A robust bootstrap change point test for high-dimensional location parameter

TL;DR

This paper introduces a robust, data-driven change point detection method for high-dimensional data that is resistant to outliers and heavy tails, with strong theoretical guarantees and practical extensions.

Contribution

It develops a novel, robust, and tuning-free change point test based on multivariate U-statistics and a tailored bootstrap, improving detection in high-dimensional, heavy-tailed settings.

Findings

The proposed test is robust against outliers and heavy tails.

The bootstrap calibration accurately approximates the test statistic distribution.

Numerical studies demonstrate the method's effectiveness in multiple change point scenarios.

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 $\ell^{\infty}$-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.