The coupling method in extreme value theory

TL;DR

This paper introduces a coupling method in univariate extreme value theory that offers an alternative to traditional tail process approaches, providing new bounds and insights into estimators like Hill and Weissman.

Contribution

It develops a coupling approach for extreme value analysis, deriving sharp Wasserstein bounds and offering new perspectives on classical estimators' asymptotic behavior.

Findings

Sharp bounds for Wasserstein distance between empirical and limit distributions.

Reinterpretation of asymptotic properties of Hill and Weissman estimators.

Alternative to tail empirical process methods in extreme value theory.

Abstract

A coupling method is developed for univariate extreme value theory , providing an alternative to the use of the tail empirical/quantile processes. Emphasizing the Peak-over-Threshold approach that approximates the distribution above high threshold by the Generalized Pareto distribution, we compare the empirical distribution of exceedances and the empirical distribution associated to the limit Generalized Pareto model and provide sharp bounds for their Wasser-stein distance in the second order Wasserstein space. As an application , we recover standard results on the asymptotic behavior of the Hill estimator, the Weissman extreme quantile estimator or the probability weighted moment estimators, shedding some new light on the theory.