Several algorithms for constructing copulas via ⁎-product decompositions.

For two given measure-preserving functions defined on the unit interval f , g : I → I , the function given by C f , 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 C f , 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 C f , g = C , for a given C , is equivalent to a suitable decomposition of such copula in the form C = C f , id ⁎ C id , 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. [ABSTRACT FROM AUTHOR]