Enveloped Huber Regression

TL;DR

The paper introduces enveloped Huber regression (EHR), a robust and more efficient estimation method that leverages the envelope assumption and the Generalized Method of Moments, especially effective with heavy-tailed errors.

Contribution

It proposes a novel EHR method combining envelope models with Huber regression, improving efficiency over traditional HR and ENV models under heavy-tailed error distributions.

Findings

EHR is more efficient than HR and ENV with heavy-tailed errors.

EHR maintains near-normal efficiency with normal errors.

Simulation studies confirm theoretical advantages.

Abstract

Huber regression (HR) is a popular robust 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. Extensive simulation studies confirm the messages from the asymptotic theory. EHR is further illustrated on a real dataset.