  Nonparametric regression quantiles obtained by inverting a kernel estimator
of the conditional distribution of the response are long established in
statistics. Attention has been, however, restricted to ordinary quantiles
staying away from the tails of the conditional distribution. The purpose of
this paper is to extend their asymptotic theory far enough into the tails. We
focus on extremal quantile regression estimators of a response variable given a
vector of covariates in the general setting, whether the conditional
extreme-value index is positive, negative, or zero. Specifically, we elucidate
their limit distributions when they are located in the range of the data or
near and even beyond the sample boundary, under technical conditions that link
the speed of convergence of their (intermediate or extreme) order with the
oscillations of the quantile function and a von-Mises property of the
conditional distribution. A simulation experiment and an illustration on real
data were presented. The real data are the American electric data where the
estimation of conditional extremes is found to be of genuine interest.
