Empirical Bayes Estimation of the Multivariate Normal Covariance Matrix

Abstract

Let $\\mathbf{S}_{p \\times p}$ have a Wishart distribution with scale matrix $\\Sigma$ and $k$ degrees of freedom. Estimators of $\\Sigma$ are given for each of the loss functions $L_1(\\hat{\\Sigma}, \\Sigma) = \\operatorname{tr} (\\hat{\\Sigma}\\Sigma^{-1}) - \\log \\det (\\hat{\\Sigma}\\Sigma^{-1}) - p$ and $L_2(\\hat{\\Sigma}, \\Sigma) = \\operatorname{tr} (\\hat{\\Sigma}\\Sigma^{-1} - I)^2$. The obvious estimators of $\\Sigma$ are the scalar multiples of $\\mathbf{S}$, i.e., $a\\mathbf{S}$ where $0 < a \\leqslant 1/k$. (Recall that $(1/k)\\mathbf{S}$ is unbiased.) For each problem $(\\Sigma, \\hat{\\Sigma}, L_i), i = 1, 2$, we provide empirical Bayes estimators which dominate $a\\mathbf{S}$ by a substantial amount. It is seen that the uniform reduction in the risk function determined by $L_2$ is at least $100(p + 1)/(k + p + 1){\\tt\\%}$. Dominance results for $L_1$ and $L_2$ were first given by James and Stein.