  In several different fields, there is interest in analyzing the upper or
lower tail quantile of the underlying distribution rather than mean or center
quantile. However, the investigation of the tail quantile is difficult because
of data sparsity. In this paper, we attempt to develop nonparametric quantile
regression for the extremal quantile level. In extremal quantile regression,
there are two types of technical conditions of the order of convergence of the
quantile level: intermediate order or extreme order. For the intermediate order
quantile, the ordinary nonparametric estimator is used. On the other hand, for
the extreme order quantile, we provide a new estimator by extrapolating the
intermediate order quantile estimator. The performance of the estimator is
guaranteed by asymptotic theory and extreme value theory. As a result, we show
the asymptotic normality and the rate of convergence of the nonparametric
quantile regression estimator for both intermediate and extreme order
quantiles. A simulation is presented to confirm the behavior of the proposed
estimator. The data application is also assessed.
