On bivariate Archimax copulas: Level sets, mass distributions and related results

Abstract

Motivated by the results in n [Mai and Scherer, 2011; Trutschnig et al., 2016], which examine the way bivariate Extreme Value copulas distribute their mass, we extend these findings to the larger family of bivariate Archimax copulas $\mathcal{C}_{am}$. Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas $C \in \mathcal{C}_{am}$ and show that the support of $C$ is determined by some functions $f^0$, $g^L$ and $g^R$. Additionally, we prove that the discrete component (if any) of $C$ concentrates its mass on the graphs of certain convex functions $f^s$ or non-decreasing functions $g^t$. Investigating the level sets $L^t$ of Archimax copulas $C \in \mathcal{C}_{am}$, we establish that these sets can also be characterized in terms of the afore-mentioned functions $f^s$ and $g^t$. Furthermore, recognizing the close relationship between the level sets $L^t$ of a copula $C$ and its Kendall distribution function $F_C^K$, we provide an alternative proof for the representation of $F_C^K$ for arbitrary Archimax copulas $C\in \mathcal{C}_{am}$ and derive simple expressions for the level set masses $μ_C(L^t)$. Building upon the fact that Archimax copulas $C \in \mathcal{C}_{am}$ can be represented via two univariate probability measures $γ$ and $\vartheta$ - so-called Williamson and Pickands dependence measures - we show that absolute continuity, discreteness and singularity properties of these measures $γ$ and $\vartheta$ carry over to the corresponding Archimax copula $C_{γ, \vartheta}$. Finally, we derive conditions on $γ$ and $\vartheta$ such that the support of the absolutely continuous, discrete or singular component of $C_{γ, \vartheta}$ coincides with the support of $C_{γ, \vartheta}$.