**Sankhya:
The Indian Journal of Statistics**

2004, Volume 66, Pt. 1, 109--139

**Bayesian Versus Frequentist
Shrinkage in Multivariate Normal Problems**

By

Eric M. Vestrup, DePaul University, Chicago, USA & Francisco J. Samaniego, University of California at Davis, USA

SUMMARY. In estimating a multivariate normal mean, both the celebrated James-Stein estimator and the Bayes estimator relative to generalized squared error loss and a conjugate prior distribution shrink the sample mean toward a distinguished point. In comparing the performance of these two shrinkage estimators, we postulate the existence of a (possibly degenerate) ``true prior distribution'', and we utilize as a criterion the Bayes risk of each estimator relative to the true prior and squared error loss. Our goal is to provide characterizations of classes of ``operational priors'' available to the Bayesian for which the corresponding Bayesian shrinkage provides better performance than James-Stein shrinkage. A definitive comparison is provided in the special case in which the covariance matrices are diagonal and the true parameter value is a fixed vector of constants unknown to the statistician. There, the subclass of Bayes rules which dominate the James-Stein rule is completely characterized through results which identify its precise dependence on the dimension of the problem, the relative variances of the sampling distribution and the prior, and the magnitude of the error in the prior mean. In the final section, we give some guidance for deciding, in advance, which type of shrinkage one might wish to employ in a particular application.

*AMS (1991) subject classification}. *Primary 62H12; secondary
62F15,62F10.

*Key words and phrases. *Squared error loss, multivariate normal, Bayes
rule, James-Stein rule, operational prior, Bayes risk, non-central $\chi^2$.