Nonparametric intermediate order regression quantiles

This paper studies nonparametric estimation of d-dimensional conditional quantile functions and their derivatives in the tails. We investigate asymptotic properties of the local and global nonparametric quantile regression estimators proposed by Chaudhuri (1991a, b), respectively, under the intermediate order quantile asymptotics: as the sample size n goes to infinity, the quantile αn and a bandwidth parameter δn satisfy αn → 0 and nδd nαn → ∞ (or αn → 1 and nδd n(1−αn) →∞). We derive the pointwise convergence rate and asymptotic distribution of the local nonparametric quantile regression estimator, and the sup-norm convergence rate of the global nonparametric quantile regression estimator. Our results complement the papers by Chaudhuri (1991a, b), where the quantile αn does not vary with n, and Chernozhukov (1998), where the quantile αn satisfies αn → 0 and nδd nαn → 0.