Abstract
Several extensions of the family of (bivariate) Eyraud-Farlie-Gumbel-Morgenstern copulas (EFGM copulas) are considered. Some of them are well-known from the literature, others have recently been suggested (copulas based on quadratic constructions, based on some forms of convexity, and polynomial copulas). For each of these extensions we analyze which properties of EFGM copulas are preserved (or even improved) and which are (partly) lost. Such properties can be structural (order theoretical or topological) in nature, or algebraic (symmetry or being a polynomial) or analytic (absolute continuity). Other examples are forms of convexity, quadrant dependence, and symmetry with respect to copula transformations. The last group of properties considered here is related to some dependence parameters.
Highlights
The class of bivariate distributions we call Eyraud-Farlie-Gumbel-Morgenstern distributions was studied first by Eyraud [29], and by Morgenstern [65] and by Gumbel [38], and it was further generalized by Farlie [30]
Extensions of EFGM copulas based on quadratic constructions on copulas are discussed in Section 5, and extensions based on some forms of convexity are the topic of Section 6
This fact is one of the main reasons to look for suitable extensions of the family of EFGM copulas preserving the positive properties of the EFGM copulas and where the corresponding dependence parameters cover greater subintervals of [−1, 1]
Summary
The class of bivariate distributions we call Eyraud-Farlie-Gumbel-Morgenstern distributions was studied first by Eyraud [29], and by Morgenstern [65] (for Cauchy marginals) and by Gumbel [38] (for exponential marginals), and it was further generalized by Farlie [30]. Note that in most papers the EFGM copulas were called Farlie-Gumbel-Morgenstern copulas (FGM copulas for short), since the results of [29] had been forgotten for many years. Their quotation in the monograph [28] (see [14,15,64]) contributed to a movement to rename this family of copulas and to pay proper credit to the early achievements of Eyraud. We will join this movement and consistently speak about Eyraud-Farlie-GumbelMorgenstern (EFGM) copulas. We present polynomial copulas (in particular of degree 5, compare [81]) which by construction are natural extensions of EFGM copulas (Section 7)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
