It is well known that the conventional cumulative sum (CUSUM) test suffers from low power and large detection delay. In order to improve the power of the test, we propose two alternative statistics. The backward CUSUM detector considers the recursive residuals in reverse chronological order, whereas the stacked backward CUSUM detector sequentially cumulates a triangular array of backwardly cumulated residuals. A multivariate invariance principle for partial sums of recursive residuals is given, and the limiting distributions of the test statistics are derived under local alternatives. In the retrospective context, the local power of the tests is shown to be substantially higher than that of the conventional CUSUM test if a break occurs in the middle or at the end of the sample. When applied to monitoring schemes, the detection delay of the stacked backward CUSUM is found to be much shorter than that of the conventional monitoring CUSUM procedure. Furthermore, we propose an estimator of the break date based on the backward CUSUM detector and show that in monitoring exercises this estimator tends to outperform the usual maximum likelihood estimator. Finally, an application of the methodology to COVID-19 data is presented.

众所周知，传统的累积和（CUSUM）测试存在功耗低、检测延迟大的问题。为了提高测试的功效，我们提出了两种替代统计方法。后向 CUSUM 检测器以逆时间顺序考虑递归残差，而堆叠后向 CUSUM 检测器顺序累积向后累积残差的三角形阵列。给出了递归残差部分和的多元不变性原理，并在局部备选方案下导出了检验统计量的极限分布。在回顾性背景下，如果断裂发生在样本的中间或末端，则测试的局部功效显着高于传统的 CUSUM 检验。当应用于监测方案时，发现堆叠后向 CUSUM 的检测延迟比传统的监控 CUSUM 程序的检测延迟要短得多。此外，我们提出了一种基于后向累积和检测器的中断日期估计器，并表明在监控练习中，该估计器往往优于通常的最大似然估计器。最后，介绍了该方法在 COVID-19 数据中的应用。