Universal Inadmissibility of Least Squares Estimator

For a p-dimensional normal distribution with mean vector θ and covariance matrix Ip, it is known that the maximum likelihood estimator θ of θ with p⩾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(|δ−θ|Q), where L is nondecreasing and |δ−θ|Q denotes the Mahalanobis distance [(δ−θ)t Q(δ−θ)]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⩾q3⩾…⩾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 θ universally dominates θ if and only if p⩾4. This paper further extends Brown and Hwangʹs 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 θ which universally dominates θ if and only if p⩾4. For a general Q, we gives a lower bound on p that implies the least squares estimators is universally inadmissible.