用于ARL控制的e-检测器的普适性

中文摘要 AI 辅助

针对预变化类$\u2118$的e-检测器是一个非负过程$M$，满足对所有停时$\u03c4$及所有$P \u2208 \u2118$，有$\u2115_P[M_\u03c4] \u2264 \u2115_P[\u03c4]$。对e-检测器进行阈值处理可控制平均运行长度（ARL）：在$M$首次超过$b$的时刻$T_b$声明发生变化，可保证$\u222b_{P \u2208 \u2118} \u2115_P[T] \u2265 b$。但e-检测器的作用远不止控制ARL，它们还满足**可选时域不等式**：对每个依赖数据的停时（监测时域）$\u03c3$及$P \u2208 \u2118$，有$P(T_b \u2264 \u03c3) \u2264 \u2115_P[\u03c3]/b$。特别地，每个基于e-检测器的过程在每个固定时刻$t$都满足$P(T \u2264 t) \u2264 t/b$，从而避免早期虚警。值得注意的是，其逆命题也成立：每个满足可选时域不等式的停时$T$，必然由对某个e-检测器进行阈值处理得到。我们还推导了仅满足ARL控制的停时的普适表示，这些停时由**弱**e-检测器表示，该检测器仅要求$\u2115_P[M_\u03c4] \u2264 \u2115_P[\u03c4]$在所有阈值停时$T_b$处成立。附录给出了其他（较不常见）变化检测指标的普适表示。

英文摘要

An e-detector for a pre-change class $\mathcal P$ is a nonnegative process $M$ such that $\mathbb E_P[M_τ] \leq \mathbb E_P[τ]$ for all stopping times $τ$ and all $P \in \mathcal P$. Thresholding e-detectors controls the average run length (ARL): declaring a change at the first time $T_b$ when $M$ crosses $b$ ensures that $\inf_{P \in \mathcal P}\mathbb E_P[T] \geq b$. But e-detectors do substantially more than control the ARL; they also satisfy a \emph{optional-horizon inequality}: \[ P(T_b\leqσ)\leq \mathbb E_P[σ]/b \] for every data-dependent stopping time (monitoring horizon) \(σ\) and $P\in \mathcal P$. In particular, every e-detector-based procedure obeys $P(T\leq t)\leq t/b$ at each fixed $t$, thus avoiding early false alarms. Remarkably, the converse also holds: every stopping time $T$ that satisfies the optional-horizon inequality must in fact arise from thresholding an e-detector. We also derive a universal representation of stopping times that satisfy (only) ARL control. These are represented by \emph{weak} e-detectors, that only require $\mathbb E_P[M_τ] \leq \mathbb E_P[τ]$ to hold at all threshold stopping times $T_b$. Appendices present universal representations for other (less common) change detection metrics.

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47 pages