Asymptotically optimal sequential change detection for bounded means

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.