Affine Rational Transformations of Copulas and Quasi-Copulas

Pitiwat Lueangwitchajaroen, Santi Tasena

Abstract

The transformation-based methods are indeed the convenient ways for constructing a new copula using known copulas. In this research, we characterize linear rational transformations for  multivariate copulas and multivariate quasi-copulas. This is an extension to the already known results in the bivariate case. We found that this type of transformations extended naturally for the multivariate quasi-copulas. Yet, the only linear rational transformation of multivariate copulas is the identity function which is different from the bivariate case. This means that the set of linear rational functions that transform a multivariate copula  varies depending on the copula itself. As an example, we also characterize such sets in the case of the trivariate product copula.