High-dimensional inference robust to outliers with l1-norm penalization

TL;DR

This paper introduces a robust high-dimensional inference method that effectively handles outliers by combining l1-norm penalization and a two-step estimation process, achieving efficiency and computational simplicity.

Contribution

It proposes a novel two-step inference procedure using square-root lasso and OLS that is robust to outliers and attains semiparametric efficiency in high-dimensional settings.

Findings

Asymptotic normality of the two-step estimator established.

Method attains semiparametric efficiency bound in outlier-free models.

Computationally efficient with convex optimization solutions.

Abstract

This paper studies inference in the high-dimensional linear regression model with outliers. Sparsity constraints are imposed on the vector of coefficients of the covariates. The number of outliers can grow with the sample size while their proportion goes to 0. We propose a two-step procedure for inference on the coefficients of a fixed subset of regressors. The first step is a based on several square-root lasso l1-norm penalized estimators, while the second step is the ordinary least squares estimator applied to a well chosen regression. We establish asymptotic normality of the two-step estimator. The proposed procedure is efficient in the sense that it attains the semiparametric efficiency bound when applied to the model without outliers under homoscedasticity. This approach is also computationally advantageous, it amounts to solving a finite number of convex optimization programs.