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 < [alpha] < 1. Assume that [theta] is a smooth function with order of smoothness p > 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 (1 <= q < [infinity]) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T([theta]) that achieves the optimal rate (n/log n)-r in L[infinity]-norm restricted to compacts.