We propose a location-adaptive self-normalization (SN) based test for change points in time series. The SN technique has been extensively used in change-point detection for its capability to avoid direct estimation of nuisance parameters. However, we find that the power of the SN-based test is susceptible to the location of the break and may suffer from a severe power loss, especially when the change occurs at the early or late stage of the sequence. This phenomenon is essentially caused by the unbalance of the data used before and after the change point when one is building a test statistic based on the cumulative sum (CUSUM) process. Hence, we consider leaving out the samples far away from the potential locations of change points and propose an optimal data selection scheme. Based on this scheme, a new SN-based test statistic adaptive to the locations of breaks is established. The new test can significantly improve the power of the existing SN-based tests while maintaining a satisfactory size. It is a unified treatment that can be readily extended to tests for general quantities of interest, such as the median and the model parameters. The derived optimal subsample selection strategy is not specific to the SN-based tests but is applicable to any method that relies on the CUSUM process, which may provide new insights in the area for future research.

翻译：我们提出了一种基于位置自适应自归一化（SN）的变点检验方法。SN技术因其能避免直接估计冗余参数而在变点检测领域得到广泛应用。然而，我们发现基于SN的检验的统计功效易受断点位置影响，尤其在序列早期或晚期发生变点时可能出现严重的功效损失。这一现象实质上源于构建基于累积和（CUSUM）过程的检验统计量时，变点前后数据的不均衡性。因此，我们考虑剔除远离潜在变点位置的样本，并提出一种最优数据选择方案。基于该方案，建立了适应断点位置的新SN检验统计量。新检验方法在维持良好检验水平的同时，能显著提升现有SN检验的功效。该统一处理框架可便捷地推广至中位数、模型参数等一般关注量的检验。所推导的最优子样本选择策略并非SN检验特用，而是适用于任何依赖CUSUM过程的方法，可为该领域的未来研究提供新视角。