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Polynomial bivariate copulas of degree five: characterization and some particular inequalities

Author

Listed:
  • Šeliga Adam

    (Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia)

  • Kauers Manuel

    (Institute for Algebra, Johannes Kepler University, Linz, Austria)

  • Saminger-Platz Susanne

    (Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria)

  • Mesiar Radko

    (Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia)

  • Kolesárová Anna

    (Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology, Bratislava, Slovakia)

  • Klement Erich Peter

    (Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Linz, Austria)

Abstract

Bivariate polynomial copulas of degree 5 (containing the family of Eyraud-Farlie-Gumbel-Morgenstern copulas) are in a one-to-one correspondence to certain real parameter triplets (a, b, c), i.e., to some set of polynomials in two variables of degree 1: p(x, y) = ax + by + c. The set of the parameters yielding a copula is characterized and visualized in detail. Polynomial copulas of degree 5 satisfying particular (in)equalities (symmetry, Schur concavity, positive and negative quadrant dependence, ultramodularity) are discussed and characterized. Then it is shown that for polynomial copulas of degree 5 the values of several dependence parameters (including Spearman’s rho, Kendall’s tau, Blomqvist’s beta, and Gini’s gamma) lie in exactly the same intervals as for the Eyraud-Farlie-Gumbel-Morgenstern copulas. Finally we prove that these dependence parameters attain all possible values in ]−1, 1[ if polynomial copulas of arbitrary degree are considered.

Suggested Citation

  • 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.
    • Handle: RePEc:vrs:demode:v:9:y:2021:i:1:p:13-42:n:2
    DOI: 10.1515/demo-2021-0101
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    References listed on IDEAS

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