A Robust Generalized Bayes Estimator and Confidence Region for a Multivariate Normal Mean

Abstract

It is observed that in selecting an alternative to the usual maximum likelihood estimator, $\delta^0$, of a multivariate normal mean, it is important to take into account prior information. Prior information in the form of a prior mean and a prior covariance matrix is considered. A generalized Bayes estimator is developed which is significantly better than $\delta^0$ if this prior information is correct and yet is very robust with respect to misspecification of the prior information. An associated confidence region is also constructed, and is shown to have very attractive size and probability of coverage.