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
本文证明检验超鞅的e-幂不提供有限时域第二类错误保证，而Chernoff-Stein指数决定其上限，并精确刻画最优e-变量及时域选择。
AI 中文摘要
在基于检验超鞅的安全假设检验中，Ville不等式为每个显著性水平 $\alpha\in(0,1]$ 提供了任意时刻有效的第一类错误保证，即当财富过程首次超过 $1/\alpha$ 时拒绝原假设。由于固有的不对称性，第二类错误的表现有所不同。对于简单原假设和备择假设，我们证明了关于第二类错误的两点。第一，均值增长率 $\mathbb{E}_{P_1}[\log E]$（即e-幂，Kelly投注和增长率最优e-变量所最大化的量）本身不提供任何界限。对于每个水平 $c>0$、每个 $\alpha$ 和时域 $t$，我们构造条件e-幂恰好为 $c$ 的e-变量，其在 $t$ 时刻之前不拒绝的概率任意接近1。它强制最终拒绝，但不提供任何有限时域保证。第二，真正控制第二类错误的量是e-变量的Chernoff-Stein指数 $\Lambda(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$，其取值范围被精确确定：$\sup_E \Lambda(E)=\mathrm{KL}(P_0\\|P_1)$，即经典Chernoff-Stein指数，因此也是其自身逐e-变量形式的上限。每一步应用一次Hölder不等式即可得到该结果，适用于任意滤过空间上的每个检验超鞅，无需独立性或乘积结构；i.i.d.情形补充说明该上限可被匹配，但无法被任何e-变量达到。e-幂有其自身上限 $\mathrm{KL}(P_1\\|P_0)$，该上限可由似然比 $R$ 达到，且在 $P_0$-a.s.意义下唯一。两个最优值是同一散度在相反参数下的取值，位于扁平化族 $R^\beta/\mathbb{E}_{P_0}[R^\beta]$ 的两端：上限在 $\beta\downarrow0$ 处，$R$ 在 $\beta=1$ 处。哪个 $\beta$ 最优由时域精确决定：$R$ 仅在 $t=\log(1/\alpha)/\mathrm{KL}(P_1\\|P_0)$ 时最优，低于该时域时锐化（$\beta>1$）更优，高于该时域时扁平化更优。
英文摘要
In safe hypothesis testing with test supermartingales, 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 $1/α$. Due to an inherent asymmetry, the type-II error behaves differently. We prove two things about the latter, for a simple null and alternative. First, the mean growth rate $\mathbb{E}_{P_1}[\log E]$, the e-power, that Kelly betting and growth-rate-optimal e-variables maximise, bounds nothing on its own. For every level $c>0$, every $α$ and horizon $t$ we construct e-variables of conditional e-power exactly $c$ whose probability of not rejecting by $t$ is arbitrarily close to one. It forces eventual rejection, but no finite-horizon guarantee follows. Second, the quantity that does control the type-II error is the Chernoff-Stein exponent of an e-variable, $Λ(E)=\sup_{s\ge0}\{-\log \mathbb{E}_{P_1}[E^{-s}]\}$, whose range is exactly determined: $\sup_E Λ(E)=\mathrm{KL}(P_0\|P_1)$, the classical Chernoff-Stein exponent, and so the ceiling of its own per-e-variable form. One conditional application of Hoelder's inequality per step gives it, for every test supermartingale on an arbitrary filtered space, with no independence or product structure; the i.i.d. case adds that it is matched, and attained by nothing. The e-power has its own ceiling, $\mathrm{KL}(P_1\|P_0)$, and that one is attained, $P_0$-a.s. uniquely, by the likelihood ratio $R$. The two optima are the same divergence in opposite arguments, at opposite ends of the flattened family $R^β/\mathbb{E}_{P_0}[R^β]$: the ceiling as $β\downarrow0$, $R$ at $β=1$. Which $β$ is best is settled by the horizon, exactly: $R$ is optimal at $t=\log(1/α)/\mathrm{KL}(P_1\|P_0)$ alone, beaten by sharpening $(β>1)$ below it and by flattening above.
