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

Abstract

This article 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 ${\ell }_1$-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.

Keywords:

• Robust regression
• high-dimensional regression
• ${\ell }_1$-norm penalization
• unknown variance

MSC 2020 SUBJECT CLASSIFICATION:

• Primary 62J07
• secondary 62J05
• 62F35

Acknowledgments

The author wishes to thank the reviewer for his valuable suggestions and comments.