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-$ψ$ 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.

💡 Research Summary

The paper addresses a fundamental open problem in the theory of e‑variables: a complete characterization of all e‑variables for a broad class of null hypotheses defined by constraints. An e‑variable is a non‑negative random variable whose expectation does not exceed one under every distribution belonging to a given null set 𝒫. Because any non‑asymptotic level‑α test can be obtained by thresholding an e‑variable, describing the full set of e‑variables is equivalent to describing all valid tests for 𝒫. The authors consider null hypotheses generated by an arbitrary collection Φ of measurable constraint functions, allowing any number of constraints (finite or infinite), discontinuities, and without imposing topological restrictions on the underlying sample space.

The central result (Theorem 2.2) shows that a measurable function f satisfies ∫ f dμ ≤ 0 for all μ∈𝒫 if and only if f belongs to the weak closure of the convex cone
C = cone(Φ) − L⁺_Φ,
where cone(Φ) denotes all finite conic combinations of the constraint functions and L⁺_Φ is the set of functions that are quasi‑surely non‑negative under 𝒫. Consequently, every e‑variable can be written as
Z = 1 + f, f ∈ C,
and any non‑negative measurable function that is quasi‑surely equal to 1 + f for some f∈C is an e‑variable. This representation holds without any continuity, compactness, or separability assumptions, relying only on the dual pairing between measurable functions and signed measures and the bipolar theorem for weak topologies.

The paper then instantiates the abstract theorem for three important families of hypotheses:

1. Finitely generated constraints – When Φ = {g₁,…,g_d} is finite, the authors introduce the notion of a non‑redundant support and define a closed set K⊂ℝ⁺ᵈ of coefficient vectors whose support is not redundant. They prove (Theorem 3.1) that any f satisfying the null‑expectation condition can be expressed as f ≤ ∑_{i=1}^d π_i g_i quasi‑surely for some π∈K, and thus every e‑variable is of the form 1 + ∑ π_i g_i. This recovers classical moment‑based tests (e.g., mean and variance constraints) and extends them to arbitrary measurable constraints.

2. One‑sided sub‑ψ distributions – For a convex increasing function ψ, the null hypothesis consists of all distributions whose moment‑generating function satisfies E