Constrained Minimax Estimation of the Mean of the Normal Distribution with Known Variance

Abstract

In this paper we shall discuss the estimation of the mean of a normal distribution with variance 1. The main question in this work is the existence and computation of a least favorable distribution among all the prior distributions satisfying a given set of constraints. In the following we show that if this distribution is bounded from above on some even moment, then the least favorable distribution exists and it is either normal or discrete. The support of the discrete distribution function does not have any accumulation point. The least favorable distribution is normal if and only if the second moment is bounded from above, without any other relevant constraint. These theorems shed light on the James-Stein estimator as the minimax estimator for a prior with unknown bounded variance.