How can we monitor, in real time, whether one uncertain prospect has any upside over another? To answer this question, we develop a novel family of sequential, anytime-valid tests for stochastic dominance (SD; also known as stochastic ordering), a classical and popular notion for comparing entire distribution functions. The problem is distinct from the popular problem of testing for dominance in means, which would not capture distributional differences beyond the first moment. We first derive powerful, nonparametric e-processes that quantify evidence against the null hypothesis that one prospect is dominated by another. For first-order SD, these e-processes are constructed as a mixture of asymptotically growth-rate optimal e-variables and yield a test of power one. The approach further generalizes to sequential testing for SD beyond the first order, including any higher-order SD. Empirically, we demonstrate that the resulting sequential tests are competitive with existing non-sequential SD tests in terms of power, while achieving validity under continuous monitoring that existing methods do not. Finally, we sketch the complementary and challenging problem of testing the non-SD null hypothesis, which asks whether a prospect has a definite upside, and describe the conditions under which we can derive a nontrivial anytime-valid test.

翻译：如何实时监测一个不确定的前景是否比另一个更具优势？为回答此问题，我们针对随机占优（亦称随机序）——一种比较完整分布函数的经典且广受欢迎的概念——构建了全新系列的序列化随时有效检验。该问题区别于广为人知的均值占优检验问题（后者无法捕捉一阶矩以外的分布差异）。我们首先推导出强大的非参数化e过程，用于量化反对"某前景被另一前景占优"这一原假设的证据。针对一阶随机占优，这些e过程通过混合渐近增长率最优的e变量构建，并产生幂次为1的检验。该方法进一步推广至一阶以上随机占优的序列化检验，涵盖任意高阶随机占优。实验表明，所提出的序列检验在统计功效上与现有非序列随机占优检验相媲美，同时实现了现有方法无法获得的连续监测有效性。最后，我们勾勒了互补且具有挑战性的非随机占优原假设检验问题——即检验某前景是否具有明确优势，并描述了可推导出非平凡随时有效检验的条件。