Nonparametric Smoothing for Extremal Quantile Regression with Heavy Tailed Data

Abstract

In several different fields, it is interested 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 somewhat difficult because of data sparsity. This paper challenges to develop the nonparametric quantile regression for extremal quantile level. In the extremal quantile regression, there are two situation of technical conditions of order of convergence of the quantile level that 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 the new estimator by the extrapolating the intermediate order quantile estimator. The performance of the estimator is guaranteed by the asymptotic theory and the extreme value theory. As the result, we show the asymptotic normality and the rate of convergence of the nonparametric quantile regression estimator for both intermediate and extreme order quantile. Simulation is addressed to confirm the behavior of the proposed estimator. The data application is also assessed.