On an asymmetric extension of multivariate Archimedean copulas

Archimedean copulas are copulas determined by a specific real function, called the generator. Composited with the copula at a given point, this generator can be expressed as a linear form of generators of the considered point components. In this paper, we discuss the case where this function is expressed as a quadratic form (called here multivariate Archimatrix copulas). This allows extending Archimedean copulas, in order for example to build asymmetric copulas. Parameters of this new class of copulas are grouped within a matrix, thus facilitating some usual applications as level curve determination or estimation. Some choices as sub-model stability help associating each parameter to one bivariate projection of the copula. We also give some admissibility conditions for the considered Archimatrix copulas. We propose different examples as some natural multivariate extensions of FarlieGumbel-Morgenstern, Gumbel-Barnett, or particular Archimax copulas.