Abstract

Just over 50 years ago Stein published a startling statistical result. (Stein (1956).) When three or more normal means are to be estimated the sample mean is not an admissible estimator. It is better to ‘shrink’ towards the origin, or some other predetermined point. At first this fact appeared to many as a mathematical peculiarity, with no particular practical significance. Publication five years later of the James-Stein (1961) estimator demonstrated that the difference in performance could be quite substantial between the sample mean and a suitable shrinkage estimator. It has become understood in the intervening decades how this minimax surprise is intimately related to a variety of other practical statistical methodologies and its principles applicable in a wide range of practical settings.

Stein’s original paper included a geometrical explanation as to why such a paradoxical result is inevitable when estimating sufficiently many separate means, as well a relatively simple proof that 3 is sufficiently many. I’ll supplement his geometric argument with a simple geometric diagram and then sketch his proof. I can also remark about several generalizations (Brown (1966)) of this proof that show this abnormal result is not only a result about the normal distribution and squared error loss (as some statisticians at the time had suspected).

I don’t specifically plan on discussing any further technicalities, but if time permits I can sketch where this seminal paper has led, including especially James-Stein (1961), mentioned above, and Brown (1971) which attempts to show that there are several related mathematical situations where 3>>2.

[Incidentally, my Kuwait Lecture on Tuesday provides a specific illustration of how shrinkage ideas can be used in high dimensional data settings.]

Stein’s original paper is available at

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bsmsp/1200501656