Empirical and hierarchical bayes competitors of preliminary test estimators in two sample problems

We consider the problem of estimation of [mu]1 when it is suspected that [mu]1 [approximate] [mu]2 based on independent samples from Np([mu]1, [sigma]2V1) and Np([mu]2, [sigma]2V2). We assume V1, V2 known but [sigma]2 unknown. First, the EB estimator is derived and its Bayesian and frequentist properties are studied. Second, a modified EB estimator is proposed and shown to dominate a preliminary test estimator. Finally, a hierarchical Bayes approach is proposed as an alternative to EB estimators.