Optimal e-value testing for properly constrained hypotheses

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.