为复杂假设检验的停止时间提供了紧致上下界，突破了传统参数假设限制。
On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds
- 提出两种通用下界：沃尔德与法雷尔设定下，基于最小KL散度分析停止时间。
- 在非参数复合假设下，停止时间至少为 $\operatorname{KL^{-1}_{inf}} \log \log \operatorname{KL^{-1}_{inf}}$ 阶。
- 适用于无参考测度的复杂场景，对非参数检验有重要理论支撑。
本文针对任意复合零假设 $\mathcal P$ 与备择假设 $\mathcal Q$ 之间的序贯检验，给出了两个通用的停止时间下界。第一类下界适用于“沃尔德情形”：当类型-1错误率 $α \to 0$ 而备择假设 $Q \in \mathcal Q$ 固定时，停止时间下界为 $\log(1/α)$ 除以 $\mathcal P$ 与 $Q$ 间最小KL散度 $\operatorname{KL_{inf}}$。第二类下界适用于“法雷尔情形”：当 $α$ 固定而 $\operatorname{KL_{inf}} \to 0$ 沿某序列变化时，所需期望样本量至少为 $\operatorname{KL^{-1}_{inf}} \log \log \operatorname{KL^{-1}_{inf}}$。核心贡献在于这些界具有极强的普适性，无需依赖主导参考测度，显著推广了已有参数情形结果。同时给出匹配上界成立的充分条件，并验证其在多个非平凡非参数案例中可满足。
原文摘要 · Abstract (English)
We present two general lower bounds for stopping times of sequential tests between arbitrary composite nulls $\mathcal P$ and alternatives $\mathcal Q$. The first lower bound is for the ``Wald setting'' where the type-1 error level $α$ approaches zero for a fixed alternative $Q \in \mathcal Q$, and equals $\log(1/α)$ divided by a certain infimum KL divergence between $\mathcal P$ and $Q$, termed $\operatorname{KL_{inf}}$. The second lower bound applies to the ``Farrell setting'', where $α$ is fixed and $\operatorname{KL_{inf}}$ approaches $0$ along a sequence of alternatives such that the required expected sample size along that sequence is of order at least $\operatorname{KL^{-1}_{inf}} \log \log \operatorname{KL^{-1}_{inf}}$. Our main contribution is the generality of these bounds, which hold in non-parametric, composite settings, without requiring a dominating reference measure, substantially generalizing the known parametric results. We also provide sufficient conditions for matching upper bounds and show that these are met in several nontrivial non-parametric cases.
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