Inference on Consensus Ranking of Distributions

Instead of testing for unanimous agreement, I propose learning how broad of a consensus favors one distribution over another (of earnings, productivity, asset returns, test scores, etc.). Specifically, given a sample from each of two distributions, I propose statistical inference methods to learn about the set of utility functions for which the first distribution has higher expected utility than the second distribution. With high probability, an “inner” confidence set is contained within this true set, while an “outer” confidence set contains the true set. Such confidence sets can be formed by inverting a proposed multiple testing procedure that controls the familywise error rate. Theoretical justification comes from empirical process results, given that very large classes of utility functions are generally Donsker (subject to finite moments). The theory additionally justifies a uniform (over utility functions) confidence band of expected utility differences, as well as tests with a utility-based “restricted stochastic dominance” as either the null or alternative hypothesis. Simulated and empirical examples illustrate the methodology.

相较于检验是否达成完全一致的共识，本文转而研究在多大程度上存在共识，使得某一分布（如收入、生产率、资产收益、考试分数等）相较于另一分布更占优。具体而言，给定两个分布各自的样本集，本文提出统计推断方法，用以刻画满足「第一分布的期望效用高于第二分布」的效用函数集合。在高概率意义下，「内层」置信集将被包含于该真实效用函数集合中，而「外层」置信集则包含该真实集合。这类置信集可通过对本文提出的多重检验流程进行反转得到，该流程可控制家族错误率（familywise error rate）。其理论依据源自经验过程理论，鉴于绝大多数大类效用函数均属于当斯克尔（Donsker）类（满足有限矩条件）。该理论同时可用于构建效用函数空间上一致的期望效用差异置信带，也可支撑以基于效用的「受限随机占优（restricted stochastic dominance）」作为原假设或备择假设的假设检验。本文通过模拟实验与实证案例对所提方法论进行了演示说明。