An empirical Bayes estimation problem

Let x be a random variable such that, given theta, x is Poisson with mean theta, while theta has an unknown prior distribution G. In many statistical problems one wants to estimate as accurately as possible the parameter E(thetax = a) for some given a = 0,1,.... If one assumes that G is a Gamma prior with unknown parameters alpha and beta, 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.