We study the parametric online changepoint detection problem, where the underlying distribution of the streaming data changes from a known distribution to an alternative that is of a known parametric form but with unknown parameters. We propose a joint detection/estimation scheme, which we call Window-Limited CUSUM, that combines the cumulative sum (CUSUM) test with a sliding window-based consistent estimate of the post-change parameters. We characterize the optimal choice of the window size and show that the Window-Limited CUSUM enjoys first-order asymptotic optimality as average run length approaches infinity under the optimal choice of window length. Compared to existing schemes with similar asymptotic optimality properties, our test can be much faster computed because it can recursively update the CUSUM statistic by employing the estimate of the post-change parameters. A parallel variant is also proposed that facilitates the practical implementation of the test. Numerical simulations corroborate our theoretical findings.

翻译：我们研究参数化在线变化点检测问题，其中流式数据的潜在分布从已知分布转变为具有已知参数形式但参数未知的替代分布。我们提出一种联合检测/估计方案，称为窗口受限累积和（Window-Limited CUSUM），该方案将累积和（CUSUM）检验与基于滑动窗口的变化后参数一致估计相结合。我们刻画了窗口大小的最优选择，并证明在最优窗口长度下，当平均运行长度趋近于无穷时，窗口受限CUSUM具有一阶渐近最优性。与具有类似渐近最优性的现有方案相比，我们的检验方法可通过递归更新CUSUM统计量并利用变化后参数的估计值，实现更快的计算速度。我们还提出一种并行变体，以促进该检验方法的实际实施。数值模拟验证了我们的理论发现。