  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.
