Penalty M Estimation Of Adaptive Elastic Net Based On Standard Error Adjustment

Variable selection can help us extract valuable information from a large number of data and improve the prediction accuracy of the model.It is an important issue in statistical inference that how to efficiently select important variables from a large number of covari-ates.Tibshirani proposed the important method named Lasso in 1996,this method can produce the sparse model,but it does not have Oracle property.Therefore,Zou proposed adaptive Lasso,which applies different degrees to different coefficients.This method can consistently select important variables and enjoys Oracle property.There are correlations among variables in high-dimensional data,but Lasso may not effectively reflect the rela-tionship between variables.Hence the elastic-net method is proposed,it can make highly correlated variables enter or remove from the model at the same time,but this method does not have Oracle property.Therefore,based on elastic-net and adaptive Lasso,the adaptive olastic-net method enjoying Oracle property is proposed.Those mothods are all based on penalized least squares methodWhen there are outliers or heavy tailed distributions in the data,the traditional least squares with penalty function is no longer applicable,and a robust estimation method needs to be sought.M-estimation has been extensively studied in the field of robust statistical inference.In addition,with the rapid development of science and technology,a lot of data,enjoying high dimension,strong correlation and redundancy,has been generated in real life.So on the basis of robust estimation,it is necessary to find an effective variable selection method for dealing with collinearity.In order to solve the above problemy,a penalized M-estimation method based on standard error adjusted adaptive elastic-net is proposed in this paper.In this method,M-estimators and their standard errory are used in weights,and the consistency and asymptotic normality of this method are proved theoretically.For the regularization in higli-dimensional space,more iteration steps can be used to maintain the accuracy of estimation and deal with the problem of multiple collinearity,each iteration uses a separate regularization parameter.This is the idea of MSA-Enet(the multi-step adaptive elastic-net).In this paper,we use this method to reduce the dimension pn of ultra-high dimensional data to a relatively large scale that is less than the sample size n,and then use the proposed penalized M-estimation method to select variables and estimate parameters.Finally,we carry out numerical simulation studies and real data analysis to examine the finite sample performance of the proposed method.The results show that the proposed method has some advantages over other commonly used variable selection methods.