addition, empirical Bayes rules for the quadratic loss mean estimation problem which are derived from the estimated mixing distributions are shown to be asymptotically optimal in average compound risk and Bayes risk (if it is defined), provided the means are assumed to lie within a fixed range. A brief discussion of computational issues related to this estimation procedure is also included, along with some small-sample simulation results for the compound decision problem.