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

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.

References in corpus (6)

• Simultaneous analysis of Lasso and Dantzig selector
• Confidence Intervals and Hypothesis Testing for High-Dimensional Regression
• Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming
• Sparse least trimmed squares regression for analyzing high-dimensional large data sets
• Penalized Weighted Least Squares for Outlier Detection and Robust Regression
• High-dimensional robust regression and outliers detection with SLOPE