On the Admissibility of Invariant Estimators of One or More Location Parameters

Abstract

Many statistical estimation problems possess certain naturalsymmetries. The location parameter estimation problem is an important example.It is symmetric, or, to use the usual terminology, invariant with respectto translations of the sample space. This strongly suggests that the statisticianshould use an estimation procedure which also has the property of being invariant.Until recently it seemed reasonable to expect that the best invariant estimatoris a "good" estimator. In particular, it seemed reasonable to expect that it isadmissible-that is, that no other estimator has a risk which is never larger andis sometimes smaller than the risk of the best invariant estimator. However,Stein (1956) gave an example of a very simple problem in which the best invariantestimator is inadmissible.