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 optimality in algorithmic mean estimation, including the case of heavy-tailed random variables.

翻译：基于e变量的假设检验可被构建为一个序贯赌博游戏，其中玩家每轮选择一个e变量。优秀玩家的策略将产生一种有效的统计检验，一旦出现充分证据即可拒绝原假设。基于最新研究进展，本文探讨如何限制e变量集合以简化策略设计而不影响检验效能。我们拓展了Clerico(2024)的研究成果，通过刻画适用于广泛非参数假设检验的最优e变量集合特征，该检验类别由有限个正则约束定义。作为应用，我们讨论了算法均值估计中的最优性问题，包括重尾随机变量的情形。