Multivariate extreme value theory

TL;DR

This paper introduces classical multivariate extreme value theory focusing on multivariate excesses over high thresholds, using point process perspectives and limit distributions like max-stable laws.

Contribution

It provides an accessible overview of multivariate extreme value modeling, emphasizing the role of generalized Pareto distributions and failure sets in the sample space.

Findings

Multivariate excesses can be modeled using generalized Pareto distributions.

Point process framework offers a unifying perspective on extremes.

Max-stable distributions arise as limits of componentwise maxima.

Abstract

When passing from the univariate to the multivariate setting, modelling extremes becomes much more intricate. In this introductory exposition, classical multivariate extreme value theory is presented from the point of view of multivariate excesses over high thresholds as modelled by the family of multivariate generalized Pareto distributions. The formulation in terms of failure sets in the sample space intersecting the sample cloud leads to the over-arching perspective of point processes. Max-stable or generalized extreme value distributions are finally obtained as limits of vectors of componentwise maxima by considering the event that a certain region of the sample space does not contain any observation.