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 of Y given X , where α is a fixed number such that 0 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 ( X 1 , Y 1 ), …, ( X n , Y n ), that achieves the optimal nonparametric rate of convergence n − r in L q -norms (1 ≤ q T n of T ( θ ) that achieves the optimal rate ( n log n ) −r in L ∞ -norm restricted to compacts.