The calculation of nonparametric quantile regression curve estimates is often computational intensive, as typically an expensive nonlinear optimization problem is involved. This paper proposes a fast and easy-to-implement method for computing such estimates. The main idea is to approximate the costly nonlinear optimization by a sequence of well-studied penalized least-squares type nonparametric mean regression estimation problems. The new method can be paired with different nonparametric smoothing methods and can also be applied to higher dimensional settings. Therefore, it provides a unified framework for computing different types of nonparametric quantile regression estimates, and it also greatly broadens the scope of the applicability of quantile regression methodology. This wide-applicability and the practical performance of the proposed method are illustrated with smoothing spline and wavelet curve estimators, for both uni- and bivariate settings. Results from numerical experiments suggest that estimates obtained from the proposed method are superior to many competitors.
