  Recently, high-dimensional heterogeneous data have attracted a lot of
attention and discussion. Under heterogeneity, semiparametric regression is a
popular choice to model data in statistics. In this paper, we take advantages
of expectile regression in computation and analysis of heterogeneity, and
propose the regularized partially linear additive expectile regression with
nonconvex penalty, for example, SCAD or MCP for such high-dimensional
heterogeneous data. We focus on a more realistic scenario: the regression error
is heavy-tailed distributed and only has finite moments, which is violated with
the classical sub-gaussian distribution assumption and more common in practise.
Under some regular conditions, we show that with probability tending to one,
the oracle estimator is one of the local minima of our optimization problem.
The theoretical study indicates that the dimension cardinality of linear
covariates our procedure can handle with is essentially restricted by the
moment condition of the regression error. For computation, since the
corresponding optimization problem is nonconvex and nonsmooth, we derive a
two-step algorithm to solve this problem. Finally, we demonstrate that the
proposed method enjoys good performances in estimation accuracy and model
selection through Monto Carlo simulation studies and a real data example.
What's more, by taking different expectile weights $\alpha$, we are able to
detect heterogeneity and explore the entire conditional distribution of the
response variable, which indicates the usefulness of our proposed method for
analyzing high dimensional heterogeneous data.
