Laplace copulas of multifactor gamma distributions are new
generalized Farlie-Gumbel-Morgenstern copulas

arXiv:1611.07242v1 [] 22 Nov 2016

Philippe Bernardoff
Université de Pau et des Pays de l’Adour
Laboratoire de Mathématiques et de leurs Applications, UMR 5142
avenue de l’Université
64000 Pau, France.
e.mail : philippe.bernardoff@univ-pau.fr
November 10, 2016

Abstract
This paper provides bifactor gamma distribution, trivariate gamma distribution and two copula
families on [0, 1]n obtained from the Laplace transforms of the multivariate gamma distribution and
the multi-factor gamma distribution given by [P (θ)]−λ and
Q
−(λi −λ)
[P (θ)]−λ n
respectively, where P is an affine polynomial with respect to the n
i=1 (1 + pi θi )
variables θ1 , . . . , θn .
These copulas are new generalized Farlie-Gumbel-Morgenstern copulas and allow in particular
to obtain multivariate gamma distributions for which the cumulative distribution functions and the
probability distribution functions are known.

KEY WORDS: Archimedean copula, cumulative distribution function, copula, exponential
families, infinitely divisible distribution, generalized Farlie-Gumbel-Morgenstern copulas, generalized hypergeometric function, generalized Lauricella functions, Horn function, Kendall’s tau, Laplace
copula, Laplace transform, multi-factor gamma distribution, multivariate gamma distribution, Spearmann’s rho.

1

1

Introduction

This paper is motivated by the opportunity to produce explicit cumulative distribution functions (cdf)
and probability distribution functions (pdf) of multivariate gamma distributions and multi-factor gamma
distributions. In this way, we present two new classes of multivariate copulas generalizing the FarlieGumbel-Morgenstern Copulas. From Sklar (Sklar, 1959) who states that the cdf F of a random vector
X = (X1 , . . . , Xn ) with continuous marginal cdfs can be uniquely written in the form
F (x1 , . . . , xn ) = C [F1 (x1 ) , . . . , Fn (xn )] , x = (x1 , . . . , xn ) ∈ Rn ,

(1)

n

where n is the dimension of the random vector X, where C : [0, 1] → [0, 1] is a copula and where
F1 , . . . , Fn are the marginal cdfs of X.
If f is the pdf of the random vector X and f1 , . . . , fn the marginal pdfs of X, then
c (u1 , . . . , un ) =

∂n
C (u1 , . . . , un ) ,
∂u1 . . . ∂un

and we have the following equality
f (x1 , . . . , xn ) = c [F1 (x1 ) , . . . , Fn (xn )] f1 (x1 ) . . . fn (xn ) .

(2)

From Equality (1) it is possible to express the cdf F of X. This expression cannot simply explicitly
give the copula C for the multivariate gamma distribution associated with (P, λ) and the multi-factor
gamma distribution associated with (P, Λ) . But with Joe (Joe, 1997), we can give copulas deduced of
the Laplace transform of the multivariate gamma distribution associated with (P, λ) and the Laplace
transform of the multi-factor gamma distribution associated with (P, Λ) . By applying the formulas (1)
and (2) we can give a new multivariate gamma distribution associated with (P, λ) and a new multi-factor
gamma distribution associated with (P, Λ) for which we have an explicit formula for its cdf and its pdf.
The paper is organized as follows. Section 2 gives deﬁnitions of multivariate gamma distributions and
multi-factor gamma distributions for which Laplace transform is given, and considers the bidimentional
and tridimentional case. Section 3 deﬁnes the Laplace copula. Section 4 states the two main results.
Section 5 applies the main cases to bidimensional and tridimensional cases. In particular, the Joe’s
family BB10 is generalized. For ease of the ﬂuent exposition of the paper, proofs are collected in the
Appendix.

2

2

Multivariate gamma distributions and multi-factor gamma
distributions

In the literature, the multivariate gamma distributions on Rn have several non-equivalent deﬁnitions.
Many authors require only that the marginal distributions are ordinary gamma distributions (Balakrishnan et al., 1997). In the present paper, we use the extension of the classical one-dimensional deﬁnition to
Rn obtained as follows: we consider an aﬃne polynomial P (θ) in θ = (θ1 , . . . , θn ) where ‘aﬃne’ means
that, for j = 1, . . . , n, ∂ 2 P/∂θj2 = 0. We also assume that P (0) = 1. For instance, for n = 2, we have
P (θ1 , θ2 ) = 1 + p1 θ1 + p2 θ2 + p{1,2} θ1 θ2 .
We denote by Pn = P ([n]) the family of all subsets of [n] and P∗n the family of non-empty subsets
of [n] = {1, . . . , n} . For simplicity, if n is ﬁxed and if there is no ambiguity, we denote these families by
P and P∗ , respectively.
We denote by N the set of non-negative integers. If z = (z1 , . . . , zn ) ∈ Rn and α = (α1 , . . . , αn ) ∈ Nn ,
then α! = α1 ! . . . αn !, |α| = α1 + . . . + αn , aα = aα1 ,...,αn and
zα =

n
Y

ziαi = z1α1 . . . znαn .

i=1

For T in Pn , we simplify the above notation by writing zT =

Q

t∈T

zt instead of z1T where

1T = (α1 , . . . , αn ) with αi = 1 if i ∈ T and αi = 0 if i ∈
/ T.
We also write z−T for

Q

1/zt . For a mapping a : P → R, we shall use the notation a : P → R,
P
T →
7 aT . In this notation, an aﬃne polynomial with constant term equal to 1 is P (θ) = T ∈P pT θ T ,
t∈T

with p∅ = 1. For simplicity, if T = {t1 , . . . , tk } , we denote a{t1 ,...,tk } = at1 ...tk . The indicator function of
a set S is denoted by

1S , that is, 1S (x) = 1 for x ∈ S and 0 for x ∈/ S.

