Regional quantile regression for multiple responses

In this article, we study high-dimensional multiple response quantile regression model for an interval of quantile levels, in which a common set of covariates is used to analyze multiple responses simultaneously. We assume that the underlying quantile coefficient matrix is simultaneously element-wise and row-wise sparse. We address high dimensional issues to identify globally relevant variables for multiple responses when any τth conditional quantile is considered, where τ∈Δ, and Δ is an interval of quantile levels of interest. We develop a novel penalized globally concerned quantile regression with double group Lasso penalties and propose an information criterion for penalty parameter choice. We prove that the proposed method consistently selects both element-wise and row-wise sparsity patterns of the regression coefficient matrix function and that it achieves the oracle convergence rate. Numerical examples and applications to Cancer Cell Line Encyclopedia data illustrate the advantages of the proposed method over separate penalized quantile regression on each response.