Testing Hypotheses Generated by Constraints

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-[Formula: see text] 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. Funding: This research was supported by the National Science Foundation [Grants NSF DMS-2310718 and NSF DMS-2510965] and a Sloan Fellowship.