All Admissible Linear Estimates of the Mean Vector

Abstract

Let $y$ be a single observation on a $p \times 1$ random vector which is distributed according to the multivariate normal distribution with mean vector $\theta$ and covariance matrix $I$. Consider the problem of estimating $\theta$ when the loss function is the sum of the squared errors in estimating the individual components of $\theta$. Let $G$ be a $p \times p$ real matrix. Then we will prove that the estimate $Gy$ is admissible if and only if $G$ is symmetric, and the characteristic roots of $G, g_i$ say, $i = 1, 2, \cdots, p$, satisfy $0 \leqq g_i \leqq 1$, with equality at one for at most two of the roots. The proof concerning the characteristic roots uses the results of Karlin [4] and James and Stein [3]. We give two proofs of the fact that $Gy$ is inadmissible when $G$ is asymmetrical. In the first proof we give an estimate $G^\ast y$ that is better than $Gy$, where $G^\ast$ is a symmetric matrix. This not only adds to the practicality of the result, but also enables us to resolve the question of which estimates are admissible in the restricted class of estimates of the form $Gy$. The method of the second proof, which utilizes a theorem of Sacks [6], leads to the following finding: If $Gy$ is admissible, then $Gy$ must be a generalized Bayes procedure, where the unique generalized prior distribution must be either a multivariate normal distribution with mean vector zero and a specified covariance matrix determined by $G$, or the product of a distribution which is multivariate normal over a subspace of the parameter space and a distribution which is uniformly distributed over a subspace of the parameter space. This latter finding generalizes the well known one dimensional case. In Section 2 then the main results are proved. In Section 3 some remarks concerned with generalizing the main results are given. We remark here that the decision theory terminology used is more or less that of Blackwell and Girshick [1].