Problem
Research questions and friction points this paper is trying to address.
e-detector
average run length
change detection
optional-horizon inequality
stopping time
Universality of e-detectors for ARL control
🤖 AI Summary
This work addresses the challenge of rigorously controlling the average run length (ARL) in change-point detection while avoiding premature false alarms. To this end, the authors propose the e-detector framework, which guarantees a lower bound on ARL through thresholding of non-negative processes and satisfies an optional stopping inequality, ensuring false alarm control under arbitrary data-dependent monitoring times. The study further establishes the equivalence between stopping times satisfying this inequality and e-detectors, and introduces a weaker notion—weak e-detectors—that only requires the condition to hold at threshold crossing times. Leveraging martingale theory and stopping time analysis, the paper develops a general representation theorem for ARL-controlling procedures, proving that any detector meeting the ARL constraint can be characterized by a weak e-detector, thereby providing a unified and rigorous theoretical foundation for change-point detection.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
e-detector
average run length
optional-horizon inequality
change detection
universal representation
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