An empirical Bayes estimation problem

Abstract

Let x be a random variable such that, given θ, x is Poisson with mean θ, while θ has an unknown prior distribution G . In many statistical problems one wants to estimate as accurately as possible the parameter E (θǀ x = a ) for some given a = 0,1,.... If one assumes that G is a Gamma prior with unknown parameters α and β, then the problem is straightforward, but the estimate may not be consistent if G is not Gamma. On the other hand, a more general empirical Bayes estimator will always be consistent but will be inefficient if in fact G is Gamma. It is shown that this dilemma can be more or less resolved for large samples by combining the two methods of estimation.