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Maximal asymmetry of bivariate copulas and consequences to measures of dependence

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  • Griessenberger Florian
  • 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,YX,Y either in the sense of the uniform metric d∞{d}_{\infty } or the conditioning-based metrics Dp{D}_{p}, and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ1{\zeta }_{1} and ξ\xi . Considering maximal d∞{d}_{\infty }-asymmetry we obtain ζ1∈56,1{\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ∈23,1\xi \in \left[\frac{2}{3},1\right], and in the case of maximal D1{D}_{1}-asymmetry we obtain ζ1∈34,1{\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ∈12,1\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 D1{D}_{1}-asymmetry and derive some surprising properties for maximal Dp{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. 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.
    4. 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.
    5. Junker, Robert R. & Griessenberger, Florian & Trutschnig, Wolfgang, 2021. "Estimating scale-invariant directed dependence of bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).
    6. Roger Nelsen, 2007. "Extremes of nonexchangeability," Statistical Papers, Springer, vol. 48(4), pages 695-695, October.
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