An Integrated Functional Weissman Estimator for Conditional Extreme Quantiles

It is well-known that estimating extreme quantiles, namely, quantiles lying beyond the range of the available data, is a nontrivial problem that involves the analysis of tail behavior through the estimation of the extreme-value index. For heavy-tailed distributions, on which this paper focuses, the extreme-value index is often called the tail index and extreme quantile estimation typically involves an extrapolation procedure. Besides, in various applications, the random variable of interest can be linked to a random covariate. In such a situation, extreme quantiles and the tail index are functions of the covariate and are referred to as conditional extreme quantiles and the conditional tail index, respectively. The goal of this paper is to provide classes of estimators of these quantities when there is a functional (i.e. possibly infinite-dimensional) covariate. Our estimators are obtained by combining regression techniques with a generalization of a classical extrapolation formula. We analyze the asymptotic properties of these estimators, and we illustrate the finite-sample performance of our conditional extreme quantile estimator on a simulation study and on a real chemometric data set.