  High-dimensional data subject to heavy-tailed phenomena and heterogeneity are
commonly encountered in various scientific fields and bring new challenges to
the classical statistical methods. In this paper, we combine the asymmetric
square loss and huber-type robust technique to develop the robust expectile
regression for ultrahigh dimensional heavy-tailed heterogeneous data. Different
from the classical huber method, we introduce two different tuning parameters
on both sides to account for possibly asymmetry and allow them to diverge to
reduce bias induced by the robust approximation. In the regularized framework,
we adopt the generally folded concave penalty function like the SCAD or MCP
penalty for the seek of bias reduction. We investigate the finite sample
property of the corresponding estimator and figure out how our method plays its
role to trades off the estimation accuracy against the heavy-tailed
distribution. Also, noting that the robust asymmetric loss function is
everywhere differentiable, based on our theoretical study, we propose an
efficient first-order optimization algorithm after locally linear approximation
of the non-convex problem. Simulation studies under various distributions
demonstrates the satisfactory performances of our method in coefficient
estimation, model selection and heterogeneity detection.
