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Maximal asymmetry of bivariate copulas and consequences to measures of dependence

Author

Listed:
  • Griessenberger Florian

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

  • Trutschnig Wolfgang

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

Abstract

In this article, we focus on copulas underlying maximal non-exchangeable pairs ( X , Y ) \left(X,Y) of continuous random variables X , Y X,Y either in the sense of the uniform metric d ∞ {d}_{\infty } or the conditioning-based metrics D p {D}_{p} , and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ 1 {\zeta }_{1} and ξ \xi . Considering maximal d ∞ {d}_{\infty } -asymmetry we obtain ζ 1 ∈ 5 6 , 1 {\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ ∈ 2 3 , 1 \xi \in \left[\frac{2}{3},1\right] , and in the case of maximal D 1 {D}_{1} -asymmetry we obtain ζ 1 ∈ 3 4 , 1 {\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ ∈ 1 2 , 1 \xi \in \left(\frac{1}{2},1\right] , implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximal D 1 {D}_{1} -asymmetry and derive some surprising properties for maximal D p {D}_{p} -asymmetric copulas.

Suggested Citation

  • Griessenberger Florian & Trutschnig Wolfgang, 2022. "Maximal asymmetry of bivariate copulas and consequences to measures of dependence," Dependence Modeling, De Gruyter, vol. 10(1), pages 245-269, January.
  • Handle: RePEc:vrs:demode:v:10:y:2022:i:1:p:245-269:n:2
    DOI: 10.1515/demo-2022-0115
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    References listed on IDEAS

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    1. Karl Siburg & Pavel Stoimenov, 2011. "Symmetry of functions and exchangeability of random variables," Statistical Papers, Springer, vol. 52(1), pages 1-15, February.
    2. Holger Dette & Karl F. Siburg & Pavel A. Stoimenov, 2013. "A Copula-Based Non-parametric Measure of Regression Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 21-41, March.
    3. Junker, Robert R. & Griessenberger, Florian & Trutschnig, Wolfgang, 2021. "Estimating scale-invariant directed dependence of bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    4. Roger Nelsen, 2007. "Extremes of nonexchangeability," Statistical Papers, Springer, vol. 48(4), pages 695-695, October.
    5. Piotr Mikusiński & Michael Taylor, 2010. "Some approximations of n-copulas," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 72(3), pages 385-414, November.
    6. 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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