We ﬁx λ > 0. If a random vector X = (X1 , . . . , Xn ) on Rn with probability distribution (pd) µX is
such that its Laplace transform is
−λ

E {exp [− (θ1 X1 + · · · + θn Xn )]} = [P (θ)]

,

(3)

where E denotes the expectation, for a set of θ with non-empty interior, then we denote µX = γ(P,λ) ,
and γ(P,λ) will be called the multivariate gamma distribution associated with (P, λ). These multivariate
gamma distributions occur naturally in the classiﬁcation of natural exponential families in Rn (Bar-Lev
et al., 1994).
The marginal distributions of the multivariate gamma distribution associated with (P, λ) are ordinary
gamma distributions of parameters (pi , λ) for i = 1, . . . , n, with Laplace transform (1 + pi θi )−λ , and pd
γ(pi ,λ) (d x) = xλ−1 p−λ
i /Γ (λ) exp (−x/pi ) 1(0,∞) (x)d x. We extend the ﬁrst deﬁnition to the multi-factor
3

gamma distribution associated with (P, Λ) where Λ = (λ, λ1 , . . . , λn ) and λi > λ > 0 for all i = 1, . . . , n
by its Laplace transform
E {exp [− (θ1 X1 + · · · + θn Xn )]} = [P (θ)]−λ

n
Y

(1 + pi θi )−(λi −λ) .

(4)

i=1

The marginal distributions of the multi-factor gamma distribution associated with (P, Λ) are ordinary
gamma distributions of parameters (pi , λi ) for i = 1, . . . , n.
We state ﬁrst a proposition, whose proof is obvious.
Proposition 1 A random vector X with distribution γP,Λ can be obtained in the following way:
Let Y be a vector with distribution γP,λ . Let Z = (Z1 , . . . , Zn ) be a random vector constituted of independent margins for which its pds are γ(pi ,λi ) , and such that Z and Y are independent. Then the vector
X = Y + Z has Laplace transform (4), and consequently is a multi-factor gamma distribution associated
to (P, Λ) .
For the bidimensional case, Dussauchoy and Berland, (Dussauchoy and Berland, 1972) consider the
random vector X = (X1 , X2 ) with Laplace transform
LX (θ1 , θ2 ) = (1 + p1 θ1 )
= (1 + p1 θ1 )

−λ1

−λ2

(1 + p2 θ2 )

−(λ1 −λ)



r12 p1 p2 θ1
1−
(1 + p1 θ1 ) (1 + p2 θ2 )

−(λ2 −λ)

(1 + p2 θ2 )

−λ
−λ

(1 + p1 θ1 + p2 θ2 + p12 θ1 )

,

(5)

where r12 = 1 − p12 / (p1 p2 ) > 0 and p1 , p2 , p12 > 0.
Unfortunately the Laplace transform of these pds is simple, but its pdfs and cdfs are unknown,
p
except for the case n = 2 for the multivariate gamma distribution associated with (P, λ). Let Fm
be the

generalized hypergeometric function (Slater, 1966) deﬁned by
p
Fm

∞
X
(α1 )k · · · (αp )k z k
(α1 , . . . , αp ; β1 , . . . , βm ; z) =
,
(β1 )k · · · (βm )k k!

(6)

k=0

where (a)k = Γ (a + k) /Γ (a) for a > 0 and k ∈ N, or more generally by (a)0 = 1, (a)n+1 = (a + n)(a),
0
∀n ∈ N, ∀a ∈ R, is the Pochhammer’s symbol. For simpliﬁcation, we denote Fm
by Fm . Bernardoﬀ

(Bernardoﬀ, 2006) gives the
Proposition 2 Let P (θ1 , θ2 ) = 1 + p1 θ1 + p2 θ2 + p12 θ1 θ2 be an aﬃne polynomial where p1 , p2 > 0 and
p1,2 > 0. Let µ = γ (P,λ) be the gamma distribution associated to (P, λ) . The measure µ exists if and only
if c = (p1 p2 − p12 ) /p212 = p1 p2 /p212 r12 > 0. Then we have
γ (P,λ) (dx1 , dx2 ) =

p−λ
12
Γ (λ)

p

2e

p

− p 2 x1 − p 1 x2
12

12

λ−1

(x1 x2 )

4

F1 (λ; cx1 x2 ) 1(0,∞)2 (x) d (x) .

