A General U-Statistic Framework for High-Dimensional Multiple Change-Point Analysis

Abstract

High-dimensional change-point analysis is essential in modern statistical inference. However, existing methods are often designed either for specific parameters (e.g., mean or variance) or for particular tasks (e.g., testing or estimation), making them difficult to generalize. Moreover, they typically rely on restrictive distributional assumptions, limiting their robustness to heavy-tailed data. We propose a unified framework for testing, estimating, and inferring multiple change points in high-dimensional data. Our approach leverages a two-sample U-statistic within a moving window, allowing flexible kernel function selection to accommodate structural changes in general parameters such as variance changes or robust statistics. For testing, we develop an L-infinity norm-based statistic with a high-dimensional multiplier bootstrap procedure, achieving minimax-optimal power under sparse alternatives. For estimation, we construct an initial estimator for the change-point number and locations and refine it using the U-statistic Projection Refinement Algorithm (U-PRA), attaining minimax-optimal localization rates. We further derive the asymptotic distribution of refined estimators, enabling valid confidence interval construction. Extensive numerical experiments demonstrate the better performance of our method across various settings, including heavy-tailed distributions. Applications to genomic copy number variation data highlight its practical utility. An R package implementing the proposed method, U-PRA, is publicly available at https://github.com/liubin0145/R-codes-UPRA/.