Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued and Y = (X) +, where X and are independent and the th quantile of is 0 (is fixed such that 0 0, and set r = (p - m) / (2p + d), where m is a nonnegative integer smaller than p. Let T () denote a derivative of of order m. It is proved that there exists a pointwise estimate Tₙ of T (), based on a set of i. i. d. observations (X₁, Y₁), , (Sₙ, Yₙ), that achieves the optimal nonparametric rate of convergence n^-r under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate Tₙ and this is used to obtain some useful asymptotic results.