Optimal e-value testing for properly constrained hypotheses
Eugenio Clerico
TL;DR
The paper presents a rigorous characterization of the optimal e-class for hypothesis testing via e-values under properly constrained, non-parametric hypotheses defined by finitely many regular constraints. It proves that the optimal e-class coincides with the dual e-class $\mathcal{E}_\mathcal{H}^\vee$ and establishes existence in finite domains, extends to compact and general closed domains via matching-set techniques, and extends to non-properly and loosely constrained cases. The results are directly applicable to constructing tight confidence sequences for means, including bounded and heavy-tailed settings, by restricting to the optimal e-class. This work thus provides a principled, tractable framework for designing sequential tests with e-values, with practical impact on adaptive data collection and mean-estimation tasks while connecting to classical admissibility concepts and laying groundwork for broader sequential testing beyond single-round e-variables.
Abstract
Hypothesis testing via e-variables can be framed as a sequential betting game, where a player each round picks an e-variable. A good player's strategy results in an effective statistical test that rejects the null hypothesis as soon as sufficient evidence arises. Building on recent advances, we address the question of restricting the pool of e-variables to simplify strategy design without compromising effectiveness. We extend the results of Clerico(2024), by characterising optimal sets of e-variables for a broad class of non-parametric hypothesis tests, defined by finitely many regular constraints. As an application, we discuss this notion of optimality in algorithmic mean estimation, including for heavy-tailed random variables.