This article is concerned with change point detection for object-valued data that reside in a metric space, which has attracted some recent interests in statistics and econometrics literature. The existing methods either focus on independent data or can only detect change in the Fréchet mean or variance. In this paper, we propose a self-normalization (SN, hereafter) based statistic for detecting a shift in the marginal distribution of object-valued time series. Our test is universally applicable to a wide range of object-valued data, such as distributional and network data, and can accommodate weak serial dependence. In addition the proposed test statistic is almost tuning parameter free, has pivotal limiting null distribution and only uses the pairwise distances. When combined with the Wild Binary Segmentation algorithm (WBS, hereafter), our statistic can be used to estimate the number and locations of multiple change points. Asymptotic results for our SN based statistic are derived under both null and local alternatives in the single change point setting. For the first time, the WBS estimation consistency is shown for a broad class of object-valued time series and in a nonparametric setting, which requires new non-standard theoretical arguments. Extensive numerical experiments and real data analysis are conducted to illustrate the effectiveness and broad applicability of our proposed method.

翻译：本文关注度量空间中对象值数据的变点检测问题，该问题近年来在统计学和计量经济学文献中引起了一定关注。现有方法或聚焦于独立数据，或仅能检测Fréchet均值或方差的变化。本文提出一种基于自归一化（以下简称SN）的统计量，用于检测对象值时间序列边际分布的偏移。我们的检验方法普遍适用于分布数据、网络数据等多种对象值数据，并能处理弱序列相关性。此外，所提出的检验统计量几乎无需调整参数，具有枢轴渐近零分布，且仅使用成对距离。当与Wild Binary Segmentation算法（以下简称WBS）结合时，该统计量可用于估计多个变点的数量和位置。我们在单变点设定下推导了基于SN的统计量在原假设和局部备择假设下的渐近结果。首次在非参数设定下证明了WBS对一大类对象值时间序列的估计相合性，这需要引入非标准的理论论证。通过大量数值实验和实际数据分析，验证了所提方法的有效性和广泛适用性。