Global nonparametric estimation of conditional quantile functions and their derivatives

Abstract

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 ofY given X, where α is a fixed number such that 0 lt;α lt; 1. Assume that θ is a smooth function with order of smoothness p gt; 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_nof T(θ), based on a set of i.i.d. observations (X₁, Y₁), ..., (X_n, Y_n), that achieves the optimal nonparametric rate of convergence n^-r in L_q-norms (1≤q lt; ∞) 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.