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

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.