Bayesian Statistics: An Introduction, 4th Edition

8.4 The Stein estimator

This section is about an aspect of classical statistics which is related to the aforementioned discussion, but an understanding of it is by no means necessary for developing a knowledge of Bayesian statistics per se. The Bayesian analysis of the hierarchical normal model is continued in Section 8.5.

One of the most puzzling and provocative results in classical statistics in the past half century was Stein’s startling discovery (see Stein, 1956, and James and Stein, 1961) that the ‘obvious’ estimator of the multivariate normal mean is inadmissible if . In fact if c is any constant with

then

dominates . The best value of c is r–2, leading to the James–Stein estimator

Because it may be considered as a weighted mean of and , it is often called a shrinkage estimator ...