Asymptotic Properties Of The Elastic Net Estimator Via An Iteratively Reweighted Algorithm

Lasso regression is one of the important methods to deal with multiple mutual linear problems.Different from ridge regression,it has both characteristics of shrinkage and selection.Recently,the large sample property of Lasso estimators has become a research hotspot in the field of statistics.However,the Lasso estimator based on conditional heteroscedastic models started relatively late.Wagener and Dette(2013)researched a linear regression-heteroscedastic model with Yt=Xt,n' ?n0+?(Xt,n,?n0)?t,and proved the sign consistency and the asymptotic normality of its weighted adaptive Lasso estimator.Moreover,Ziel(2016)applied an iteratively reweighted algorithm to study a linear regression-conditional heteroscedastic model with Yt?Xt,n'?n0+?t,?t =?tZt,?t =g(?n0;Ln,t0),and proved that its weighted adaptive Lasso estimator also has sign consistency and asymptotic normality.Elastic net regression is a generalization of Lasso regression.In some cases,it can combine the advantages of Lasso regression and ridge regression.In this paper,we generalize the conclusion of Ziel(2016).First of all,we put forward an iteratively reweighted algorithm for the model with Yt = Xt,n'?n0+?t,?t = ?tZt,?t = g(?n0;Ln,t0).Secondly,we prove that the weighted adaptive elastic net estimator based on above algorithm has sign consistency and asymptotic normality.In the process of proof,the KKT condition and the central limit theorem of martingale difference are mainly applied.Finally,we simulate a specific AR-ARCH model,and the sign consistency of the elastic net estimator is verified from the perspective of screening variables.