Robust Estimation of High-Dimensional Linear Regression With Changepoints

Xiaolong Cui, Haoyu Geng, Zhaojun Wang, Changliang Zou

Abstract: The identification of changes in linear models is a fundamental problem encountered in various applications. Traditional methods often encounter difficulties when attempting to identify changepoints in the presence of heavy-tailed distribution. This paper focuses on the study of high-dimensional linear models with multiple structural changes in the presence of heavy-tailed errors, especially for those errors without moment conditions. We first propose a robust method that simultaneously estimates regression coefficient and changepoint by incorporating $\ell _{1}$ norm penalized Wilcoxon rank loss minimization for single changepoint estimation. Furthermore, we extend it to multiple changepoints estimation based on a novel two-step moving window mechanism that combines fast coarse grid screening and an efficient refinement. Our method exhibits robustness against heavy-tailed random errors while maintaining high efficiency for normal random errors. Theoretically, we establish non-asymptotic error bounds with a near-oracle rate for the estimates of both the coefficient and the changepoint under weak conditions on the random error distribution. Numerical results provide evidence for the validity and effectiveness of the proposed approach.