(7)

For the case Λ = (λ, λ, λ2 ), the multi-factor gamma distribution associated with (P, Λ) is named by
Chatelain et al. (Chatelain et al., 2008) the multisensor gamma distribution associated with (P, λ, λ2 )
and they have proved that its pd is given by the equality
−(λ −λ)

γ (P,Λ) (dx1 , dx2 ) =

2
p−λ
1 λ−1 λ2 −1 − pp2 x1 − pp1 x2
p12
12 p2
12
x2 ; cx1 x2 )
x
x2
e 12
Φ3 (λ2 − λ; λ2 ; c
Γ (λ) Γ (λ2 ) 1
p2

× 1(0,∞)2 (x1 , x2 ) dx1 dx2 ,
where
Φ3 (a; b; x, y) =

(8)

X

m,n>0

(a)m xm y n
(b)m+n m! n!

(9)

is the Horn function.
For the bidimensional general case, we have the following Theorem. Let FI be the function deﬁned
by
FI (a, b, c, z) =

∞
X

m1 ,m2 ,m3

(a)m1 (b)m2 (c)m3
z1m1 z2m2 z3m3
;
(a + c)m1 +m3 (b + c)m2 +m3 m1 ! m2 ! m3 !
=0

(10)

it is a particular generalized Lauricella function deﬁned, by example, in (Panda, 1973).
Theorem 3 The pd of γ (P,(λ,λ1 ,λ2 )) is given by the equality
−(λ −λ) −(λ −λ)

p2
p
p2 2
p−λ p 1
−
x + 1 x
x1λ1 −1 x2λ2 −1 e p12 1 p12 2
γ (P,(λ,λ1 ,λ2 )) (dx1 , dx2 ) = 12 1
Γ (λ1 ) Γ (λ2 )


p12
p12
x1 ,
x2 , cx1 x2 1(0,∞)2 (x1 , x2 ) dx1 dx2 ,
× FI λ1 − λ, λ2 − λ, λ,
p1
p2





(11)

If we get λ1 = λ in the equality (11), we obtain Chatelain and Tourneret’s result (8) because
FI (0, λ2 − λ, λ, z1 , z2 , z3 ) =
∞
X

m2 ,m3

(b)m2
z2m2 z3m3
= Φ3 (b; b + c; z2 , z3 ) .
(b + c)m2 +m3 m2 ! m3 !
=0

Bernardoﬀ (Bernardoﬀ, 2006) gives the following Proposition:
Proposition 4 Let µ be a multivariate gamma distribution on Rn associated to (P, λ) . Assume that µ is
not concentrated on a linear subspace of Rn of the form {x ∈ Rn ; xk = 0} for some k in [n] = {1, . . . , n}.
Then:
(i) For all i ∈ [n] , pi 6= 0.
k

n−k

(ii) If p1 , . . . , pk < 0 and pk+1 , . . . , pn > 0, then Supp (µ) ⊂ (−∞, 0] × [0, ∞)
(iii) If p1 , . . . , pn > 0 then p[n] > 0.

5

.

Bernardoﬀ (Bernardoﬀ, 2006) gives a necessary and suﬃcient condition for inﬁnite divisibility of the
multivariate gamma distribution associated with (P, λ), in the sense that the Laplace transform of γ(P,λ)
power t for all positive t is still the Laplace transform of a positive measure, by the following theorem:
Theorem 5 Let µ = γP,λ be a gamma distribution associated with (P, λ) , where λ > 0 and P (θ) =
P
P
T
e
eT θT be the aﬃne
T ∈Pn p
T ∈Pn pT θ is such that pi > 0, for all i ∈ [n] , and p[n] > 0. Let P (θ) =
polynomial such that peT = −pT /p[n] for all T ∈ Pn , where T = [n] r T . Let
ebS = bS (Pe ) =

|S|
X

(k − 1)!

k=1

X Y

T

∈Πk
S

T ∈T

peT ,

where |S| denotes the cardinality of the set S, and ΠkS denotes the set of all partitions of S into k
non-empty subsets of S. Then the measure µ is inﬁnitely divisible if and only if
e
pi < 0 for all i ∈ [n] ,

(12)

ebS > 0 for all S ∈ P∗ such that |S| > 2.
n

(13)

and

Corollary 6 By the properties of inﬁnite divisible distributions we conclude that the necessary and
suﬃcient conditions for inﬁnite divisibility of a multivariate gamma distribution associated to (P, λ) of
theorem (5), are also necessary and suﬃcient conditions for inﬁnite divisibility of multivariate multifactor gamma distribution associated to (P, Λ) .
To illustrate the diﬃculty to calculate the multivariate gamma distribution associated with (P, λ) we
give, for the tridimensional case, the following theorem. Let FII be the function deﬁned by
FII (λ1 , λ2 , z1 , z2 , z3 , z4 ) =

∞
X

m1 ,...,m4

1
z1m1 z2m2 z3m3 z4m4
;
(λ1 )m1 +m2 +m3 (λ2 )2m1 +m2 +m4 m1 ! m2 ! m3 ! m4 !
=0

(14)

it is still a particular generalized Lauricella function.
2
k
+
Theorem 7 In the case n = 3, pi > 0 for i ∈ [3] , pij > 0 for (i, j) ∈ [3] , ebij = − pb123

pjk pik
p2123

> 0 for

i 6= j and {i, j, k} = [3] , p123 > 0, and

eb123 = − 1 + p12 p1 + p13 p2 + p23 p1 + 2 p12 p13 p23 > 0,
p123
p2123
p2123
p2123
p3123

the inﬁnitely divisible multivariate gamma distribution γ(λ,P ) associated with (P, λ) , is given by the
formula
γ(λ,P ) (dx) =

