Local linear quantile regression

In this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting. Two such estimators are considered. One is based on localizing the characterization of a regression quantile as the minimizer of E{rho(p)(Y-a)\X = x}, where rho(p) is the appropriate "check" function. The other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters, rather than one. Our aim is to present fully operational versions of both approaches and to show that each works quite well; although either might be used in practice, we have a particular preference for the second. Our automatic smoothing parameter selection method is novel; the main regression quantile smoothing parameters are chosen by rule-of-thumb adaptations of state-of-the-art methods for smoothing parameter selection for regression mean estimation. The techniques are illustrated by application to two datasets and compared in simulations.