Geometric extremal graphical models and coefficients of extremal dependence on block graphs

TL;DR

This paper introduces geometric extremal graphical models based on gauge functions to analyze extremal dependence in high-dimensional data, especially on block graphs, advancing the modeling of complex extremal dependence structures.

Contribution

It develops the concept of geometric extremal graphical models and establishes results on extremal dependence coefficients propagation on block graphs, linking to conditional extreme value theory.

Findings

Propagation results for extremal dependence coefficients on block graphs

Connection of coefficients to conditional extreme value theory

Framework for high-dimensional extremal dependence modeling

Abstract

We introduce the concept of geometric extremal graphical models, which are defined through the gauge function of the limit set obtained from suitably scaled random vectors in light-tailed margins. For block graphs, we prove results relating to the propagation of various extremal dependence coefficients along the graph. A particular focus is placed on coefficients that link to the framework of conditional extreme value theory, which are especially interesting when variables do not all attain their most extreme values simultaneously. We also consider results related to the case when variables do exhibit joint extreme behaviour. Through the recent translation of the geometric approach for multivariate extremes to a statistical modelling framework, geometric extremal graphical models, and results relating to them, pave the way for an approach to modelling of high dimensional extremes with complex extremal dependence structures.