On two new modified tawn copulas

Abstract

At its core, copula theory focuses on constructing a copula function, which characterizes the structure of dependence between random variables. In particular, the creation of extreme value copulas is crucial because they allow accurate modeling of extreme dependence that traditional copulas can ignore. In this article, we propose theoretical developments on this subject by proposing two new extreme value copulas. They aim to extend the functionalities of the so-called Tawn copula. This is of interest because the Tawn copula is known to be a powerful tool in modeling joint distributions, particularly in capturing asymmetric and upper tail dependences, making it valuable for analyzing extreme events and tail risk. The proposed copulas are designed to go beyond these attractive features. On the mathematical side, they are derived from new Pickands dependence functions; one modifies the Pickands dependence function of the Tawn copula by using a polynomial-exponential function, and the other does the same but by introducing a power function. The proofs are based on diverse differentiation, arrangement, and inequality techniques. Overall, the created copulas are attractive because (i) they possess modulable levels of asymmetry, (ii) they depend on several tuning parameters, making them very flexible in terms of upper tail dependence in particular, and (iii) they benefit from interesting correlation ranges of values. Several figures and value tables support the theoretical findings.