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.

翻译：我们考虑在平均游程长度约束下的最快变点检测问题，其中变点前与变点后的分布律分别属于复合族 $\mathscr{P}$ 和 $\mathscr{Q}$。在此类问题中，一个重大挑战在于刻画当 $\mathscr{P}$ 中“最困难”的变点前分布律依赖于未知的变点后分布 $Q\in\mathscr{Q}$ 时可能达到的最佳检测延迟。此时，针对 Page-CUSUM 和 Shiryaev-Roberts 检验的典型简单假设似然比论证完全无法适用。为此，我们在低第一类错误（$γ\to\infty$ 情形）下，对任意满足 ARL 校准的变点检测器，推导出一个完全普适的尖锐下界，其阶为 $\log(γ)/\mathrm{KL}_{\mathrm{inf}}(Q,\mathscr{P})$。通过在重要的均值有界检测设定中证明一个紧匹配的上界（具有相同的尖锐 $\logγ$ 常数），我们表明该普适下界是可以达到的。此外，对于分离的均值偏移情形，我们还推导了该可达性在备择假设上的一致极小极大保证。