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 = 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 T̂ₙ of T(θ), 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 T̂ₙ and this is used to obtain some useful asymptotic results.