Robust censored regression with l1-norm regularization

TL;DR

This paper introduces a robust l1-norm penalized estimator for censored linear regression that effectively handles outliers, maintains efficiency, and is computationally feasible, with improved outlier detection capabilities.

Contribution

It proposes a novel l1-penalized estimator for censored regression that is robust to outliers and retains asymptotic efficiency, along with an outlier detection method.

Findings

Estimator achieves the same asymptotic variance as Stute's estimator.

The method is computationally efficient via convex optimization.

Outlier detection improves finite sample performance.

Abstract

This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the same asymptotic variance as Stute's estimator in the censored linear model without outliers. Hence, there is no loss of efficiency as a result of robustness. Tests and confidence sets can therefore rely on the theory developed by Stute. The outlined procedure is also computationally advantageous, since it amounts to solving a convex optimization program. We also propose a second estimator which uses the proposed penalized Stute estimator as a first step to detect outliers. It has similar theoretical properties but better performance in finite samples as assessed by simulations.