Universal Inadmissibility of Least Squares Estimator

✍ Scribed by Chang-Yu Lu; Ning-Zhong Shi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 140 KB
• Volume: 72
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1999.1843

✦ Synopsis

For a p-dimensional normal distribution with mean vector % and covariance matrix I p , it is known that the maximum likelihood estimator % of % 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( |$&%| Q ), where L is nondecreasing and |$&%| Q denotes the Mahalanobis distance [($&%) t Q($&%)] 1Â2 with respect to a given positive definite matrix Q, which, without loss of generality, may be assumed to be diagonal, i.e., Q=diag(q 1 , ..., q p ), q 1 >q 2 q 3 } } } q p >0. For the case where q 1 >q 2 =q 3 = } } } =q p >0, L. D. Brown and J. T. Hwang (1989, Ann. Statist. 17, 252 267) showed that there exists an estimate of % universally dominates % if and only if p 4. This paper further extends Brown and Hwang's result to the case in which q 1 >q 2 and at least there are two equal elements among q 2 , } } } , q p&1 ; namely, we show that, for this case, there exists an estimate of % which universally dominates % 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.