Universality of e-detectors for ARL control

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.