Universal Domination and Stochastic Domination-an Improved lower Bound for the Dimensionality

For a p-dimensional normal distribution with mean vector and covariance matrix Ip, it is known that the maximum likelihood estimator <^> of with p = 3 is inadmissible under the squared loss. The present article considers possible extensions of the result to the case where the loss is a member of a general class of the form L - Q , where L is non decreasing, - Q denotes the Mahalanobis distance - tQ - 1 2 with respect to a given positive definite matrix Q, which, with loss of generality, may be assumed to be diagonal, i. e., Q = diag q1 qp q1 = q2 = = qn > qn+ 1 = = qp > 0. Brown and Hwang (1989) showed that there exists an estimator stochastic dominates <^> if p is " large enough", and further more, for the case n = 1, they give a expression about the lower bound on p. This article further extends Brown and Hwang's result to the situation of general n = 1, and We give a plain expression about the lower bound than that of Brown and Hwang's (1989).