−λ
p123

[Γ (λ)]3

exp(e
p1 x1 + pe2 x2 + pe3 x3 ) (x1 x2 x3 )λ−1

× FII (λ, λ, eb13 x1 x3eb23 x2 x3 , eb123 x1 x2 x3 , eb12 x1 x2 , eb13 x1 x3 + eb23 x2 x3 )1(0,∞)3 (x) dx.
6

(15)

Remark 8 The case p123 = 0 is solved by Letac and Wesolowski (Letac and Wesolowski, 2008).
Remark 9 If eb12 = eb13 = eb23 = 0, Theorem 7 gives
γ(λ,P ) (dx) =

p−λ
123

[Γ (λ)]

3

λ−1

exp(e
p1 x1 + pe2 x2 + pe3 x3 ) (x1 x2 x3 )



F2 λ, λ; eb123 x1 x2 x3 1(0,∞)3 (x) dx,

and if we put λ = 1 in this last equality, we obtain the Kibble and Moran distribution given in (Balakrishnan et al., 2000).

3

Laplace copula

We recall the following theorem (Marshall and Olkin, 1988), Theorem 2.1, p. 835, and his Corollary
Theorem 10 Let H1 , . . . , Hn , be univariate cdfs, and let G be an n-variate cdf such that G (0, . . . , 0) = 1,
with univariate margins Gi (i = 1, . . . , n). Denote the Laplace transforms of G and Gi , respectively, by

φ and φi (i = 1, . . . , n). Let K be a copula. If Fi (x) = exp −φ−1
i [Hi (x)] (i = 1, . . . , n) , then
Z
Z
n
o
θ
θ
H (x1 , . . . , xn ) = · · · K [F1 (x1 )] 1 , . . . , [Fn (xn )] n dG (θ1 , . . . , θn )
(16)
is an n-variate cdf with marginals H1 , . . . , Hn .
Corollary 11 Under the conditions of the Theorem 10, with
Qn
K (x1 , . . . , xn ) = i=1 xi , (0 6 xi 6 1; i = 1, . . . , n) ,


H (x1 , . . . , xn ) = φ φ1−1 [H1 (x1 )] , . . . , φ−1
n [Hn (xn )]

(17)

deﬁnes an n-variate cdf with marginals H1 , . . . , Hn .

Therefore copula for H is associated to G and is given by the formula


−1
C (u1 , . . . , un ) = φ φ−1
1 (u1 ) , . . . φn (un ) .

(18)

Remark 12 If we know φ and φi , for i = 1, . . . , n, we can ﬁnd the copula C, that is the case for the
multivariate gamma distributions and for the multivariate multi-factor gamma distributions. For the
case Hi = Gi , we can write :

−1
H (x1 , . . . , xn ) = φ φ−1
1 [G1 (x1 )] , . . . φn [Gn (xn )] ,

and this last formula deﬁnes n-variate cdf gamma H with margin distributions G1 , . . . , Gn .
Then the copula associated to H is


C (u1 , . . . , un ) = φ φ1−1 (u1 ) , . . . , φ−1
n (un ) .
7

(19)

Now, from the Formula 18, we can give the following Deﬁnition
n

Definition 13 Let X = (X1 , X2 , . . . , Xn ) be a random vector in ]0, +∞[ with Laplace transform ϕX
deﬁned by LX (θ) = LX (θ1 , . . . , θn ) = ϕX (θ1 , . . . , θn ) = E {exp [− (θ1 X1 + · · · + θn Xn )]}, and let ϕXi be
the Laplace transform of the random variable Xi , deﬁned by LXi (θi ) = ϕXi (θi ) = E [exp (−θi Xi )] . The
function ϕXi is a one and onto, and is decreasing of [0, +∞[ onto ]0, 1]. Its inverse function is denoted
by ϕ−1
Xi . Then the function CLX deﬁned by


−1
CLX (u1 , . . . , un ) = ϕX ϕ−1
X1 (u1 ) , . . . , ϕXn (un )

is a copula. We call this copula CLX the Laplace copula associated to the random vector X. If X has pd
µX (dx) , then we denote still CLX (u1 , . . . , un ) = CLµ (u1 , . . . , un ) .
X

Proposition 14 Let F1 , . . . , Fn , be n univariate cdfs on ]0, +∞[ . The relation

−1
F (x1 , . . . , xn ) = ϕX ϕ−1
X1 [F1 (x1 )] , . . . , ϕXn [Fn (xn )]
= CLX [F1 (x1 ) , . . . , Fn (xn )]

deﬁnes a cdf F with marginal cdfs F1 , . . . , Fn . Then the copula associated to F is CLX .
If we choose, for i = 1, 2, . . . , n, Fi = FXi , the cdf of Xi , then F is the cdf of a random vector X for
which the marginal cdfs are the cdfs of Xi , and we have F 6= FX where FX calls the cdf of X. In the
case of multivariate gamma distributions and multi-factor gamma distributions, we get other multivariate
gamma distributions and multi-factor gamma distributions with given copula.
Laplace copula can be seen as a generalization of Archimedean copula (Joe, 1997,2014).

