High-Order Conditional Quantile Estimation Based on Nonparametric Models of Regression

Abstract

We consider the estimation of a high order quantile associated with the conditional distribution of a regressand in a nonparametric regression model. Our estimator is inspired by Pickands (1975) where it is 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 standardized residuals from a first stage kernel based nonparametric estimation. 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 defined quantile estimator. A Monte Carlo study provides evidence that our estimator behaves well in finite samples and is easily implementable. Our results have direct application in finance, particularly in the estimation of conditional Value-at-Risk, but other researchers in applied fields such as insurance will also find the results useful.

Keywords:

• Conditional quantile
• Extreme value theory
• Generalized Pareto distribution
• Nonparametric regression

AMS-MS Classification:

• C10
• C14
• C21

ACKNOWLEDGMENTS

We thank Essie Maasoumi, an Associate Editor, and two anonymous referees for comments that improved the paper substantially. Any remaining errors are the authors’ responsibility.

Notes

The case where (Y, X) ∈ ℜ^1+D with X ∈ ℜ ^D and D > 1 can be analyzed with arguments that are similar to those we have used. The only differences reside on how the kernel function is defined and the speed of convergence of the relevant bandwidths to zero.

‖x‖ denotes the Euclidean norm of the vector x.

Substituting k ₀ = − α⁻¹ shows that H is identical to the homonymous matrix in Eq. (11).

If m were estimated by a Nadaraya–Watson estimator then . The additional terms that involve m ⁽¹⁾(x)(X _t  − x) can be shown to be negligible in probability at the desired rate .