Change point testing is a well-studied problem in statistics. Owing to the emergence of high-dimensional data with structural breaks, there has been a recent surge of interest in developing methods to accommodate high-dimensionality. In practice, when the dimension is less than the sample size but is not small, it is often unclear whether a method that is tailored to high-dimensional data or simply a classical method that is developed and justified for low-dimensional data is preferred. In addition, the methods designed for low-dimensional data may not work well in the high-dimensional environment and vice versa. This naturally brings up the question of whether there is a change point test that can work for data of low, medium, and high dimensions. In this paper, we first propose a dimension-agnostic testing procedure targeting a single change point in the mean of multivariate time series. Our new test is inspired by the recent work of arXiv:2011.05068, who formally developed the notion of ``dimension-agnostic" in several testing problems for iid data. We develop a new test statistic by adopting their sample splitting and projection ideas, and combining it with the self-normalization method for time series. Using a novel conditioning argument, we are able to show that the limiting null distribution for our test statistic is the same regardless of the dimensionality and the magnitude of cross-sectional dependence. The power analysis is also conducted to understand the large sample behavior of the proposed test. Furthermore, we present an extension to test for multiple change points in the mean and derive the limiting distributions of the new test statistic under both the null and alternatives. Through Monte Carlo simulations, we show that the finite sample results strongly corroborate the theory and suggest that the proposed tests can be used as a benchmark for many time series data.

翻译：变点检验是统计学中一个被充分研究的问题。随着存在结构性断裂的高维数据的出现，近年来人们日益关注开发能够适应高维性的方法。在实践中，当维度小于样本量但并不小时，常常不清楚是应该选择专为高维数据设计的方法，还是选择为低维数据开发并验证的经典方法。此外，针对低维数据设计的方法可能无法在高维环境中良好运行，反之亦然。这自然引出一个问题：是否存在一种变点检验，能够适用于低维、中维和高维数据？本文首先提出了一种面向多元时间序列均值中单个变点的维度无关检验程序。我们的新检验受arXiv:2011.05068近期工作的启发，该工作正式提出了独立同分布数据中若干检验问题的"维度无关"概念。我们采用其样本分裂和投影思想，并结合时间序列的自标准化方法，构建了新的检验统计量。通过一种新颖的条件论证，我们能够证明无论数据维度和截面依赖强度如何，检验统计量的极限零分布均保持一致。我们还进行了功效分析以理解所提检验的大样本行为。此外，我们将其扩展至均值中多个变点的检验，并推导了新检验统计量在原假设和备择假设下的极限分布。通过蒙特卡洛模拟，我们证明有限样本结果与理论高度吻合，表明所提检验可作为众多时间序列数据的基准方法。