Empirical Bayes rules and Gaussian processes

In this paper we study empirical Bayes (e.B.) rules from a viewpoint which has not yet got any attention in the literature. Since an e.B. estimator can be seen as an estimate of an unknown function, namely the true Bayes estimator, it is natural to consider e.B. estimators as stochastic processes. In this paper we make a first attempt in the direction of this approach. For a certain class of e.B. estimators for the continuous one-parameter exponential family, we investigate the global behaviour on finite intervals. It is shown that the difference between the e.B. and the true Bayes estimator can be represented as a certain type of Gaussian process plus a remainder which is uniformly of smaller order. Several applications of this result are given.