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

TL;DR

This paper extends the analysis of mass distribution, level sets, and support properties from bivariate Extreme Value copulas to the broader family of bivariate Archimax copulas, providing new characterizations and representations.

Contribution

It introduces a detailed analysis of the support, level sets, and mass distribution of Archimax copulas using functions and dependence measures, extending prior results from Extreme Value copulas.

Findings

Support of Archimax copulas is determined by specific functions.

Level sets can be characterized via these functions and relate to the Kendall distribution.

Properties of dependence measures influence the copula's support and components.

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 $\mu_C(L^t)$. Building upon the fact that Archimax copulas $C \in \mathcal{C}_{am}$ can be represented via two univariate probability measures $\gamma$ and $\vartheta$ - so-called Williamson and Pickands dependence measures - we show that absolute continuity, discreteness and singularity properties of these measures $\gamma$ and $\vartheta$ carry over to the corresponding Archimax copula $C_{\gamma, \vartheta}$. Finally, we derive conditions on $\gamma$ and $\vartheta$ such that the support of the absolutely continuous, discrete or singular component of $C_{\gamma, \vartheta}$ coincides with the support of $C_{\gamma, \vartheta}$.