Enveloped Huber Regression

Abstract

Huber regression (HR) is a popular flexible alternative to the least squares regression when the error follows a heavy-tailed distribution. We propose a new method called the enveloped Huber regression (EHR) by considering the envelope assumption that there exists some subspace of the predictors that has no association with the response, which is referred to as the immaterial part. More efficient estimation is achieved via the removal of the immaterial part. Different from the envelope least squares (ENV) model whose estimation is based on maximum normal likelihood, the estimation of the EHR model is through Generalized Method of Moments. The asymptotic normality of the EHR estimator is established, and it is shown that EHR is more efficient than HR. Moreover, EHR is more efficient than ENV when the error distribution is heavy-tailed, while maintaining a small efficiency loss when the error distribution is normal. Moreover, our theory also covers the heteroscedastic case in which the error may depend on the covariates. The envelope dimension in EHR is a tuning parameter to be determined by the data in practice. We further propose a novel generalized information criterion (GIC) for dimension selection and establish its consistency. Extensive simulation studies confirm the messages from our theory. EHR is further illustrated on a real dataset. Supplementary materials for this article are available online.

KEYWORDS:

• Asymptotics efficiency
• Envelope model
• Generalized information criterion
• Heavy-tailed distributions
• Huber regression

Supplementary Materials

The supplementary file contains the theoretical proofs, additional real data example and additional discussions. It also contains all codes to replicate the simulation and real data analysis results.

Acknowledgments

We sincerely thank the AE and two referees for their helpful and constructive comments which greatly improved the quality of this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

Zhou’s research is supported in part by HKBU 162864 and 179424. Zou’s research is supported in part by NSF grants 1915842, 2015120 and 2220286.