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Extensions of Farlie-Gumbel-Morgenstern distributions: A review

A copula is a function that completely describes the dependence structure between the marginal distributions. One of the most important parametric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family. In practical applications this copula has been shown to be somewhat limited. We review several extensions of this family. A copula is a function that completely describes the dependence structure. It contains all the information to link the marginal distributions to their joint distribution. To obtain a valid multivariate distribution function, it suffices to combine several marginal distribution functions with any copula function. Thus, for the purposes of statistical modeling, it is desirable to have a large collection of copulas at one’s disposal. Many examples of copulas can be found in the literature, most are members of families with one or more real parameters. One of the most important parametric family of copulas is the Farlie-GumbelMorgenstern (FGM) family defined as C θ (u, v) = uv + θuv(1 − u)(1 − v), u, v ∈ [0, 1], (1) where θ ∈ [−1, 1]. The density function of FGM copulas is given by ∂C θ (u, v) ∂u∂v = θ (2u − 1) (2v − 1) + 1, for any u, v ∈ [0, 1]. Members of the FGM family are symmetric, i.e., C θ (u, v) = C θ (v, u) for all (u, v) in [0, 1] 2 and have the lower and upper tail dependence coefficients equal to 0. The copula given in (1) is PQD for θ ∈ (0, 1] and NQD for θ ∈ [−1, 0). In practical applications this copula has been shown to be somewhat limited, for copula dependence parameter θ ∈ [−1, 1] , Spearman’s correlation ρ ∈ [−1/3, 1/3] and Kendall’s τ ∈ [−2/9, 2/9] , for more details on copulas see, for example, [1]. To overcome this limited dependence, several authors proposed extensions of this family.