Universal Inadmissibility of Least Squares Estimator

For a p-dimensional normal distribution with mean vector [theta] and covariance matrix Ip, it is known that the maximum likelihood estimator [theta] of [theta] with p[greater-or-equal, slanted]3 is inadmissible under the squared loss. The present paper considers possible extensions of the result to the case where the loss is a member of a general class of losses of the form L([delta]-[theta]Q), where L is nondecreasing and [delta]-[theta]Q denotes the Mahalanobis distance [([delta]-[theta])t Q([delta]-[theta])]1/2 with respect to a given positive definite matrix Q, which, without loss of generality, may be assumed to be diagonal, i.e., Q=diag(q1, ..., qp), q1>q2[greater-or-equal, slanted]q3[greater-or-equal, slanted]...[greater-or-equal, slanted]qp>0. For the case where q1>q2=q3=...=qp>0, L. D. Brown and J. T. Hwang (1989, Ann. Statist.17, 252-267) showed that there exists an estimate of [theta] universally dominates [theta] if and only if p[greater-or-equal, slanted]4. This paper further extends Brown and Hwang's result to the case in which q1>q2 and at least there are two equal elements among q2, ..., qp-1; namely, we show that, for this case, there exists an estimate of [theta] which universally dominates [theta] if and only if p[greater-or-equal, slanted]4. For a general Q, we gives a lower bound on p that implies the least squares estimators is universally inadmissible.