Extreme-Value Copulas

Abstract

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.

A note on the Galambos copula and its associated Bernstein function

“…see, e.g., [8]. It is observed that the empirical value for Spearman's Rho approximates the theoretical value with increasing n, as implied by the theory.…”

When Gumbel met Galambos

“…in terms of a function : [12,15,19] whose analytical characterization is given in [3,34]. The stable tail dependence function of the Gumbel copula with parameter ρ ∈ (0, 1) is given, for all…”

On tail index estimation based on multivariate data

“…In addition the interval [0.75k opt , 1.25k opt ] seemed to be a reasonable choice to account for the error in the selection of the optimal sample fraction. We tried two different choices of copulas to describe the asymptotic dependence structure of the field, namely the Gumbel copula (Gudendorf and Segers (2010)) and the t-copula (Frahm, Junkera, and A. (2003)).…”