Proper Bayes Minimax Estimators of the Multivariate Normal Mean Vector for the Case of Common Unknown Variances

Abstract

We investigate the problem of estimating the mean vector $\mathbf{\theta}$ of a multivariate normal distribution with covariance matrix equal to $\sigma^2\mathbf{I}_p, \sigma^2$ unknown, and loss $\|\delta - \mathbf{\theta}\|^2/\sigma^2$. We first find a class of minimax estimators for this problem which enlarges a class given by Baranchik. This result is then used to show that for sufficiently large sample sizes (which never need exceed 4) proper Bayes minimax estimators exist for $\mathbf{\theta}$ if $p \geqq 5$.