Characterizing extremal dependence on a hyperplane

TL;DR

This paper explores the extremal dependence of multivariate variables by analyzing vectors on a hyperplane, enabling the use of linear techniques like PCA to understand tail dependence structures.

Contribution

It introduces a novel hyperplane-based framework for characterizing extremal dependence, connecting multivariate extremes with linear algebra methods.

Findings

Hyperplane characterization simplifies multivariate extreme analysis.

PCA can be used for tail dependence approximation.

Hüsler-Reiss family relates to Gaussian distributions on the hyperplane.

Abstract

In this paper, we characterize the extremal dependence of $d$ asymptotically dependent variables by a class of random vectors on the $(d-1)$-dimensional hyperplane perpendicular to the diagonal vector $\mathbf1=(1,\ldots,1)$. This translates analyses of multivariate extremes to that on a linear vector space, opening up possibilities for applying existing statistical techniques that are based on linear operations. As an example, we demonstrate obtaining lower-dimensional approximations of the tail dependence through principal component analysis. Additionally, we show that the widely used H\"usler-Reiss family is characterized by a Gaussian family residing on the hyperplane.