New copulas obtained by maximizing Tsallis or Rényi entropies

International audienceSklar [1] introduced the notion of copula, solving the problem studied by Fr'echet [2] and others on the determination of a joint distribution function when the one dimensional marginal cumulative distributions are prescribed. The same problem also arises in the context of image (the internal density distribution of some physical or biological quantity inside a section of the body) reconstruction in X-ray computated tomography when only two orthogonal projections are given. The two problems are mathematically equivalent when restricted to distributions with bounded support, we propose to study the solutions which maximize Shannon [3], Tsallis-Havrda-Charv'at [4, 5] or the R'enyi [6] entropies by rescaling. The case of Shannon and Tsallis or R'enyi with index q = 2 admits analytic solutions which curiously give new copula families. In this paper, we give a theorem and its corollary using the wellknown uniform transformation yielding a method for constructing new family of copulas. We also give the expression of some dependence concepts and then provide many examples of this method in practice