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

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.

A Collection of Two-Dimensional Copulas Based on an Original Parametric Ratio Scheme

“…These secondary functions thus perturb the product copula. The idea of perturbing the functionalities of the product copula is not new (see [28,29], and the references therein), but the considered weighted ratio form for P(x, y) remains the original angle of the article. Due to the mathematical complexity of the form in Equation ( 1), a general study is nearly impossible.…”