Global nonparametric estimation of conditional quantile functions and their derivatives

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and [theta](X) is the conditional [alpha]th quantile of Y given X, where [alpha] is a fixed number such that 0 0, and set r = (p - m)/(2p + d), where m is a nonnegative integer smaller than p. Let T([theta]) denote a derivative of [theta] of order m. It is proved that there exists estimate of T([theta]), based on a set of i.i.d. observations (X1, Y1), ..., (Xn, Yn), that achieves the optimal nonparametric rate of convergence n-r in Lq-norms (1regression quantiles nonparametric estimates bin smoothers optimal rates of convergence