Admissibility and Complete Classes

Abstract

The previous chapters mentioned repeatedly that the Bayes estimators were instrumental for the frequentist notions of optimality, in particular, for admissibility. This chapter provides a more detailed description of this phenomenon. In §6.1, it considers the performances of the Bayes and generalized Bayes estimators in terms of admissibility. Then, §6.2 studies Stein’s sufficient condition in order to relate the admissibility of a given estimator with a sequence of prior distributions. The notion of complete class introduced in §6.3 is also fundamental, as it provides a characterization of admissible estimators or at least a substantial reduction in the class of acceptable estimators. We show that, in many cases, the set of the Bayes estimators constitutes a complete class and that, in other cases, it is necessary to include generalized Bayes estimators. In a more general although non-Bayesian perspective, §6.4 presents a method introduced by Brown (1971) and developed by Hwang (1982b), which provides necessary admissibility conditions. For a more technical survey of these topics, see Rukhin (1994).KeywordsPrior DistributionExponential FamilyAdmissibility ConditionQuadratic LossContinuous RiskThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.