Multivariate Empirical Bayes and Estimation of Covariance Matrices

Abstract

The problem of estimating several normal mean vectors in an empirical Bayes situation is considered. In this case, it reduces to the problem of estimating the inverse of a covariance matrix in the standard multivariate normal situation using a particular loss function. Estimators which dominate any constant multiple of the inverse sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as the James-Stein estimator for a mean vector shrinks the maximum likelihood estimators toward a common value. These covariance estimators then lead to a class of multivariate estimators of the mean, each of which dominates the maximum likelihood estimator.