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Abstract:
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We consider the problem of change point detection for high-dimensional distributions in a location family when dimension can be much larger than sample size. In change point analysis, the widely used cumulative sum 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 robustness in a nonparametric setting, we formulate the detection in the multivariate U-statistics framework with anti-symmetric and nonlinear kernels. Specifically, the within-sample noise is cancelled out by anti-symmetry, while the nonlinear signal distortion can be controlled such that the between-sample 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.
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