Asymptotically optimal sequential change detection for bounded means

TL;DR

This paper establishes a universal lower bound and demonstrates an asymptotically optimal sequential change detection method for bounded means, addressing the challenge of composite pre- and post-change laws.

Contribution

It derives a universal sharp lower bound for detection delay and proves an asymptotically optimal detection procedure in the bounded mean setting.

Findings

Universal lower bound of detection delay established.

Achievability of the lower bound shown with a tight upper bound.

Optimal detection procedure proven for bounded mean shifts.

Abstract

We consider the problem of quickest changepoint detection under the Average Run Length (ARL) constraint where the pre-change and post-change laws lie in composite families $\mathscr{P}$ and $\mathscr{Q}$ respectively. In such a problem, a massive challenge is characterizing the best possible detection delay when the "hardest" pre-change law in $\mathscr{P}$ depends on the unknown post-change law $Q\in\mathscr{Q}$. And typical simple-hypothesis likelihood-ratio arguments for Page-CUSUM and Shiryaev-Roberts do not at all apply here. To that end, we derive a universal sharp lower bound in full generality for any ARL-calibrated changepoint detector in the low type-I error ($\gamma\to\infty$ regime) of the order $\log(\gamma)/\mathrm{KL}_{\mathrm{inf}}(Q,\mathscr{P})$. We show achievability of this universal lower bound by proving a tight matching upper bound (with the same sharp $\log\gamma$ constant) in the important bounded mean detection setting. In addition, for separated mean shifts, we also we derive a uniform minimax guarantee of this achievability over the alternatives.