On all Pickands Dependence Functions whose corresponding Extreme-Value-Copulas have Spearman $\rho$ (Kendall $\tau$) identical to some value $v \in [0,1]$

TL;DR

This paper characterizes the families of Pickands dependence functions corresponding to extreme-value copulas with fixed Spearman's rho or Kendall's tau, providing precise bounds and demonstrating their optimality.

Contribution

It determines the exact sets containing all such dependence functions for fixed Spearman's rho or Kendall's tau, answering an open question from 2016.

Findings

Identified compact sets ^ ho_v, ^ au_v containing all relevant functions.

Proved these sets are the best possible bounds.

Provided a complete characterization of the families for fixed dependence measures.

Abstract

We answer an open question posed by the second author at the Salzburg workshop on Dependence Models and Copulas in 2016 concerning the size of the family $\mathcal{A}^\rho_v$ ($\mathcal{A}^\tau_v$) of all Pickands dependence functions $A$ whose corresponding Extreme-Value-Copulas have Spearman $\rho$ (Kendall $\tau$) equal to some arbitrary, fixed value $v \in [0,1]$. After determining compact sets $\Omega^\rho_v, \Omega^\tau_v \subseteq [0,1] \times [\frac{1}{2},1]$ containing the graphs of all Pickands dependence functions from the classes $\mathcal{A}^\rho_v$ and $\mathcal{A}^\tau_v$ respectively, we then show that both sets are best possible.