Article

A new class of bivariate copulas

Departamento de Estadı́stica y Matemática Aplicada, Universidad de Almerı́a, Carretera de Sacramento s/n 04120, La Cañada de San Urbano, Almerı́a, Spain
Statistics [?] Probability Letters (Impact Factor: 0.6). 02/2004; 66(3):315-325. DOI: 10.1016/j.spl.2003.09.010

ABSTRACT

We study a wide class of bivariate copulas depending on two univariate functions which generalizes many known families of copulas. We measure the dependence of any copula of this class in different ways, exhibit several properties concerning symmetry, dependence concepts, and concordance ordering, and provide several examples.

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Available from: Jose Antonio Rodriguez--Lallena, Oct 06, 2014
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    • "Therefore to determine the case in which C is a copula the result from [21] can be adapted. We may also remark that in [21], the authors have proposed the following family: C(u, v) = uv + θΦ(u)ψ(v), θ ∈ [−1, 1] (12) where Φ, ψ are absolutely continuous distributions on [0] [1] and their derivatives are bounded for almost every values taken in [0] [1], with the conditions "
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    ABSTRACT: Sklar [1] introduced the notion of copula, solving the problem studied by Fréchet [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-Charvát [4, 5] or the Rényi [6] entropies by rescaling. The case of Shannon and Tsallis or Rényi 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 well-known 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.
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    • "Here, it should be recalled that, roughly speaking, each FGM 2-copula is a " perturbation " of the independence 2-copula Π 2 , but its ability to model dependence is somehow limited since, for example, Spearman's correlation coefficients for FGM 2-copulas take values in [− 1 3 , 1 3 ]. However—as stressed in [1] [28]—a 2-copula C in of the form (16) can model a wider range of dependence. Notice that C in is not a symmetric function of its arguments when g = f , a feature of interest when it makes little sense to assume that the random variables of interest are exchangeable: see [8] for a discussion about this aspect. "
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