We give an exact characterization of admissibility in statistical decision problems in terms of Bayes optimality in a so-called nonstandard extension of the original decision problem, as introduced by Duanmu and Roy. Unlike the consideration of improper priors or other generalized notions of Bayes optimalitiy, the nonstandard extension is distinguished, in part, by having priors that can assign "infinitesimal" mass in a sense that can be made rigorous using results from nonstandard analysis. With these additional priors, we find that, informally speaking, a decision procedure $\delta_0$ is admissible in the original statistical decision problem if and only if, in the nonstandard extension of the problem, the nonstandard extension of $\delta_0$ is Bayes optimal among the extensions of standard decision procedures with respect to a nonstandard prior that assigns at least infinitesimal mass to every standard parameter value. We use the above theorem to give further characterizations of admissibility, one related to Blyth's method, one to a condition due to Stein which characterizes admissibility under some regularity assumptions; and finally, a characterization using finitely additive priors in decision problems meeting certain regularity requirements. Our results imply that Blyth's method is a sound and complete method for establishing admissibility. Buoyed by this result, we revisit the univariate two-sample common-mean problem, and show that the Graybill--Deal estimator is admissible among a certain class of unbiased decision procedures.
翻译:与Duamu 和 Roy提出的对Bayes 最优性统计决定问题的可接受性,在所谓的原决定问题的非标准延期中,我们确切地描述在Bayes 最优性统计决定问题的可接受性,这是Duamu 和 Roy提出的。与对Bayes 最优性不适当的前期或其他普遍概念的考虑不同,非标准延期部分地区别于对非标准性决定程序的可严格地分配“无限”质量的前期,这种前期可以使用非标准性分析的结果,而这种前期分析可以严格地加以严格。我们发现,在非正式地说,在原统计决定问题的非标准延期中,只有在问题的非标准延期中,非标准性延长 $\del_0美元是巴耶斯最优性的,非标准性扩展部分是通过非标准性的前期决定程序,至少给每个标准参数值分配无限性质量。我们用以上理论来进一步描述可接受性的定性,一个与Blyth 方法有关的前期方法,一个因斯坦在正常性假设下具有可接受性特点的条件;最后,使用固定性标准性标准性评估性程序的定性分析方法表明我们通常性的结果。