Some advances in the study of the compatibility of three bivariate copulas

Summary In this paper we study the problem of the compatibility of three bivariate copulas, i.e., we look for conditions which allow us to assure the existence of a three-copula whose two-dimensional margins are given. As a particular case, we seek conditions for two bivariate copulasC ₁ andC ₂ under whichC ₂[C₁ (x, y), z] is a three-copula. We specifically study the compatibility of the copulasM, W andII with other copulas both in general and in the particular case. We also study the compatibility of a two-copula with convex linear combinations of other two-copulas. Several examples illustrate the results obtained in each case, and some applications are given.