Nonparametric smoothing for extremal quantile regression with heavy tailed distributions

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.