Empirical Bayes and conditional inference with many nuisance parameters

Abstract

SUMMARY This paper concerns the use of empirical Bayes methods to improve the efficiency of a parameter of interest, θ, in the presence of many nuisance parameters, {ø i }, one from each data stratum. A class of distributions is introduced such that for fixed θ, the minimal sufficient statistic t i (θ) for ø i is from an exponential family and hence complete. By imposing the assumption that the ø i 's are generated from an unspecified distribution function Q, we show that for this class of distributions the conditional score function for θ (Lindsay, 1982) with oslhas i estimated by E Q {oslhas i /t i (θ)} is optimal in a sense similar to that of Cox & Reid (1987) and Liang (1987). Two empirical Bayes estimates of ø, along with the maximum likelihood estimate of ø i are compared through simulations in terms of θ estimation.