The Best Uniform Convergence Rate of Linear Empirical Bayes Estimators.

In this paper the thought on the uniform convergence of an empirical Bayes estimator or a linear empirical Bayes (L.e.B.) estimator is advanced. Under two different models the Le.B. estimators of the parameter are constructed respectively. It is proved that the conver gence rates and uniform convergence rates of these l.e.B. estimators are all one with respect to the corresponding prior families. It is shown that the uniform convergence rate one in the best, under mild assumptions imposed on the conditional density of the sample.