  High-dimensional data can often display heterogeneity due to heteroscedastic
variance or inhomogeneous covariate effects. Penalized quantile and expectile
regression methods offer useful tools to detect heteroscedasticity in
high-dimensional data. The former is computationally challenging due to the
non-smooth nature of the check loss, and the latter is sensitive to
heavy-tailed error distributions. In this paper, we propose and study
(penalized) robust expectile regression (retire), with a focus on iteratively
reweighted $\ell_1$-penalization which reduces the estimation bias from
$\ell_1$-penalization and leads to oracle properties. Theoretically, we
establish the statistical properties of the retire estimator under two regimes:
(i) low-dimensional regime in which $d \ll n$; (ii) high-dimensional regime in
which $s\ll n\ll d$ with $s$ denoting the number of significant predictors. In
the high-dimensional setting, we carefully characterize the solution path of
the iteratively reweighted $\ell_1$-penalized retire estimation, adapted from
the local linear approximation algorithm for folded-concave regularization.
Under a mild minimum signal strength condition, we show that after as many as
$\log(\log d)$ iterations the final iterate enjoys the oracle convergence rate.
At each iteration, the weighted $\ell_1$-penalized convex program can be
efficiently solved by a semismooth Newton coordinate descent algorithm.
Numerical studies demonstrate the competitive performance of the proposed
procedure compared with either non-robust or quantile regression based
alternatives.
