Testing Hypotheses Generated by Constraints

• 成果类型: Article; Early Access
• 署名作者: Larsson, Martin; Ramdas, Aaditya; Ruf, Johannes
• 署名单位: Carnegie Mellon University; Carnegie Mellon University; University of London; London School Economics & Political Science
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2025.1129
• 发表日期: 2026-05-12
• 关键词: e-variables; Hypothesis Testing; moment constraints; nonparametric testing; model uncertainty; bipolar; optimization; time-uniform

摘要

E-variables are nonnegative random variables with expected value at most one under any distribution from a given null hypothesis. Every nonasymptotically valid test can be obtained by thresholding some e-variable. As such, e-variables arise naturally in applications in statistics and operations research, and a key open problem is to characterize their form. We provide a complete solution to this problem for hypotheses generated by constraints-a broad and natural framework that encompasses many hypothesis classes occurring in practice. Our main result is an abstract representation theorem that describes all e-variables for any hypothesis defined by an arbitrary collection of measurable constraints. We instantiate this general theory for three important classes: hypotheses generated by finitely many constraints, one-sided sub-iJ distributions (including sub-Gaussian distributions), and distributions constrained by group symmetries. In each case, we explicitly characterize all e-variables as well as all admissible e-variables. Numerous examples are treated, including constraints on moments, quantiles, and conditional value-at-risk (CVaR). Building on these, we prove the existence and uniqueness of optimal e-variables under a large class of expected utility-based objective functions used for optimal decision making, in particular covering all criteria studied in the e-variable literature to date.