Matrix representation of copulas and quasi-copulas defined on non-square grids of the unit square

Abstract

In this paper a matrix representation for discrete copulas and quasi-copulas defined on a non-square grid I n × I m of [ 0 , 1 ] 2 is given. Two special cases are studied, irreducible discrete copulas and quasi-copulas (those that cannot be expressed as non-trivial convex combination of other ones) and those of minimal range. This study is divided into two cases depending on whether n divides m or not. In the first case, it is proved that copulas and quasi-copulas with minimal range admit a representation through a special kind of matrices, that they are always irreducible and that this implication becomes an equivalence in the case of copulas. Moreover, an algorithm to express any discrete copula as a convex combination of irreducible ones is given. However, the case when n does not divide m is quite more complex and it is proved that the previous results do not longer hold in this case. Similar results are given only for a subclass of copulas and quasi-copulas, those whose associated matrices are given by blocks.