Asymptotically optimal sequential change detection for bounded means

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 mathscrP and mathscrQ respectively. In such a problem, a massive challenge is characterizing the best possible detection delay when the "hardest" pre-change law in mathscrP depends on the unknown post-change law Q∈mathscrQ. 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→∞ regime) of the order log(γ)/KLinf(Q,mathscrP). 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.