Quantile regression for the single-index coefficient model

We consider quantile regression incorporating polynomial spline approximation for single-index coefficient models. Compared to mean regression, quantile regression for this class of models is more technically challenging and has not been considered before. We use a check loss minimization approach and employed a projection/orthogonalization technique to deal with the theoretical challenges. Compared to previously used kernel estimation approach, which was developed for mean regression only, spline estimation is more computationally expedient and directly produces a smooth estimated curve. Simulations and a real data set is used to illustrate the finite sample properties of the proposed estimator.