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 θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ 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(θ) denote a derivative of θ of order m. It is proved that there exists estimate T ̂n of T(θ), 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 < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate T ̂n of T(θ) that achieves the optimal rate ( n log n)-r in L∞-norm restricted to compacts. © 1991.