Polynomial bivariate copulas of degree five: characterization and some particular inequalities
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 parameter...
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De Gruyter
2021
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oai:doaj.org-article:25fdaa58c61d4d5fa4c43ece7c732d342021-12-05T14:10:45ZPolynomial bivariate copulas of degree five: characterization and some particular inequalities2300-229810.1515/demo-2021-0101https://doaj.org/article/25fdaa58c61d4d5fa4c43ece7c732d342021-03-01T00:00:00Zhttps://doi.org/10.1515/demo-2021-0101https://doaj.org/toc/2300-2298Bivariate 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.Šeliga AdamKauers ManuelSaminger-Platz SusanneMesiar RadkoKolesárová AnnaKlement Erich PeterDe Gruyterarticlecopulapolynomial inequalitycylindrical algebraic decompositiondependence parameterschur concavityultramodularityprimary 26b25, 62e10secondary 39b62, 60e05, 62h20Science (General)Q1-390MathematicsQA1-939ENDependence Modeling, Vol 9, Iss 1, Pp 13-42 (2021) |
| institution |
DOAJ |
| collection |
DOAJ |
| language |
EN |
| topic |
copula polynomial inequality cylindrical algebraic decomposition dependence parameter schur concavity ultramodularity primary 26b25, 62e10 secondary 39b62, 60e05, 62h20 Science (General) Q1-390 Mathematics QA1-939 |
| spellingShingle |
copula polynomial inequality cylindrical algebraic decomposition dependence parameter schur concavity ultramodularity primary 26b25, 62e10 secondary 39b62, 60e05, 62h20 Science (General) Q1-390 Mathematics QA1-939 Šeliga Adam Kauers Manuel Saminger-Platz Susanne Mesiar Radko Kolesárová Anna Klement Erich Peter Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| description |
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. |
| format |
article |
| author |
Šeliga Adam Kauers Manuel Saminger-Platz Susanne Mesiar Radko Kolesárová Anna Klement Erich Peter |
| author_facet |
Šeliga Adam Kauers Manuel Saminger-Platz Susanne Mesiar Radko Kolesárová Anna Klement Erich Peter |
| author_sort |
Šeliga Adam |
| title |
Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| title_short |
Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| title_full |
Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| title_fullStr |
Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| title_full_unstemmed |
Polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| title_sort |
polynomial bivariate copulas of degree five: characterization and some particular inequalities |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/25fdaa58c61d4d5fa4c43ece7c732d34 |
| work_keys_str_mv |
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| _version_ |
1718371758921220096 |