The half-quadratic approach for high-dimensional robust M-estimation

In this paper we investigate a unified framework, the half-quadratic (HQ) approach, for regularised robust M-estimation. This approach streamlines both numerical and theoretical computations. We introduce augmented objective functions to facilitate robust parameter estimation in both fixed- and high-dimensional settings. These objective functions serve the dual purpose of estimating parameters robustly and detecting influential data points. Specifically, the HQ approach is scrutinised for high-dimensional robust regression, examining the $ l_{1} $ l1- and $ l_{2} $ l2-estimation errors of the proposed regression estimator across various loss functions. Nonasymptotic upper bounds are derived for estimation errors in high-dimensional scenarios. We demonstrate that optimal estimation accuracy can be attained by employing loss functions with bounded derivatives, even in the presence of influential data points. These results remain hold even with heavy-tailed error distributions. Furthermore, the proposed HQ approximation method is compared with existing methods through numerical studies. Additionally, a real dataset is analysed using this proposed methodology.