The Stochastic Fluctuation of the Quantile Regression Curve

Let (X1, Y1), . . ., (Xn, Yn) be i.i.d. rvs and let l(x) be the unknownp-quantile regression curve of Y on X. A quantile-smootherln(x) is a localised, nonlinear estimator of l(x). The strong uniformconsistency rate is established under general conditions. In many applicationsit is necessary to know the stochastic fluctuation of theprocess {ln(x) − l(x)}. Using strong approximations of the empiricalprocess and extreme value theory allows us to consider the asymptoticmaximal deviation sup0<=x<=1 |ln(x)−l(x)|. The derived result helps inthe construction of a uniform confidence band for the quantile curvel(x). This confidence band can be applied as a model check, e.g. ineconometrics. An application considers a labour market discriminationeffect.