4

Main results

Now, we are in the capacity to give the two main results of this paper
Theorem 15 Let P an aﬃne polynomial in the n variables θi , i = 1, . . . , n, with P (0) = 1, then the
Laplace copula of the multivariate gamma distribution γ(P,λ) associated to (P, λ) is
CLγ(P,λ) (v) = v[n] [1 +

X

|T |

(−1)

Y
1
1
P (− 1T )
(1 − vtλ )]−λ ,
p

(20)

t∈T

T ⊂[n],|T |>1

where v = (v1 , . . . , vn ) , |T | is the cardinality of T and the vector


/ T , for i ∈ {1, 2, . . . , n}.
i ∈ T , p1 1T = 0 if i ∈

1
p 1T

is deﬁned by



1
p 1T



i

=

1
pi

if

i

This family is given, by example, for the simpler case P (θ) =

Qn

i=1 (1+pi θi )]−βp

1, corresponding to the family gived by (Fang et al., 2000), namely C (v) = v
8

[n]

[n] [n]

θ , with 0 6 β <
1
Qn
[1 − β i=1 (1 − viλ )]−λ .

Theorem 16 Let P an aﬃne polynomial in the n variables θi , i = 1, . . . , n, with P (0) = 1, let Λ =
(λ, λ1 , . . . , λn ) with λi > λ > 0, for i ∈ {1, 2, . . . , n} , then the Laplace copula of the multivariate multifactor gamma distribution γ(P,Λ) associated to (P, Λ) is
CLγ(P,Λ) (v) = v[n] [1 +

X

|T |

(−1)

1
Y
1
P (− 1T )
(1 − vtλt )]−λ .
p

(21)

t∈T

T ⊂[n],|T |>1

This copula family is a new generalization of Farlie-Gumbel-Morgenstern Copulas.
Bekrizadeh et al., 2012 propose a similar formula only for λ being a negative integer.
We note that, if the conditions of Theorem 5 are checked, formulas (20) and (21) deﬁne copulas.
For these copulas, if we inject univariate gamma distributions γ(pi ,λ) in the ﬁrst case or γ(pi ,λi ) in
the second case, then we obtain other multivariate gamma distributions and other multivariate multifactor gamma distributions in the sense that their marginal distributions are respectively γ(p1 ,λ) and
γ(p1 ,λi ) . Their cdf and pdf are given by (1) and (2) respectively, and have a link with multivariate
gamma distribution associated with (P, λ) and multi-factor gamma distribution associated with (P, Λ)
respectively.

5

The bidimensional and tridimensional cases

For the bidimensional case, Theorem (15) and Theorem (16) give the following corollary
−λ

Corollary 17 For the bivariate gamma distribution such that Lγ(P,λ) (θ) = (P (θ))
1

, we have

1

CLγ(P,λ) (v1 , v2 ) = v1 v2 [1 − r12 (1 − v1λ )(1 − v2λ )]−λ ,
−1
where, for X = (X1 , X2 ) with pd γ(P,λ) , r12 = −P (−p−1
1 , −p2 ) = 1 − p12 / (p1 p2 ) is the linear correlation

coeﬃcient of the random variables X1 , X2 ; it checks 0 6 r12 6 1. It is the copula of the BB10 family
p. 154 in the Joe’s book (Joe 1997). This result is due to the following equalities : E (Xi ) = λpi,
V ar (Xi ) = λp2i , Cov(X1 , X2 ) = λ (−p12 + p1 p2 ) obtained by derivating Lγ(P,λ) (θ) at 0.
Q
For the bivariate multi-factor gamma distribution such that Lγ(P,Λ) (θ) = (P (θ))−λ 2i=1 (1 + pi θi )−(λi −λ)

, we have a more general family

1

1

CLγ(P,Λ) (v1 , v2 ) = v1 v2 [1 − r12 (1 − v1λ1 )(1 − v2λ2 )]−λ ,

(22)

where, for X = (X1 , X2 ) = (Y1 , Y2 ) + (Z1 , Z2 ) = Y + Z as deﬁned in Proposition 1 with pd γ(P,Λ) , and
0 6 r12 = −P (−p1−1 , −p2−1 ) = 1 − p12 / (p1 p2 ) 6 1, is the linear correlation coeﬃcient of the random
variables Y1 , Y2 in the bivariate gamma distribution such that LY (θ) = [P (θ)]

−λ

.

This bivariate family is not given in the Joe’s books (Joe, 1997, 2014). Bekrizadeh et al., 2012 give
a similar formula for λ being a negative integer.
9

We recall the formulas (Joe, 2014) for the computation of τ, the Kendall’s tau, and ρS , the Spearmann’s rho
τ =1−4
ZZ
ρS = 12

ZZ

[0,1]

2

∂C
∂C
(u, v)
(u, v) dudv,
∂u
∂v

(23)

C (u, v) dudv − 3.

