Abstract
This article considers change-point testing and estimation for a sequence of high-dimensional data. In the case of testing for a mean shift for high-dimensional independent data, we propose a new test which is based on U-statistic in Chen and Qin (Ann. Statist. 38 (2010) 808–835) and utilizes the self-normalization principle (Shao J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 (2010) 343–366; Shao and Zhang J. Amer. Statist. Assoc. 105 (2010) 1228–1240). Our test targets dense alternatives in the high-dimensional setting and involves no tuning parameters. To extend to change-point testing for high-dimensional time series, we introduce a trimming parameter and formulate a self-normalized test statistic with trimming to accommodate the weak temporal dependence. On the theory front we derive the limiting distributions of self-normalized test statistics under both the null and alternatives for both independent and dependent high-dimensional data. At the core of our asymptotic theory, we obtain weak convergence of a sequential U-statistic based process for high-dimensional independent data, and weak convergence of sequential trimmed U-statistic based processes for high-dimensional linear processes, both of which are of independent interests. Additionally, we illustrate how our tests can be used in combination with wild binary segmentation to estimate the number and location of multiple change points. Numerical simulations demonstrate the competitiveness of our proposed testing and estimation procedures in comparison with several existing methods in the literature.
Funding Statement
Shao’s research is partially supported by NSF Grants DMS-1807023 and DMS-2014018. Vogulshev’s research is partially supported by a discovery grant from NSERC of Canada.
Acknowledgments
We would like to thank the three reviewers for their constructive comments which led to substantial improvements. We are grateful to Dr. Farida Enikeeva for sending us the code used in Enikeeva and Harchaoui (2019). Runmin Wang is Assistant Professor at Southern Methodist University, Department of Statistical Science (e-mail: runminw@smu.edu); Changbo Zhu is Postdoctoral Scholar, Department of Statistics, University of California at Davis; Stanislav Volgushev is Assistant Professor at Department of Statistical Sciences, University of Toronto (e-mail: stanislav.volgushev@utoronto.ca); Xiaofeng Shao is Professor, Department of Statistics, University of Illinois at Urbana–Champaign (e-mail: xshao@illinois.edu). Wang and Zhu are joint first authors and made equal contributions to the paper.
Citation
Runmin Wang. Changbo Zhu. Stanislav Volgushev. Xiaofeng Shao. "Inference for change points in high-dimensional data via selfnormalization." Ann. Statist. 50 (2) 781 - 806, April 2022. https://doi.org/10.1214/21-AOS2127
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