# Inference robust to outliers with l1-norm penalization

## TL;DR

This paper introduces a robust linear regression method using l1-norm penalization that effectively handles outliers, achieves standard asymptotic properties, and simplifies inference procedures.

## Contribution

It develops a square-root lasso-based estimator for outlier-robust inference with theoretical guarantees and practical computational advantages.

## Key findings

- Estimator achieves asymptotic normality with standard variance.
- Method handles growing number of outliers with sample size.
- Provides a practical alternative to OLS for robust inference.

## Abstract

This paper considers the problem of inference in a linear regression model with outliers where the number of outliers can grow with sample size but their proportion goes to 0. We apply the square-root lasso estimator penalizing the l1-norm of a random vector which is non-zero for outliers. We derive rates of convergence and asymptotic normality. Our estimator has the same asymptotic variance as the OLS estimator in the standard linear model. This enables to build tests and confidence sets in the usual and simple manner. The proposed procedure is also computationally advantageous as it amounts to solving a convex optimization program. Overall, the suggested approach constitutes a practical robust alternative to the ordinary least squares estimator.