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Sklar’s theorem derived using probabilistic continuation and two consistency results

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  • Faugeras, Olivier P.

Abstract

We give a purely probabilistic proof of Sklar’s theorem by using a simple continuation technique and sequential arguments. We then consider the case where the distribution function F is unknown but one observes instead a sample of i.i.d. copies distributed according to F: we construct a sequence of copula representers associated with the empirical distribution function of the sample which convergences a.s. to the representer of the copula function associated with F. Eventually, we are surprisingly able to extend the last theorem to the case where the marginals of F are discontinuous.

Suggested Citation

  • Faugeras, Olivier P., 2013. "Sklar’s theorem derived using probabilistic continuation and two consistency results," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 271-277.
    • Handle: RePEc:eee:jmvana:v:122:y:2013:i:c:p:271-277
    DOI: 10.1016/j.jmva.2013.07.010
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    References listed on IDEAS

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    1. Genest, Christian & Nešlehová, Johanna, 2007. "A Primer on Copulas for Count Data," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 475-515, November.
    2. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504, January.
    3. Denuit, Michel & Lambert, Philippe, 2005. "Constraints on concordance measures in bivariate discrete data," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 40-57, March.
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    Cited by:

    1. Faugeras, Olivier P., 2015. "Maximal coupling of empirical copulas for discrete vectors," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 179-186.
    2. Faugeras Olivier P., 2017. "Inference for copula modeling of discrete data: a cautionary tale and some facts," Dependence Modeling, De Gruyter, vol. 5(1), pages 121-132, January.
    3. Šeliga Adam & Kauers Manuel & Saminger-Platz Susanne & Mesiar Radko & Kolesárová Anna & Klement Erich Peter, 2021. "Polynomial bivariate copulas of degree five: characterization and some particular inequalities," Dependence Modeling, De Gruyter, vol. 9(1), pages 13-42, January.
    4. Saminger-Platz Susanne & Kolesárová Anna & Šeliga Adam & Mesiar Radko & Klement Erich Peter, 2021. "New results on perturbation-based copulas," Dependence Modeling, De Gruyter, vol. 9(1), pages 347-373, January.
    5. Tsionas, Mike G. & Andrikopoulos, Athanasios, 2020. "On a High-Dimensional Model Representation method based on Copulas," European Journal of Operational Research, Elsevier, vol. 284(3), pages 967-979.

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