Functional kernel estimators of large conditional quantiles

TL;DR

This paper develops functional kernel estimators for large conditional quantiles, providing asymptotic properties and a Weissman estimator for high-order quantiles, with practical illustrations.

Contribution

It introduces a novel approach to estimate large conditional quantiles with functional data, including asymptotic Gaussian results and a new Weissman estimator.

Findings

Asymptotic Gaussian distribution of estimators established.

Conditions for convergence rate of quantile order to one identified.

Finite sample illustrations demonstrate estimator performance.

Abstract

We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate large conditional quantiles of arbitrary large order. These results are illustrated on finite sample situations.