A CLASS OF MULTIPLE SHRINKAGE ESTIMATORS

C. S. WITHERS

Applied Mathematics Division, Department of Scientific and Industrial Research, P. O. Box 1335, Wellington, New Zealand

(Received May 9, 1988; revised December 11, 1989)

Abstract. Based on a sample of size n, we investigate a class of estimators of the mean θ of a p-variate normal distribution with independent components having unknown covariance. This class includes the James-Stein estimator and Lindley’s estimator as special cases and was proposed by Stein. The mean squares error improves on that of the sample mean for p ≥ 3. Simple approximations for this improvement are given for large n or p. Lindley’s estimator improves on that of James and Stein if either n is large, and the “coefficient of variation” of θ is less than a certain increasing function of p, or if p is large. An adaptive estimator is given which for large samples always performs at least as well as these two estimators.

Key words and phrases: Shrinkage estimates, multivariate normal, loss.