Robust penalized least squares of depth trimmed residuals regression for high-dimensional data

TL;DR

This paper introduces a robust penalized regression method for high-dimensional data that effectively handles outliers and contaminated points, outperforming existing methods in estimation and prediction accuracy.

Contribution

A novel robust penalized regression approach based on depth-trimmed residuals, addressing robustness issues in high-dimensional settings with outliers.

Findings

The new method outperforms competitors in simulations.

Most existing penalized methods are vulnerable to single outliers.

The approach is effective on real high-dimensional datasets.

Abstract

Challenges with data in the big-data era include (i) the dimension $p$ is often larger than the sample size $n$ (ii) outliers or contaminated points are frequently hidden and more difficult to detect. Challenge (i) renders most conventional methods inapplicable. Thus, it attracts tremendous attention from statistics, computer science, and bio-medical communities. Numerous penalized regression methods have been introduced as modern methods for analyzing high-dimensional data. Disproportionate attention has been paid to the challenge (ii) though. Penalized regression methods can do their job very well and are expected to handle the challenge (ii) simultaneously. Most of them, however, can break down by a single outlier (or single adversary contaminated point) as revealed in this article. The latter systematically examines leading penalized regression methods in the literature in terms of their robustness, provides quantitative assessment, and reveals that most of them can break down by a single outlier. Consequently, a novel robust penalized regression method based on the least sum of squares of depth trimmed residuals is proposed and studied carefully. Experiments with simulated and real data reveal that the newly proposed method can outperform some leading competitors in estimation and prediction accuracy in the cases considered.