基于MCP的非对称最小二乘估计  

The Asymmetric Least Squares Estimator Based on Minimax Concave Penalty

作　　者：张晓琴 卫夏利 米子川[1] 李顺勇[3] ZHANG Xiaoqin;WEI Xiali;MI Zichuan;LI Shunyong(School of Statistics,Shanxi University of Finance and Economics,Taiyuan 030006;School of Economics and Management,Shanxi University,Taiyuan 030006;School of Mathematical Sciences,Shanxi University,Taiyuan 030006)

机构地区：[1]山西财经大学统计学院,太原030006 [2]山西大学经济与管理学院,太原030006 [3]山西大学数学科学学院,太原030006

出　　处：《系统科学与数学》2022年第5期1344-1360,共17页Journal of Systems Science and Mathematical Sciences

基　　金：国家社会科学基金项目(17BTJ010);山西省自然科学基金(201901D111320)资助课题。

摘　　要：作为一种流行的非凸惩罚,极小极大凹惩罚(MCP)在变量选择中被广泛使用.非对称最小二乘回归(ALS)区别于最小二乘回归,能够研究响应变量的整个条件分布.文章基于MCP惩罚,提出带有MCP惩罚的稀疏非对称最小二乘回归模型(MCP-ALS),并得到了相应估计量的性质.文章证明:首先,在一定的正则化条件下,当协变量维度固定时,诱导估计量具有Oracle性质.在高维模型中,当回归误差具有有限阶矩时,诱导估计量具有弱化Oracle性质.其次,通过采取不同的非对称权重值,文章提出的方法能够识别出引起异方差的协变量.数值模拟表明,文章提出的方法在变量选择上有优良的表现,并且能有效检测异方差.最后,将所提方法应用于糖尿病数据集中,实例分析表明,所提方法在实现变量选择的同时,能够挖掘解释变量与响应变量之间的潜在关系,以期对糖尿病人病情的预测和控制提供借鉴.As a promising nonconvex penalty,the minimax concave penalty(MCP)has been a widely used technique in variable selection.Asymmetric least squares regression is proposed as an alternative regression to investigate the whole conditional distribution of the response variable.In this paper,we investigate the minimax concave penalty in sparse asymmetric least squares regression models(MCP-ALS).Under some regular conditions,we prove that the MCP-ALS estimator enjoys oracle property when the covariate dimension is fixed.In high dimensional model,we obtain the weaken oracle property of the estimator when the error has finite moments.As a by-product,our proposed method is able to detect heteroscedasticity by taking different asymmetric weight values.The results from simulation show that the proposed method has good performance on variable selection and can detect heteroscedasticity efficiently.Finally,the proposed method is applied to the diabetes dataset.The real analysis shows that the proposed method can mine the potential relationship between explanatory variables and response variables while realizing variable selection to provide a reference for the prediction and control of the condition of diabetic patients.

关 键 词：非对称最小二乘回归 MCP 异方差 变量选择 高维数据 

分 类 号：O212.1[理学—概率论与数理统计]