M-estimation in high-dimensional linear model

TL;DR

This paper investigates the properties of M-estimators in high-dimensional linear regression, demonstrating their robustness and versatility through theoretical analysis and numerical simulations.

Contribution

It introduces a framework for M-estimation with local linear approximation penalties, unifying various regression methods and analyzing their properties in high-dimensional settings.

Findings

M-estimators exhibit good robustness properties.

The proposed method unifies multiple regression techniques.

Numerical simulations confirm the effectiveness of the approach.

Abstract

We mainly study the M-estimation method for the high-dimensional linear regression model, and discuss the properties of M-estimator when the penalty term is the local linear approximation. In fact, M-estimation method is a framework, which covers the methods of the least absolute deviation, the quantile regression, least squares regression and Huber regression. We show that the proposed estimator possesses the good properties by applying certain assumptions. In the part of numerical simulation, we select the appropriate algorithm to show the good robustness of this method