MINIMAX ESTIMATORS FOR LOCATION VECTORS IN ELLIPTICAL DISTRIBUTIONS WITH

UNKNOWN SCALE PARAMETER AND ITS APPLICATION TO VARIANCE REDUCTION IN SIMULATION

M. TAN^{1} AND L. J. GLESER^{2}

1Department of Biostatistics and Epidemiology/P88, The Cleveland Clinic Foundation,

9500 Euclid Avenue, Cleveland, OH 44195-5196, U.S.A. 2Department of Mathematics and Statistics, University of Pittsburgh, 4200 Fifth Avenue, Pittsburgh, PA 15260-0001, U.S.A.

(Received December 17, 1990; revised October 14, 1991)

Abstract. In this paper, we give an ever wider and new class of minimax estimators for the location vector of an elliptical distribution (a scale mixture of normal densities) with an unknown scale parameter. Then its application to variance reduction for Monte Carlo simulation when control variates are used is considered. The results obtained thus extend (i) Berger’s result concerning minimax estimation of location vectors for scale mixtures of normal densities with known scale parameter and (ii) Strawderman’s result on the estimation of the normal mean with common unknown variance.

Key words and phrases: Minimax estimators, shrinkage estimators, elliptical distributions, scale mixture of normal, Monte Carlo simulation, variance reduction.