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Maximal coupling of empirical copulas for discrete vectors

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  • Faugeras, Olivier P.

Abstract

For a vector X with a purely discrete multivariate distribution, we give simple short proofs of uniform a.s. convergence on their whole domain of two versions of genuine empirical copula functions, obtained either via probabilistic continuation, i.e. kernel smoothing, or via the distributional transform. These results give a positive answer to some delicate issues related to the convergence of copula functions in the discrete case. They are obtained under the very weak hypothesis of ergodicity of the sample, a framework which encompasses most types of serial dependence encountered in practice. Moreover, they allow to derive, as simple corollaries, almost sure consistency results for some recent extensions of concordance measures attached to discrete vectors. The proofs are based on a maximal coupling construction of the empirical cdf, a result of independent interest.

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  • Faugeras, Olivier P., 2015. "Maximal coupling of empirical copulas for discrete vectors," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 179-186.
    • Handle: RePEc:eee:jmvana:v:137:y:2015:i:c:p:179-186
    DOI: 10.1016/j.jmva.2015.02.013
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    References listed on IDEAS

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    1. Neslehová, Johanna, 2007. "On rank correlation measures for non-continuous random variables," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 544-567, March.
    2. Mesfioui, Mhamed & Quessy, Jean-François, 2010. "Concordance measures for multivariate non-continuous random vectors," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2398-2410, November.
    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.
    4. 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.
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    Cited by:

    1. 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.
    2. Genest, Christian & Nešlehová, Johanna G. & Rémillard, Bruno, 2017. "Asymptotic behavior of the empirical multilinear copula process under broad conditions," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 82-110.
    3. Faugeras, Olivier P. & Rüschendorf, Ludger, 2021. "Functional, randomized and smoothed multivariate quantile regions," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    4. Olivier Paul Faugeras & Ludger Rüschendorf, 2021. "Functional, randomized and smoothed multivariate quantile regions," Post-Print hal-03352330, HAL.
    5. Wei, Zheng & Kim, Daeyoung, 2021. "Measure of asymmetric association for ordinal contingency tables via the bilinear extension copula," Statistics & Probability Letters, Elsevier, vol. 178(C).

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