李顺勇1, 卫夏利1, 张晓琴2,等.异方差下正则化Expectile回归的变量选择[J].河南理工大学学报（自然科学版）,2020,39(4):125-132.

LI Shunyong1, WEI Xiali1, ZHANG Xiaoqin2,et al.Variable selection in regularized expectile regression with heteroscedasticity[J].Journal of Henan Polytechnic University(Natural Science) ,2020,39(4):125-132.

异方差下正则化Expectile回归的变量选择

李顺勇1, 卫夏利1, 张晓琴2

1.山西大学数学科学学院，山西 太原 030006；2.山西财经大学统计学院，山西 太原 030006

摘要:为了解决异方差存在时最小二乘回归不适用的问题，基于最小化非对称L₂范数的Expectile 回归，通过引入一种非凸惩罚（minimax concave penalty ， MCP），提出带有MCP惩罚项的正则化Expectile回归模型，可以同时实现模型的变量选择和异方差检测，挖掘自变量与因变量之间更完整关系。传统方法假设随机误差项独立同分布且具有有限阶矩，本文方法将该假设弱化为误差项独立但不同分布，具有有限阶矩。证明了在一定条件下，带有MCP惩罚项的Expectile回归得到的估计量具有Oracle性质。数值模拟结果表明，该方法在变量选择上具有优良的表现，且通过不同Expectile权重值时的自变量集合变化，能有效检测出异方差。

关键词:Expectile回归;独立但不同分布;异方差;非凸惩罚;变量选择;Oracle性质

doi:10.16186/j.cnki.1673-9787.2020.4.18

基金项目:国家自然科学基金资助项目（81803962 ）；山西省基础研究计划项目（201701D121004 ）；山西省回国留学人员科研项目 （2017-020）；山西省科技计划研发项目（2018140105000084 ）；山西省基础研究计划项目（201901D111320）

收稿日期:2019/11/19

修回日期:2020/01/16

出版日期:2020/07/15

Variable selection in regularized expectile regression with heteroscedasticity

LI Shunyong1, WEI Xiali1, ZHANG Xiaoqin2

1.School of Mathmatical Sciences，Shanxi University，Taiyuan  030006，Shanxi，China；2.School of Statistics，Shanxi University of Finance and Economics,Taiyuan  030006，Shanxi， China

Abstract:Ordinary least square regression is invalid when the heteroscedasticity exists. To solve the problem of regression model construction and variable selection when the heteroscedasticity existed， based on expectile regression which relied on asymmetric L₂ norm the regularized expectile regression with minimax concave penalty（MCP）was proposed to achieve the variable selection and heteroscedasticity detection simultaneously，and more complete relationship between the response variable and predictors was captured. Existing achievements mostly based on the assumption that the error followed normal distribution or it was independent and identically distributed. The proposed method weakened this assumption to the assumption that the random errors were independent but different distributions（non-iid）and had finite moments. Under some conditions，the Oracle property of the proposed estimator was obtained.Numerical simulation showed that the proposed method had good performance in variable selection and could detect heteroscedasticity efficiently according to the change of covariates set under different expectile values.

Key words:Expectile regression;minmax concave;heteroscedasticity;non-concave penalty;variable selection;Oracle property

异方差下正则化Expectile回归的变量选择_李顺勇.pdf