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 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
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
of T(θ) that achieves the optimal rate (n/log n)−r in L∞-norm restricted to compacts. |