Shrinkage estimation with a matrix loss function

Abstract

Consider estimating an $n\times p$ matrix of means $\Theta$, say, from an $n\times p$ matrix of observations $X$, where the elements of $X$ are assumed to be independently normally distributed with $E(x_{ij})=\theta_{ij}$ and constant variance, and where the performance of an estimator is judged using a $p\times p$ matrix quadratic error loss function. A matrix version of the James-Stein estimator is proposed, depending on a tuning constant $a$. It is shown to dominate the usual maximum likelihood estimator for some choices of $a$ when $n\geq 3$. This result also extends to other shrinkage estimators and settings.