Asymptotically optimal sequential change detection for bounded means
Ashwin Ram, Aaditya Ramdas
TL;DR
The paper addresses quickest changepoint detection under an Average Run Length (ARL) constraint with composite pre-change and post-change distributions. It derives a universal lower bound on the worst-case detection delay in terms of the least-favorable KL divergence $I(Q;\mathcal{P})$ and then constructs a practically implementable, asymptotically optimal detector for the bounded-mean setting using a mixture Shiryaev–Roberts (e-detector) approach, achieving $\lim_{\gamma\to\infty} \mathcal{C}_Q(T^{\mathrm{BM}}_\gamma)/\log\gamma = 1/\mathrm{KL}_{\inf}(Q; m)$. A uniform minimax guarantee is shown over separated alternatives, establishing tight first-order constants under the ARL constraint. The key insight is that the first-order change-detection delay is governed by the information quantity $\mathrm{KL}_{\inf}(Q;\mathcal{P})$, enabling robust, distribution-ambiguous detection rules with explicit, sharp asymptotics. The work lays groundwork for extending the framework beyond IID and bounded-mean settings, and connects to broader e-processes and robust sequential inference methods.
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 ($γ\to\infty$ regime) of the order $\log(γ)/\mathrm{KL}_{\mathrm{inf}}(Q,\mathscr{P})$. We show achievability of this universal lower bound by proving a tight matching upper bound (with the same sharp $\logγ$ 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.