Minimax Estimation of a Normal Mean Vector for Arbitrary Quadratic Loss and Unknown Covariance Matrix

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqq 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\not\Sigma$. It is desired to estimate $\theta$ under the quadratic loss $L(\delta, \theta, \not\Sigma) = (\delta - \theta)^tQ(\delta - \theta)/\operatorname{tr} (Q\not\Sigma)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta^c(X, W) = (I - c\alpha Q^{-1}W^{-1}/(X^tW^{-1}X))X,$ where $W$ is a $p \times p$ random matrix with a Wishart $(\not\Sigma, n)$ distribution (independent of $X$), $\alpha$ is the minimum characteristic root of $(QW)/(n - p - 1)$ and $c$ is a positive constant. For appropriate values of $c, \delta^c$ is shown to be minimax and better than the usual estimator $\delta^0(X) = X$.