Abstract Investigating the level sets Lᵗ of Archimax copulas C C₀₌, 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 C₀₌. Working with Markov kernels (conditional distributions), we analyze the mass distributions of Archimax copulas C C₀₌ 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 C₀₌ and derive simple expressions for the level set masses C (Lᵗ). Building upon the fact that Archimax copulas C C₀₌ can be represented via two univariate probability measures and — so-called Williamson and Pickands dependence measures — we show that absolute continuity, discreteness, and singularity properties of these measures and carry over to the corresponding Archimax copula C,. Finally, we derive conditions on and such that the support of the absolutely continuous, discrete, or singular component of C, coincides with the support of C,.
Nicolas Dietrich (Tue,) studied this question.