Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix

Abstract

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ based on $\mathbf{X}$ and $\mathbf{A}$ through $\mathbf{X}$ and $\mathbf{X}'\mathbf{A}^{-1}\mathbf{X}$. It is shown that there are minimax estimators of the form $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ which are also generalized Bayes. A special case where $\Sigma = \sigma^2\mathbf{I}$ is also considered.