Nonparametric estimates of regression quantiles and their local Bahadur representation

Abstract

Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and Y=θ(X)+ε, where X and ε are independent and the αth quantile of ε is 0 (α is fixed 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 a pointwise estimate Tˆn of T(θ), based on a set of i.i.d. observations (X₁,Y₁),⋯,(S_n,Y_n), 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 Tˆn and this is used to obtain some useful asymptotic results.