🤖 AI Summary
This study addresses the lack of a systematic approach for modeling distorted copulas and stable tail dependence functions (stdfs) within the Morillas-type framework. The authors propose a stochastic representation and a general sampling algorithm for such transformations, establishing a distortion framework tailored to stdfs. By leveraging monomial distortions and convex combinations, the method substantially enhances modeling flexibility for extremal dependence structures. Notably, they prove that monomial distortions with exponent less than one preserve the stdf property—a result that enables, for the first time, effective sampling from Morillas-type copulas. Furthermore, the work introduces a new class of distortion functions that maintain the extremal copula structure, extends modeling capabilities to non-regularly varying settings, provides efficient sampling schemes for Archimedean and Archimax copulas, and clarifies the impact of distortions on limits of max-domains of attraction.
📝 Abstract
A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.