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2022 A robust bootstrap change point test for high-dimensional location parameter
Mengjia Yu, Xiaohui Chen
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Electron. J. Statist. 16(1): 1096-1152 (2022). DOI: 10.1214/21-EJS1915

Abstract

We consider the problem of change point detection for high-dimensional distributions in a location family when the dimension can be much larger than the sample size. In change point analysis, the widely used cumulative sum (CUSUM) statistics are sensitive to outliers and heavy-tailed distributions. In this paper, we propose a robust, tuning-free (i.e., fully data-dependent), and easy-to-implement change point test that enjoys strong theoretical guarantees. To achieve the robust purpose in a nonparametric setting, we formulate the change point detection in the multivariate U-statistics framework with anti-symmetric and nonlinear kernels. Specifically, the within-sample noise is canceled out by anti-symmetry of the kernel, while the signal distortion under certain nonlinear kernels can be controlled such that the between-sample change point signal is magnitude preserving. A (half) jackknife multiplier bootstrap (JMB) tailored to the change point detection setting is proposed to calibrate the distribution of our -norm aggregated test statistic. Subject to mild moment conditions on kernels, we derive the uniform rates of convergence for the JMB to approximate the sampling distribution of the test statistic, and analyze its size and power properties. Extensions to multiple change point testing and estimation are discussed with illustration from numerical studies.

Funding Statement

Research partially supported by NSF DMS-1404891, NSF CAREER Award DMS-1752614, and University of Illinois at Urbana-Champaign (UIUC) Research Board Awards (RB17092, RB18099).

Acknowledgments

This work is completed in part with the high-performance computing resource provided by the Illinois Campus Cluster Program at UIUC. The authors are grateful to the editor, associate editor, and referee for their insightful comments.

Citation

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Mengjia Yu. Xiaohui Chen. "A robust bootstrap change point test for high-dimensional location parameter." Electron. J. Statist. 16 (1) 1096 - 1152, 2022. https://doi.org/10.1214/21-EJS1915

Information

Received: 1 June 2020; Published: 2022
First available in Project Euclid: 3 February 2022

arXiv: 1904.03372
Digital Object Identifier: 10.1214/21-EJS1915

Subjects:
Primary: 62F40 , 62G35
Secondary: 62E17

Keywords: bootstrap , Change point analysis , Gaussian approximation , High-dimensional data , U-statistics

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