Estimating the Mean of a Multivariate Normal Population with General Quadratic Loss Function

Abstract

Let $X$ be a $p$-variate random vector, $p \geqq 3$, having a normal distribution with mean vector $\theta$ and covariance matrix $\sigma^2I$. For estimating $\theta$ with loss function $L_1(\theta,\hat{\theta}) = (\theta - \hat{\theta})'(\theta - \hat{\theta}) = \|\theta - \hat{\theta}\|^2,$ and known $\sigma^2$, Stein [2] has shown the inadmissibility of the usual estimator $X$ by considering an alternative estimator with uniformly smaller risk than that of $X$, the improvement being substantial for $\theta$ close to the origin. The problem of estimating $\theta$ with the same loss function when $\sigma^2$ is unknown has been treated similarly by James and Stein [1], when an observation is available on another random variable which is distributed as $\sigma^2\chi^2_n$ independently of $X$. James and Stein have also demonstrated the inadmissibility of the usual estimator for $\theta$ under the loss function \begin{equation*}\tag{1}L_2(\theta,\hat{\theta}) = (\theta - \hat{\theta})'D(\theta - \hat{\theta})\end{equation*} for the case of known $\sigma^2$, where $D$ is a diagonal matrix with unequal diagonal elements. This result has been proved even without the normality assumption, but no explicit formulas for alternative estimators have been given which improve on the usual estimator in some parts of the parameter space under normality. In this note an estimator for $\theta$ is obtained for the case when $\sigma^2$ is unknown and the loss function is $L_2$. An upper bound for the risk function of this estimator is given, which always remains below the risk function of the usual estimator and is substantially smaller for $\theta$ close to the origin. This estimator coincides with the estimator given by James and Stein when the diagonal elements of $D$ become equal. An application of this estimator gives an improvement on the least squares estimator of the parameter vector in a usual linear observational model with normal errors.