The coupling method in extreme value theory

One of the main goal of extreme value theory is to infer probabilities of extreme events for which only limited observations are available and require extrapolation of the tail distribution of the observations. One major result is Balkema-de Haan-Pickands theorem that provides an approximation of the distribution of exceedances above high threshold by a Generalized Pareto distribution. We revisit these results with coupling arguments and provide quantitative estimates for the Wasserstein distance between the empirical distribution of exceedances and the limit Pareto model. In a second part of the talk, we extend the results to the analysis of a proportional tail model for quantile regression closely related to the heteroscedastic extremes framework developed by Einmahl et al. (JRSSB 2016). We introduce coupling arguments relying on total variation and Wasserstein distances for the analysis of the asymptotic behavior of estimators of the extreme value index and integrated skedasis function.

Joint work with B. Bobbia and D. Varron (Université de Franche Comté).