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

Abstract Investigating the level sets $$\varvec{L^t}$$ of Archimax copulas $$\varvec{C \in \mathcal {C}_{am}}$$ , we establish that these sets can be characterized in terms of certain convex functions $$\varvec{f^s}$$ and non-decreasing functions $$\varvec{g^t}$$ . Motivated by the results in Mai and Scherer (Extremes 14 , 311-324 2011) and Trutschnig et al. (Extremes 19 , 405-427 2016), which examine the way bivariate Extreme Value copulas distribute their mass, we extend these findings to the larger family of bivariate Archimax copulas $$\varvec{{C}_{am}}$$ . Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas $$\varvec{C \in \mathcal {C}_{am}}$$ and show that the support of $$\varvec{C}$$ is determined by some functions $$\varvec{f^0}$$ , $$\varvec{g^L}$$ , and $$\varvec{g^R}$$ . Additionally, we prove that the discrete component (if any) of $$\varvec{C}$$ concentrates its mass on the graphs of the afore-mentioned functions $$\varvec{f^s}$$ or $$\varvec{g^t}$$ . Recognizing the close relationship between the level sets $$\varvec{L^t}$$ of a copula $$\varvec{C}$$ and its Kendall distribution function $$\varvec{F_C^K}$$ , we provide an alternative proof for the representation of $$\varvec{F_C^K}$$ for arbitrary Archimax copulas $$\varvec{C \in \mathcal {C}_{am}}$$ and derive simple expressions for the level set masses $$\varvec{\mu _C(L^t)}$$ . Building upon the fact that Archimax copulas $$\varvec{C \in \mathcal {C}_{am}}$$ can be represented via two univariate probability measures $$\gamma$$ and $$\varvec{\vartheta }$$ — so-called Williamson and Pickands dependence measures — we show that absolute continuity, discreteness, and singularity properties of these measures $$\varvec{\gamma }$$ and $$\varvec{\vartheta }$$ carry over to the corresponding Archimax copula $$\varvec{C_{\gamma , \vartheta }}$$ . Finally, we derive conditions on $$\varvec{\gamma }$$ and $$\varvec{\vartheta }$$ such that the support of the absolutely continuous, discrete, or singular component of $$\varvec{C_{\gamma , \vartheta }}$$ coincides with the support of $$\varvec{C_{\gamma , \vartheta }}$$ .