Adaptive elastic net-penalized quantile regression for variable selection

Abstract

There has been much attention on the high-dimensional linear regression models, which means the number of observations is much less than that of covariates. Considering the fact that the high dimensionality often induces the collinearity problem, in this article, we study the penalized quantile regression with the elastic net (EnetQR) that combines the strengths of the quadratic regularization and the lasso shrinkage. We investigate the weak oracle property of the EnetQR under mild conditions in the high dimensional setting. Moreover, we propose a two-step procedure, called adaptive elastic net quantile regression (AEnetQR), in which the weight vector in the second step is constructed from the EnetQR estimate in the first step. This two-step procedure is justified theoretically to possess the weak oracle property. The finite sample properties are performed through the Monte Carlo simulation and a real-data analysis.

Keywords:

• Adaptive elastic net
• high-dimensional linear regression
• quantile regression
• variable selection
• weak oracle property

Acknowledgments

The authors thank the editor and the anonymous referee for their valuable comments and suggestions.

Additional information

Funding

Ailing Yan’s research was supported by National Natural Science Foundation of China (grant nos. 11671116, 11801130), Natural Science Foundation of Hebei Province for Youth (grant no. QN2016044). Fengli Song’s research was supported by National Natural Science Foundation of China (grant no. 11771215), Natural Science Foundation of Jiangsu Province (grant no. BK20161530), and Major Project of the National Social Science Fund of China (grant nos. 16ZDA047, 17ZDA092).