ASYMPTOTIC RISK BEHAVIOR OF MEAN VECTOR AND VARIANCE ESTIMATORS

AND THE PROBLEM OF POSITIVE NORMAL MEAN

ANDREW L. RUKHIN

Department of Mathematics and Statistics, The University of Maryland, Baltimore County Campus, Baltimore, MD 21228-5398, U.S.A. and University of Münster

(Received August 20, 1990; revised February 20, 1991)

Abstract. Asymptotic risk behavior of estimators of the unknown variance and of the unknown mean vector in a multivariate normal distribution is considered for a general loss. It is shown that in both problems this characteristic is related to the risk in an estimation problem of a positive normal mean under quadratic loss function. A curious property of the Brewster-Zidek variance estimator of the normal variance is also noticed.

Key words and phrases: Bowl-shaped loss function, Brewster-Zidek estimator of normal variance, James-Stein estimator of normal mean, relative risk reduction, positive normal mean, Stein estimator of normal variance.