发表机构
AI4Science Lab; Korteweg-de Vries Institute for Mathematics; University of Amsterdam（AI4Science实验室; 科尔特韦格-德弗里斯数学研究所; 阿姆斯特丹大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究检验超鞅下尾概率假设对序贯检验II型误差的影响，通过主不等式和Legendre变换导出多种误差界，并涵盖经验Bernstein变体。
AI 中文摘要
在使用检验超鞅的安全假设检验中，Ville不等式为每个显著性水平$\alpha\in(0,1]$提供了随时有效的I型误差保证，前提是每当财富过程首次超过$\frac{1}{\alpha}$时拒绝原假设。由于固有的不对称性，II型误差不具有此类保证：概率在对数增量的下尾上的集中可能导致一次灾难性下注，从而抵消任何累积的证据。本文研究关于这些下尾概率的不同假设如何导致序贯检验的II型误差的不同界。它们都归结为一个主不等式，该不等式在固定时间范围和序贯设置下，以$\alpha$水平的II型误差为界，其形式为e变量（逆）矩生成函数的单侧Legendre变换，在该函数的一个数值处求值：累积e幂的下界超过$\log\frac{1}{\alpha}$的量。并且，该步骤是无损的，因为它精确地提取了一个受约束的信息投影。这里呈现的每个界都是通过上述函数的某个特定优函数获得的推论。这些假设包括：有限的负矩；具有获胜侧矩的指数小崩溃概率；具有条件方差的财富下限及其Bernstein变体，该变体在由方差设定的高斯区间和由尺度设定的指数区间之间插值；次高斯或有界倾斜下尾；有界对数增量；以及独立同分布增量，其中优函数是真实情况。我们还提供了经验Bernstein变体。每个假设既可以理解为对现有e变量的条件，也可以理解为使用似然比的近似而非似然比本身进行下注的价格，该近似免费满足最弱的条件。
英文摘要
In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}α$. Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence. This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test. They all reduce to one master inequality, which bounds the type-II error at level $α$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}α$. And, the step is lossless, in the sense, that it extracts exactly a constrained information projection. Every bound presented here is a corollary, obtained by a certain majorant of the above function. The hypotheses are: a finite negative moment; an exponentially small crash probability with a moment on the winning side; a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale; a sub-Gaussian or bounded-tilt lower tail; bounded log-increments; and i.i.d. increments, where the majorant is the truth. We also provide an empirical-Bernstein variant. Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.
