On the structure of exchangeable extreme-value copulas

TL;DR

This paper characterizes the structure of exchangeable extreme-value copulas, showing their stable tail dependence functions form a simplex, and provides conditions for their marginals and conditional independence.

Contribution

It establishes that symmetric stable tail dependence functions form a simplex and identifies extremal boundaries, advancing understanding of exchangeable extreme-value copulas.

Findings

Stable tail dependence functions form a simplex.

Proper subset of marginals for higher dimensions.

Necessary condition for bivariate copulas to be marginals.

Abstract

We show that the set of $d$-variate symmetric stable tail dependence functions, uniquely associated with exchangeable $d$-dimensional extreme-value copulas, is a simplex and determine its extremal boundary. The subset of elements which arises as $d$-margins of the set of $(d+k)$-variate symmetric stable tail dependence functions is shown to be proper for arbitrary $k \geq 1$. Finally, we derive an intuitive and useful necessary condition for a bivariate extreme-value copula to arise as bi-margin of an exchangeable extreme-value copula of arbitrarily large dimension, and thus to be conditionally iid.