A copula is a function that completely describes the dependence structure between the marginal distributions. One of the most important para-metric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family. In practical applications this copula has been shown to be somewhat limited. We propose a new extension of this family based on the introduction of an univariate function

Stoica, Emil

English

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An extension of FGM distributions based on an
univariate function
Emil Stoica
To cite this version:
Emil Stoica. An extension of FGM distributions based on an univariate function. 2017. ￿hal-01535617￿
An extension of FGM distributions based on an
univariate function
Emil Stoica
Dragan European University of Lugoj
Abstract
A copula is a function that completely describes the dependence struc-
ture between the marginal distributions. One of the most important para-
metric family of copulas is the Farlie-Gumbel-Morgenstern (FGM) family.
In practical applications this copula has been shown to be somewhat lim-
ited. We propose a new extension of this family based on the introduction
of an univariate function.
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-Gumbel-
Morgenstern (FGM) family defined as
CFGMθ (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
∂2CFGM
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., CFGM
θ
(u, v) = CFGM
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 practi-
cal 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.
1
• Theoretical issues: [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28]
• Applications: [29, 30, 31, 32, 33, 34, 35, 36]
Here, we focus on the machinery introduced in [37]. Let C1 be the FGM copula:
C1(u, v) = uv + θu(1− u)v(1− v)
and C2 be the copula proposed by [12]:
C2(u, v) = min(u, v)f(max(u, v)).
Let C = C1 ⋆ C2 where ⋆ is the copula-product, see for instance [1]. Routine
calculations show that
C(u, v) = uv + θu(1− u)ψ(v)
where
ψ(v) = f(v)v(1− v) + v
∫
1
v
(1− 2t)f ′(t)dt
= f(v)v(1− v) + v(f(1)− f(v))− 2v
∫
1
v
tf ′(t)dt
= f(v)v(1− v) + v(1− f(v))− 2v
∫
1
v
tf ′(t)dt
= v(1− vf(v))− 2v
∫
1
v
tf ′(t)dt
It is thus clear that C is an extension of C1 since, for a given function f (not
constant equal to 1), one has ψ(v) 6= v(1− v).
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4
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5


An extension of FGM distributions based on an univariate function

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