Robust Variable Selection With Outliers Based On Combined Quantile Regression

As we all know,the ordinary least squares(OLS)is the most classical parameter esti-mation method,but it is very sensitive to the outliers.The least absolute deviation(LAD)method and quantile regression(QR)are particularly apposite to the heavy-tailed error.Compared with OLS,the relative efficiency of the LAD can be arbitrarily small.And the QR method is more efficientin terms of heavy-tail error.The composite quantile regression(CQR)method further improves upon the usual QR,and it possess the robustness of QR estimator.However,it is sensitive to the outliers in explanatory variables.If the outliers occur in explanatory variables,the performance of the CQR estimators is no longer better.In addition,the shrinkage method has been one of the hot topics in recent years because of its simplicity and robustness.There generally existmultiple collinearity between variables with the parameter diverging.Such that,the variable selection method with group effects needs to be considered.In this paper,we propose a robustly weighted CQR(WCQR),which gives every obser-vation a weight to downweight the leverage points and hence reduce the effect of the outliers on the estimation procedure.In addition,we use the principal component analysis method to reduce the dimensions of the dataset in ultra-high dimensional data,and then select the weight.Combined with the WCQR function and SCAD-L2 penalty,this article presents a robust WCQR-SCAD-L2 method.The proposed method can achieve robust parameter estimation and variable selection in regression simultaneously.Under some mild assumptions,we give oracle properties(s-parsity and asymptotic normality)and their rigorous theoretical proof.The selection of weight adopts "the decontamination subset" method.The simulation process uses the local quadratic approximation and the MM(Majorize-Minimization)algorithm.Simulation stud-ies and real examples demonstrated the performance of the proposed method.It shows that when there are outliers in the explanatory variable and the response variable,the proposed method performs well better than the non-weighted method.