(24)

[0,1]2

Then, we can give the following result
Proposition 18 The Kendall’s tau and the Spearman’s rho of the Laplace copula of the bivariate multifactor gamma distribution are given by the following formulas
τ = 1 − F23 (2λ, 1, 1; 2λ1 + 1, 2λ2 + 1; r12 )
4λ
r12 F23 (2λ + 1, 1, 2; 2λ1 + 2, 2λ2 + 2; r12 )
(2λ1 + 1) (2λ2 + 1)
λ2
−
r2 F 3 (2λ + 2, 2, 2; 2λ1 + 3, 2λ2 + 3; r12 ) ,
(2λ1 + 1) (2λ2 + 1) (λ1 + 1) (λ2 + 1) 12 2
+

(25)

and


ρS = 3 F23 (1, 1, λ; 2λ1 + 1, 2λ2 + 1; r12 ) − 1
=

(26)

3λ
r12 F23 (1, 2, λ + 1; 2λ1 + 2, 2λ2 + 2; r12 ).
(2λ1 + 1) (2λ2 + 1)

(27)

For the tridimensional case, Theorem (15) and Theorem (16) give the following corollary
−λ

Corollary 19 For the trivariate gamma distribution γ(P,λ) such that Lγ(P,λ) (θ) = (P (θ))
1

1

1

, we have

1

CLγ(P,λ) (v1 , v2 , v3 ) = v1 v2 v3 [1 − r12 (1 − v1λ )(1 − v2λ ) − r13 (1 − v1λ )(1 − v3λ )
1

1

1

1

1

− r23 (1 − v2λ )(1 − v3λ ) + 2r123 (1 − v1λ )(1 − v2λ )(1 − v3λ )]−λ ,

where, for X = (X1 , X2 , X3 ) with pd γ(P,λ) , we denote by rij = −P (− p1 1{i,j} ) = 1 − pij /(pi pj ), 1 6 i 6=
j 6 3, the linear correlation coeﬃcient of the random variables Xi , Xj , they check 0 6 rij 6 1, and we
denote by r123 the number deﬁned by
r123 = E{

3
Y

[Xi − E (Xi )]}/

3
Y
3
(E{[Xi − E (Xi )] })1/3
i=1

i=1

−1
−1
= − 21 P (−p−1
1 , −p2 , −p3 )

(to compare with r12 = E{

Q2

i=1

[Xi − E (Xi )]}/

Q2

i=1 (E{[Xi

2

− E (Xi )] })1/2 ).

This last result is due to the following equalities : E (Xi ) = λpi, E{[Xi − E (Xi )]3 } = 2λp3i ,
Q3
−1
−1
E{ i=1 [Xi − E (Xi )]} = −λp1 p2 p3 P (−p−1
1 , −p2 , −p3 ) obtained by derivating Lγ(P,λ) (θ) at 0.
10

(28)

−λ

For the trivariate multi-factor gamma distribution such that Lγ(P,Λ) (θ) = [P (θ)]
(λi > λ, i = 1, 2, 3 ), we have
1

1

Q3

−(λi −λ)

i=1

1

(1 + pi θi )

1

CLγ(P,Λ) (v1 , v2 , v3 ) = v1 v2 v3 [1 − r12 (1 − v1λ1 )(1 − v2λ2 ) − r13 (1 − v1λ1 )(1 − v3λ3 )
1

1

1

1

1

− r23 (1 − v2λ2 )(1 − v3λ3 ) + 2r123 (1 − v1λ1 )(1 − v2λ2 )(1 − v3λ3 )]−λ ,
where, for X = (X1 , X2 , X3 ) = (Y1 , Y2 , Y3 ) + (Z1 , Z2 , Z3 ) = Y + Z as deﬁned in Proposition 1 with pd
pij
pi pj ,

γ(P,Λ) , we denote by rij = −P (− p1 1{i,j} ) = 1 −

1 6 i 6= j 6 3, the linear correlation coeﬃcient
−λ

of the random variables Yi , Yj in the trivariate gamma distribution such that LY (θ) = (P (θ))

, they

check 0 6 rij 6 1, and r123 is deﬁned by equality (28).

6
6.1

Appendix
Proof of Theorem 3

Let X = (X1 , X2 ) and Y = (Y1 , Y2 ) be independent random variables. The pdf of Z = (X1 + Y1 , X2 + Y2 )
where Xi has pd γ (pi , λi − λ) , i = 1, 2 and Y has pd γ (P, λ), is obtained by convolution. By changing
−2
−1
variables 1 − vi /zi = ui , i = 1, 2, and with the notation c = b12 (Pe ) = −p−1
12 + p1 p2 p12 , di = cp12 pi ,
2

2

i = 1, 2 we obtain the pdf fZ of Z by fZ (z1 , z2 ) = 0 if (z1 , z2 ) ∈
/ (0, ∞) and for (z1 , z2 ) ∈ (0, ∞)
Z

z1

Z

z2

1

−



p2
p12

p

v1 + p 1 v2



λ−1

(v1 v2 )
F1 (λ; cv1 v2 )
2
[Γ (λ)] pλ12
z −v
λ −λ − z2p−v2 
λ −λ
− 1 1 
2
e p1
z1 − v1 1
z2 − v2 2
dv2
e
dv1
×
Γ (λ1 − λ)
p1
Γ (λ2 − λ)
p2
z1 − v1 z2 − v2

fZ (z1 , z2 ) =

0

e

12

0



p2

p1



−
z +
z
X (cz1 z2 )k
z1λ1 −1 z2λ2 −1 e p12 1 p12 2
=
(Γ (λ)) Γ (λ1 − λ) Γ (λ2 − λ) pλ12 p1λ1 −λ p2λ2 −λ k>0 Γ (λ + k) k!
Z 1
Z 1
λ+k−1
λ+k−1
×
ed1 z1 u1 u1λ1 −λ−1 (1 − u1 )
du1
ed2 z2 u2 uλ2 2 −λ−1 (1 − u2 )
du2 .
0

