The coupling method in extreme value theory

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