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

Abstract

The problem of finding classes of estimators which dominate the usual estimator $X$ of the mean vector $\mu$ of a $p$-variate normal distribution $(p \geq 3)$ under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of $\mu$ in this case depend upon $X$ and an independent Wishart matrix $W$. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to yield unbiased estimates of risk difference (versus $X$) for certain classes of estimators, defined indirectly through a "seed" function $h(X, W)$. An application of this technique produces a new class of minimax estimators of $\mu$.