Robust Inference for High-Dimensional Linear Models via Residual Randomization

TL;DR

This paper introduces a residual randomization method for robust inference in high-dimensional linear models, effectively handling heavy-tailed errors, covariates, and clustered errors, outperforming existing methods in challenging scenarios.

Contribution

It develops a novel residual randomization procedure that extends robust inference capabilities to heavy-tailed and clustered error settings in high-dimensional linear models.

Findings

Outperforms state-of-the-art methods in heavy-tailed and small-sample scenarios.

Maintains competitiveness in standard settings.

Validated through extensive simulations.

Abstract

We propose a residual randomization procedure designed for robust Lasso-based inference in the high-dimensional setting. Compared to earlier work that focuses on sub-Gaussian errors, the proposed procedure is designed to work robustly in settings that also include heavy-tailed covariates and errors. Moreover, our procedure can be valid under clustered errors, which is important in practice, but has been largely overlooked by earlier work. Through extensive simulations, we illustrate our method's wider range of applicability as suggested by theory. In particular, we show that our method outperforms state-of-art methods in challenging, yet more realistic, settings where the distribution of covariates is heavy-tailed or the sample size is small, while it remains competitive in standard, "well behaved" settings previously studied in the literature.