i.i.d. 模型中序贯可检验性与变化可检测性的完整刻画
A complete characterization of sequential testability and change detectability in i.i.d. models
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中文总结 AI 辅助
本文给出独立同分布复合检验中功效为一序贯检验存在性的充要条件，并证明相同条件刻画变化可检测性，构造出最优期望样本量与平均运行长度的检测器。
中文摘要 AI 辅助
我们给出了在独立同分布复合检验问题中存在功效为一的序贯检验的充要条件。当且仅当备择假设与零假设被可数个有限块事件分离时，才存在一个水平为 \\(\alpha\\) 且对每个备择假设功效均为一的检验。我们利用随机化固定样本检验、有界有限块得分、e 过程、降滤过检验鞅，以及一个有限块弱-* 闭凸包在全变差距离下正分离的可数覆盖，提供了其他等价条件。作为额外收获，构造性证明表明检验具有逐点期望样本量 \\(O_Q(\log(1/\alpha))\\)。完全相同的条件也刻画了在可选视界平均运行长度控制下的 i.i.d. 变化可检测性：对于每个 \\(\eta>0\\)，它们等价于一个报警族 \\((T_\gamma)_{\gamma\ge1}\\) 满足对每个零假设分布和每个停时 \\(\sigma\\) 有 \\(\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma\\)。事实上，当这些条件成立时，我们可以构造一个单一的 e 检测器，使得每个零假设平均运行长度介于 \\(\gamma\\) 和 \\((1+\eta)\gamma+1\\) 之间，并具有稳健的 Lorden 延迟 \\(O_Q(\log\gamma)\\)。
英文摘要
We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-\(α\) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size \(O_Q(\log(1/α))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(η>0\), they are equivalent to an alarm family \((T_γ)_{γ\ge1}\) satisfying \(\Prob_{P^\infty}(T_γ\leσ)\le \E_{P^\infty}σ/γ\) for every null law and every stopping time \(σ\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(γ\) and \((1+η)γ+1\), and having robust Lorden delay \(O_Q(\logγ)\).
发表机构
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