(29)

0

As we have, with the notation B (α, β) =
the Euler’s Beta function,
Z

1

R1
0

β−1

eδu uα−1 (1 − u)

β−1

uα−1 (1 − u)

du = B (α, β)

0

11

du = Γ (α) Γ (β) /Γ (α + β), α, β > 0 for

∞
X

(α)n δ n
,
(α + β)n n!
n=0

we obtain
fZ (z) =
×

=

−
z1λ1 −1 z2λ2 −1 e



p2
p12

p

z1 + p 1 z2
12



Γ (λ) Γ (λ1 − λ) Γ (λ2 − λ) pλ12 p1λ1 −λ p2λ2 −λ
∞
X

n

X

k>0
∞
X

(cz1 z2 )k Γ (λ1 − λ) Γ (λ + k)
Γ (λ + k) k!
Γ (λ1 + k)

(λ1 − λ)n (d1 z1 ) Γ (λ2 − λ) Γ (λ + k)
(λ2 − λ)m (d2 z2 )m
(λ1 + k)n n!
Γ (λ2 + k)
(λ2 + k)m m!
n=0
m=0


p
p2
z1 + p 1 z2 X
∞
∞ X
∞ X
λ1 −1 λ2 −1 − p12
12
z1
z2
e
Γ (λ1 ) Γ (λ2 ) pλ12 p1λ1 −λ p2λ2 −λ k=0 n=0 m=0

(λ1 − λ)n (λ2 − λ)m (λ)k (d1 z1 )n (d2 z2 )m (cz1 z2 )k
.
(λ1 )k+n (λ2 )k+m
n!
m!
k!

Hence, we have proved the formula (11).

6.2

Proof of Theorem 7

We start from the Laplace transform of the trivariate gamma distribution γ(λ,P ) associated to (λ, P ).
First, we write

Lγ(λ,P ) (θ) = (1 + p1 θ1 + p2 θ2 + p3 θ3 + p12 θ1 θ2 + p13 θ1 θ3 + p23 θ2 θ3 + p123 θ1 θ2 θ3 )−λ

−λ
p1 + p13 θ3
p2 + p23 θ3
p12 + p123 θ3
−λ
1+
= (1 + p3 θ3 )
.
θ1 +
θ2 +
θ1 θ2
1 + p 3 θ3
1 + p 3 θ3
1 + p 3 θ3
Let Q (θ1 , θ2 ) = 1+q1 θ1 +q2 θ2 +q12 θ1 θ2 where q1 = (p1 + p13 θ3 ) / (1 + p3 θ3 ) , q2 = (p2 + p23 θ3 ) / (1 + p3 θ3 ) ,
and q12 = (p12 + p123 θ3 ) / (1 + p3 θ3 ). With these notations, we have
Lγ(λ,P ) (θ) = (1 + p3 θ3 )

−λ

−λ

(1 + q1 θ1 + q2 θ2 + q12 θ1 θ2 )

.

p1 pe2 pe3 , then we have
Let peT = −pT /p123 ,ebij = peij + pei pej , and eb123 = pe123 + pe1 pe23 + pe2 pe13 + pe3 pe12 + 2e
e = −q12 + q1 q2
b1,2 (Q)
2
q12
eb123
eb13eb23
.
= eb12 +
+
(θ3 − pe3 ) (θ3 − pe3 )2

Let γ(λ,Q) be the bivariate gamma distribution associated to (λ, Q), its Laplace transform is
(1 + q1 θ1 + q2 θ2 + q12 θ2 θ3 )−λ . We have
γ(λ,Q) (dx) =

−λ
q12

[Γ (λ)]

q

2e

q

− q 2 x1 − q 1 x2
12

12

(x1 x2 )

λ−1

e 1 x2 ]1
F1 [λ, b1,2 (Q)x
(0,∞)2 (x1 , x2 ) dx1 dx2 .

Second, we are looking for the unidimensional distribution for which its Laplace transform with respect

12

to the variable θ3 is equal to
(1 + p3 θ3 )−λ
p−λ
123

=

[Γ (λ)]

−λ
q12

q

(Γ (λ))

2e

q

− q 2 x1 − q 1 x2
12

12

e 1 x2 ]1
(x1 x2 )λ−1 F1 [λ, b1,2 (Q)x
(0,∞)2 (x1 , x2 ) dx1 dx2

λ−1

2

exp (e
p1 x1 + pe2 x2 ) (x1 x2 )

× F1 {λ, [eb12 +

(θ3 − pe3 )

−λ

exp(

eb13 x1 + eb23 x2
)
θ3 − pe3

eb13eb23
eb123
2
+
2 ]x1 x2 }1(0,∞) (x1 , x2 ) dx1 dx2 .
θ3 − pe3
(θ3 − pe3 )

(30)

From equality (30) and (Hladik, 1969), we obtain the unidimensional distribution for which its Laplace
transform is

p−λ
123
[Γ (λ)]

2

exp (e
p1 x1 + pe2 x2 + pe3 x3 ) (x1 x2 )

× F1 [λ, (

λ−1

θ3−λ exp(

eb13 x1 + eb23 x2
)
θ3

eb13eb23 eb123
+
+ eb12 )x1 x2 ]1(0,∞)2 (x1 , x2 ) dx1 dx2 .
θ32
θ3

(31)

