High order conditional quantile estimation based on nonparametric models of regression

We consider the estimation of a high order conditional quantile associated with the distribution of the regressand in a nonparametric regression model. Our estimator is inspired by Pickands (1975) which has shown that arbitrary distributions which lie in the domain of attraction of an extreme value type have tails that, in the limit, behave as generalized Pareto distributions (GPD). Smith (1987) has studied the asymptotic properties of maximum likelihood (ML) estimators for the parameters of the GPD in this context, but in our paper the relevant random variables used in estimation are residuals from a  rst stage kernel based nonparametric estimator. We obtain convergence in probability and distribution of the residual based ML estimator for the parameters of the GPD as well as the asymptotic distribution for a suitably de ned quantile estimator. A Monte Carlo study provides evidence that our estimator behaves well in nite samples and is easily implementable. Our results have direct application in nance, particularly in the estimation of conditional Value-at-Risk, but other researchers in applied elds such as insurance and hydrology will nd the results useful.