Group selection by adaptive elastic-net method in a quantile model with diverging number of variable groups

In real applications, in a linear model, the explanatory variables are very often naturally grouped, the most common example being the multivariate variance analysis. For a quantile model with structure group, with the possibility that the number of groups diverges with sample size, since the accurate estimation of parameters is an important problem, we introduce and study the adaptive elastic-net estimation method. This method automatically selects, with a probability converging to one, the significant groups and, moreover, the non zero parameter estimators are asymptotically normal. The convergence rate of the adaptive elastic-net group quantile estimator is obtained, it depending on the number of groups. To put the estimation method into practice, an algorithm based on the subgradient method is proposed and implemented for computing this estimator. The Monte Carlo simulations show that the adaptive elastic-net group quantile estimations are more accurate that other existing group estimations in the literature. Moreover, the numerical study confirms also the theoretical results and the usefulness of the proposed estimation method.