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

Šeliga Adam (), Kauers Manuel (), Saminger-Platz Susanne (), Mesiar Radko (), Kolesárová Anna () and Klement Erich Peter ()
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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

Dependence Modeling, 2021, vol. 9, issue 1, 13-42

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.

Keywords: Copula; polynomial inequality; cylindrical algebraic decomposition; dependence parameter; Schur concavity; ultramodularity (search for similar items in EconPapers)
Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:vrs:demode:v:9:y:2021:i:1:p:13-42:n:2

DOI: 10.1515/demo-2021-0101

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