Quantile regression shrinkage and selection via the Lqsso

Quantile regression has recently received a considerable attention due to its remarkable development in enriching the variety of regression models. Many efforts have been made to blend different penalty and loss function to extend or develop novel regression models that are unique from different perspectives. Bearing in mind that the lasso quantile regression model ignores the randomness of the realizations in the penalty part, we propose a new penalty for the quantile regression models. Similar to the adaptive lasso quantile regression model, the proposed model simultaneously does estimation and variable selection tasks. We call the new model 'lqsso-QR', standing for the least quantile shrinkage and selection operator quantile regression. In this article, we present a sufficient and necessary condition for the variable selection of the lasso quantile regression to enjoy the consistent property. We show that the lqsso-QR follows oracle properties under some mild conditions. From computational perspective, we apply an efficient algorithm, originally developed for the lasso quantile regression. Using simulation studies, we elaborate on the superiority of the proposed model compared with other lasso-type penalties, especially regarding relative prediction error. Also, an application of our method to a real-life data; the rat eye data, is reported.