Variable selection of varying coefficient models in quantile regression

Varying coefficient (VC) models are commonly used to study dynamic patterns in many scientific areas. In particular, VC models in quantile regression are known to provide a more complete description of the response distribution than in mean regression. In this paper, we develop a variable selection method for VC models in quantile regression using a shrinkage idea. The proposed method is based on the basis expansion of each varying coefficient and the regularization penalty on the Euclidean norm of the corresponding coefficient vector. We show that our estimator is obtained as an optimal solution to the second order cone programming (SOCP) problem and that the proposed procedure has consistency in variable selection under suitable conditions. Further, we show that the estimated relevant coefficients converge to the true functions at the univariate optimal rate. Finally, the method is illustrated with numerical simulations including the analysis of forced expiratory volume (FEV) data.