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


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
    • "V C (R) 0. (2.2) By using Rodríguez-Lallena and Úbeda-Flores results from [16] "
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    ABSTRACT: In 2004, Rodríguez-Lallena and Úbeda-Flores have introduced a class of bivariate copulas which generalizes some known families such as the Farlie-Gumbel-Morgenstern distributions. In 2006, Dolati and Úbeda-Flores presented multivariate generalizations of this class, also they investigated symmetry, dependence concepts and measuring the dependence among the components of each classes. In this paper, a new method of constructing binary copulas is introduced, extending the original study of Rodríguez-Lallena and Úbeda-Flores to new families of symmetric/asymmetric copulas. Several properties and parameters of newly introduced copulas are included. Among others, relationship of our construction method with several kinds of ordinal sums of copulas is clarified.
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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.
    Full-text · Conference Paper · Jul 2012
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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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    ABSTRACT: For every n≥3, a method is introduced and investigated for generating n-dimensional copulas starting with an (n−1)-dimensional copula already known. These copulas are particularly useful when the behaviour of a random vector (X 1, X 2, …, X n−1) formed by n−1 components is known, but another random variable, say X n , should be included into the model. An illustration of the usefulness of this construction is presented, showing some of its computational features.
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