Several algorithms for constructing copulas via ⁎-product decompositions

Abstract

For two given measure-preserving functions defined on the unit interval f,g:I→I, the function given byCf,g(u,v):=λ(f−1([0,u])∩g−1([0,v])) is a copula. Although the theoretical problem for constructing this copula is completely solved, in practice it is a rather difficult task. The principal problem is the reverse implication (that is, to prove that f and g are measure-preserving when Cf,g is a copula). We provide new proof of this fact with a technique that is far from the previous ones already known in the literature. Indeed, finding two measure-preserving functions f and g, such that Cf,g=C, for a given C, is equivalent to a suitable decomposition of such copula in the form C=Cf,id⁎Cid,g (the ⁎-product), where id denotes the identity function. We also provide explicit algorithms which solve this problem in various contexts such as the measure preserving functions f and g are monotonic, as well as the copula C is a diagonal copula, an extreme copula, an extremal biconic copula, an Archimedean copula, a conic copula, a copula invariant under truncations, or an α-migrative copula.