# Universality of e-detectors for ARL control

arXiv: https://arxiv.org/abs/2608.12660 · HTML: https://mlsanity.com/paper/2608.12660/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Aug 2026.

## Built on by (most cited first, 1 of 1)

- [A complete characterization of sequential testability and change detectability in i.i.d. models](https://mlsanity.com/paper/2609.05752/) (2026)

## Abstract (arXiv)

An e-detector for a pre-change class mathcal P is a nonnegative process M such that mathbb EP[Mτ] ≤ mathbb EP[τ] for all stopping times τ and all P ∈ mathcal P. Thresholding e-detectors controls the average run length (ARL): declaring a change at the first time Tb when M crosses b ensures that infP ∈ mathcal Pmathbb EP[T] ≥ b. But e-detectors do substantially more than control the ARL; they also satisfy a optional-horizon inequality: [ P(Tb≤σ)≤ mathbb EP[σ]/b ] for every data-dependent stopping time (monitoring horizon) (σ) and P∈ mathcal P. In particular, every e-detector-based procedure obeys P(T≤ t)≤ 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 weak e-detectors, that only require mathbb EP[Mτ] ≤ mathbb EP[τ] to hold at all threshold stopping times Tb. Appendices present universal representations for other (less common) change detection metrics.