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

Investigating the level sets Lᵗ of Archimax copulas C ∈ Cₐₘ , we establish that these sets can be characterized in terms of certain convex functions fˢ and non-decreasing functions gᵗ . 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 Cₐₘ . Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas C ∈ Cₐₘ and show that the support of C is determined by some functions f⁰ , gL , and gR . Additionally, we prove that the discrete component (if any) of C concentrates its mass on the graphs of the afore-mentioned functions fˢ or gᵗ . Recognizing the close relationship between the level sets Lᵗ of a copula C and its Kendall distribution function FCK , we provide an alternative proof for the representation of FCK for arbitrary Archimax copulas C ∈ Cₐₘ and derive simple expressions for the level set masses μ C(Lᵗ) . Building upon the fact that Archimax copulas C ∈ Cₐₘ can be represented via two univariate probability measures γ and ϑ — so-called Williamson and Pickands dependence measures — we show that absolute continuity, discreteness, and singularity properties of these measures γ and ϑ carry over to the corresponding Archimax copula Cγ , ϑ . Finally, we derive conditions on γ and ϑ such that the support of the absolutely continuous, discrete, or singular component of Cγ , ϑ coincides with the support of Cγ , ϑ .