Improving on Equivariant Estimators

Abstract

Techniques for improving on equivariant estimators are described. They may be applied, although without assurance of success, whatever be the family of underlying distributions. The loss function is required to satisfy an intuitively reasonable condition but is otherwise arbitrary. One of these techniques amounts to a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit. It yields, when successful, a "testimator". A second technique obtains the limit of a certain sequence of "testimator-like" estimators. The result is "smoother" than a testimator and often identical to a generalized Bayes estimator over much of its domain. Applications are presented. In the first we extend results of Stein (1964) and obtain a minimax estimator which is generalized Bayes, and in a univariate subcase, admissible within the class of scale-equivariant estimators. In the second, we extend a result of Srivastava and Bancroft (1967).