Robust Bayesian analysis of the binomial empirical Bayes problem

Empirical Bayes estimation is considered for an i.i.d. sequence of binomial parameters θ_i arising from an unknown prior distribution G(.). This problem typically arises in industrial sampling, where samples from lots are routinely used to estimate the lot fraction defective of each lot. Two related issues are explored. The first concerns the fact that only the first few moments of G are typically estimable from the data. This suggests consideration of the interval of estimates (e.g., posterior means) corresponding to the different possible G with the specified moments. Such intervals can be obtained by application of well-known moment theory. The second development concerns the need to acknowledge the uncertainty in the estimation of the first few moments of G. Our proposal is to determine a credible set for the moments, and then find the range of estimates (e.g., posterior means) corresponding to the different possible G with moments in the credible set.