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Testing Hypotheses Generated by Constraints
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DOI：10.1287/moor.2025.1129.png)
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Abstract
En 中文
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.
Keywords:
e-variables
hypothesis testing
moment constraints
nonparametric testing
model uncertainty
bipolar
optimization
Journal
M
IF:
1.9
Papers:
77
Citations:
0