Improving on the MLE of a bounded normal mean

Abstract

We consider the problem of estimating the mean of a $p$-variate normal distribution with identity covariance matrix when the mean lies in a ball of radius $m$. It follows from general theory that dominating estimators of the maximum likelihood estimator always exist when the loss is squared error. We provide and describe explicit classes of improvements for all problems $(m, p)$. We show that,for small enough $m$, a wide class of estimators, including all Bayes estimators with respect to orthogonally invariant priors, dominate the maximum likelihood estimator. When $m$ is not so small, we establish general sufficient conditions for dominance over the maximum likelihood estimator. These include, when $m \le \sqrt{p}$, the Bayes estimator with respect to a uniform prior on the boundary of the parameter space. We also study the resulting Bayes estimators for orthogonally invariant priors and obtain conditions of dominance involving the choice of the prior. Finally, these Bayesian dominance results are further discussed and illustrated with examples, which include (1) the Bayes estimator for a uniform prior on the whole parameter space and (2) a new Bayes estimator derived from an exponential family of priors.