Testing hypotheses generated by constraints

TL;DR

The paper provides a comprehensive, nonasymptotic framework to characterize all e-variables for hypotheses described by constraints, via a bipolar-duality representation. It presents an abstract result and then instantiates it for three important classes: finitely generated constraints, one-sided sub-$\psi$ distributions, and distributions invariant under group symmetries, yielding explicit forms for all e-variables and their admissible, optimal variants. It furthermore develops admissibility and minimal complete classes, proves existence and, under mild conditions, uniqueness of optimal e-variables under broad objective functions, and discusses unions of hypotheses and relaxed integrability. The results enable construction of threshold-based tests and decision rules with guaranteed type-I control and principled optimality, with concrete examples including moment/quantile constraints and CVaR. The work also outlines significant open problems, including extensions to i.i.d. samples, sequential e-processes, and broader constraint settings.

Abstract

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-$ψ$ 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 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.

Theorems & Definitions (75)

• Definition 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Remark 2.4
• Lemma 2.5
• proof
• Theorem 3.1
• Lemma 3.2
• proof