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BY 4.0 license Open Access Published by De Gruyter Open Access March 30, 2021

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

  • Adam Šeliga ORCID logo , Manuel Kauers ORCID logo , Susanne Saminger-Platz ORCID logo , Radko Mesiar ORCID logo , Anna Kolesárová ORCID logo and Erich Peter Klement ORCID logo EMAIL logo
From the journal Dependence Modeling

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.

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Received: 2020-11-21
Accepted: 2021-02-10
Published Online: 2021-03-30

© 2021 Adam Šeliga et al., published by De Gruyter

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