The statistical decision theory pioneered by Wald (1950) has used state-dependent mean loss (risk) to measure the performance of statistical decision functions across potential samples. We think it evident that evaluation of performance should respect stochastic dominance, but we do not see a compelling reason to focus exclusively on mean loss. We think it instructive to also measure performance by other functionals that respect stochastic dominance, such as quantiles of the distribution of loss. This paper develops general principles and illustrative applications for statistical decision theory respecting stochastic dominance. We modify the Wald definition of admissibility to an analogous concept of stochastic dominance (SD) admissibility, which uses stochastic dominance rather than mean sampling performance to compare alternative decision rules. We study SD admissibility in two relatively simple classes of decision problems that arise in treatment choice. We reevaluate the relationship between the MLE, James-Stein, and James-Stein positive part estimators from the perspective of SD admissibility. We consider alternative criteria for choice among SD-admissible rules. We juxtapose traditional criteria based on risk, regret, or Bayes risk with analogous ones based on quantiles of state-dependent sampling distributions or the Bayes distribution of loss.

翻译：由Wald（1950）开创的统计决策理论采用状态依赖的平均损失（风险）来衡量统计决策函数在潜在样本上的表现。我们认为，决策表现的评估应当遵循随机占优原则是显而易见的，但并未发现将关注点仅局限于平均损失的充分理由。我们认为，采用其他遵循随机占优的泛函（如损失分布的分位数）来衡量表现同样具有启发性。本文发展了尊重随机占优的统计决策理论的一般原理与示例应用。我们将Wald关于可容许性的定义修改为随机占优（SD）可容许性的概念，该概念采用随机占优而非平均抽样表现来比较不同决策规则。我们在治疗选择中两类相对简单的决策问题中研究了SD可容许性。我们从SD可容许性的视角重新评估了MLE、James-Stein估计量及其正部估计量之间的关系。我们考虑了在SD可容许规则间进行选择的替代准则。我们将基于风险、遗憾或贝叶斯风险的传统准则与基于状态依赖抽样分布分位数或损失贝叶斯分布的类似准则进行了对比分析。