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Recently, high-dimensional heterogeneous data have attracted a lot ofattention and discussion. Under heterogeneity, semiparametric regression is apopular choice to model data in statistics. In this paper, we take advantagesof expectile regression in computation and analysis of heterogeneity, andpropose the regularized partially linear additive expectile regression withnonconvex penalty, for example, SCAD or MCP for such high-dimensionalheterogeneous data. We focus on a more realistic scenario: the regression erroris heavy-tailed distributed and only has finite moments, which is violated withthe 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 linearcovariates our procedure can handle with is essentially restricted by themoment condition of the regression error. For computation, since thecorresponding optimization problem is nonconvex and nonsmooth, we derive atwo-step algorithm to solve this problem. Finally, we demonstrate that theproposed method enjoys good performances in estimation accuracy and modelselection through Monto Carlo simulation studies and a real data example.What's more, by taking different expectile weights $\alpha$, we are able todetect heterogeneity and explore the entire conditional distribution of theresponse variable, which indicates the usefulness of our proposed method foranalyzing high dimensional heterogeneous data.

Semiparametric Expectile Regression for High-dimensional Heavy-tailed and Heterogeneous Data