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A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support

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  • Sánchez Juan Fernández

    (Grupo de Investigación de Análisis Matemático, Universidad de Almería, 04120 La Cañada de San Urbano, Almería, Spain)

  • Trutschnig Wolfgang

    (Department for Artificial Intelligence & Human Interfaces, University of Salzburg, Hellbrunnerstrasse 34, 5020 Salzburg, Austria)

Abstract

Working with shuffles, we establish a close link between Kendall’s τ \tau , the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well known that Spearman’s ρ \rho of a bivariate copula A A is a rescaled version of the volume of the area under the graph of A A , in this contribution we show that the other famous concordance measure, Kendall’s τ \tau , allows for a simple geometric interpretation as well – it is inextricably linked to the surface area of A A .

Suggested Citation

  • Sánchez Juan Fernández & Trutschnig Wolfgang, 2023. "A link between Kendall’s τ, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-14, January.
    • Handle: RePEc:vrs:demode:v:11:y:2023:i:1:p:14:n:1
    DOI: 10.1515/demo-2023-0105
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    References listed on IDEAS

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    1. Fredricks, Gregory A. & Nelsen, Roger B. & Rodriguez-Lallena, Jose Antonio, 2005. "Copulas with fractal supports," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 42-48, August.
    2. Coblenz, Maximilian & Grothe, Oliver & Schreyer, Manuela & Trutschnig, Wolfgang, 2018. "On the length of copula level curves," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 347-365.
    3. Juan Fernández Sánchez & Wolfgang Trutschnig, 2015. "Conditioning-based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1311-1336, December.
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