Adaptive sup-norm regularized simultaneous multiple quantiles regression

When modelling multiple conditional quantiles of univariate and/or multivariate responses, it is of great importance to share strength among them. The simultaneous multiple quantiles regression (SMQR) technique is a novel regularization method that explores the similarity among multiple conditional quantiles and performs simultaneous model selection. However, the SMQR suffers from estimation inefficiency and model selection inconsistency because it applies the same amount of shrinkage to each predictor variable without assessing its relative importance. To overcome such a limitation, we propose the adaptive sup-norm regularized SMQR (ASMQR) method, which allows different amounts of shrinkage to be imposed on different variables according to their relative importance. We show that the proposed ASMQR method, a generalized form of the adaptive lasso regularized quantile regression (ALQR) method, possesses the oracle property and that it is a better tool for selecting a common subset of significant variables than the ALQR and SMQR methods through a simulation study.