Nonparametric estimation of extremal dependence

TL;DR

This paper reviews nonparametric methods for estimating extremal dependence in multivariate distributions, focusing on tail dependence functions and spectral measures, with applications to financial data.

Contribution

It introduces nonparametric techniques for modeling tail dependence, including measures for asymptotic independence, enhancing understanding of extreme value dependence structures.

Findings

Stable tail dependence function effectively models joint tail behavior.

Spectral measure provides detailed dependence structure in extremes.

Refined tail dependence coefficients distinguish between asymptotic dependence and independence.

Abstract

There is an increasing interest to understand the dependence structure of a random vector not only in the center of its distribution but also in the tails. Extreme-value theory tackles the problem of modelling the joint tail of a multivariate distribution by modelling the marginal distributions and the dependence structure separately. For estimating dependence at high levels, the stable tail dependence function and the spectral measure are particularly convenient. These objects also lie at the basis of nonparametric techniques for modelling the dependence among extremes in the max-domain of attraction setting. In case of asymptotic independence, this setting is inadequate, and more refined tail dependence coefficients exist, serving, among others, to discriminate between asymptotic dependence and independence. Throughout, the methods are illustrated on financial data.