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.