Current Developments on Extreme Value Copulas: Extended Pickands Dependence Functions

Abstract

Copulas are mathematical tools used to model the dependence structure between random variables. Extreme value copulas specifically focus on
capturing the tail dependence, which refers to the dependence structure between random variables when they exhibit extreme or rare events. The
Pickands dependence functions are special convex functions that play a crucial role in characterizing extreme value copulas; they quantify the
strength of their tail dependence. The creation of new Pickands dependence functions enhances our understanding of complex interdependencies,
enabling more accurate modeling and risk assessment in diverse systems. In this article, a theoretical contribution to the topic is provided; an
original strategy for generating new Pickands dependence functions based on existing ones is developed. The resulting Pickands dependence
functions have the features of using the functionalities of standard functions (exponential, trigonometric, hyperbolic, etc.) and/or depend on
several tuning parameters of various natures, which are quite uncommon in the literature. Two new extreme value copulas are derived from
our findings. Their asymmetric and tail-dependent flexibility are emphasized. Numerical and graphical illustrations are given to support some
theoretical facts.