Third, we are looking for the unidimensional distribution for which its Laplace transform with respect
to the variable θ3 is equal to
eb13eb23 eb123
eb13 x1 + eb23 x2
)F1 [λ, ( 2 +
+ eb12 )x1 x2 ]
θ3
θ3
θ3
∞
X
eb13 x1 + eb23 x2
X
1
1 eℓ eℓ em en −(λ+2ℓ+m)
exp(
=
b13 b23 b123 b12 θ3
)
(x1 x2 )k
(λ)k
ℓ!m!n!
θ3

θ3−λ exp(

k=0

(32)

ℓ+m+n=k

From (Hadlik,1986), we have the following equality

L[Γ(λ)]−1 tλ−1 F1 (λ,at)1(0,∞) (t) dt (s) = s−λ exp

a
s

(33)

We utilize the equality (33) in (32) and we obtain the following unidimensional distribution
∞
X
X
1 eℓ eℓ em en
1
−1
k
b b b b [Γ (λ + 2ℓ + m)] x3λ+2ℓ+m−1
(x1 x2 )
(λ)k
ℓ!m!n! 13 23 123 12
ℓ+m+n=k
k=0
h
i
× F1 λ + 2ℓ + m, (eb13 x1 + eb23 x2 )x3 1(0,∞) (x3 ) dx3

1(0,∞) (x3 )
= [Γ (λ)]−1 xλ−1
3
X
1
×
(eb13 x1 x3eb23 x2 x3 )ℓ (eb123 x1 x2 x3 )m (eb12 x1 x2 )n
(λ)
(λ)
ℓ!m!n!
(ℓ+m+n)
(2ℓ+m)
3
(ℓ,m,n)∈N

× F1 (λ + 2ℓ + m, eb13 x1 x3 + eb23 x2 x3 ) dx3

1(0,∞) (x3 )
= [Γ (λ)] xλ−1
3
X
1
×
(λ)
(λ)
(ℓ+m+n)
(2ℓ+m+p) ℓ!m!n!p!
4
−1

(ℓ,m,n,p)∈N

× (eb13 x1 x3eb23 x2 x3 )ℓ (eb123 x1 x2 x3 )m (eb12 x1 x2 )n (eb13 x1 x3 + eb23 x2 x3 )p dx3
−1

= [Γ (λ)]

xλ−1
FII (λ, λ, eb13 x1 x3eb23 x2 x3 , eb123 x1 x2 x3 , eb12 x1 x2 , eb13 x1 x3 + eb23 x2 x3 )1(0,∞) (x3 ) dx3 (34)
3
13

We carry this last equality (34) in (31) and we obtain the pd (15).

6.3

Proof of Theorem 15
−λ

We utilize the following notations, for i ∈ [n] , Lγ(p ,λ) (θi ) = (1 + pi θi ) = vi , 0 < λi < λ and 1 + pi θi =
i

−1/λ
−1/λ
. We remark that ui = 0 ⇔ θi =
− 1 /pi and ui = vi
ui . In particular θi = (ui − 1) /pi = vi
−1/pi .

We start from the Laplace transform of the multivariate gamma distribution associated to (P, λ).
From the deﬁnition of γ(P,λ) , we obtain
1

[Lγ(P,λ) (θ)]− λ =

X

pT θ T .

T ⊂[n]

Hence, we have
[Lγ(P,λ) (θ)]

1
−λ

X

=

pT u − 1[n]

T ⊂[n]

=

X

T

 T
1
p

qT uT = Q (u)

T ⊂[n]

where u = (u1 , . . . , un ) , and

X

Q (u) =

αT u − 1[n]

T ⊂[n]

indeed, the polynomials u − 1[n]

T

T

uT ,

uT , T ⊂ [n] form a basis of the vector space of aﬃne polynomials of

degree less or equal to n. Because ui = 0 ⇔ θi = − p1i , for i ∈ [n] , we have
1
Q (0) = P (− 1[n] ) = (−1)n α[n] ,
p
consequently
1
n
α[n] = (−1) P (− 1[n] ).
p
Since P is an aﬃne polynomial, we have
|T |

αT = (−1)

1
P (− 1T ),
p

in particular αT = 0, if |T | = 1 and α∅ = 1. Finally we obtain
P (θ) =

X

|T |

(−1)

T ⊂[n]

= u[n] +

T
1
P (− 1T ) u − 1[n] uT
p

X

T ⊂[n],|T |>1

T
1
(−1)|T | P (− 1T ) u − 1[n] uT
p

14

(35)

We deduce from equality (35)
Lγ(P,λ) (θ) = [

X

|T |

(−1)

T ⊂[n]

T
1
P (− 1T ) u − 1[n] uT ]−λ
p

−λ X

1
1
(−1)|T | P (− 1T )(1[n] − )T ]−λ
[
= u[n]
p
u
T ⊂[n]

X

= v[n] [1 +

|T |

(−1)

Y
1
1
P (− 1T )
(1 − vtλ )]−λ
p
t∈T

T ⊂[n],|T |>1

= CLγ(P,λ) (v) .

6.4

Proof of Theorem 16

We start from the Laplace transform of the multivariate multi-factor gamma distribution associated
to (P, Λ) , Λ = (λ, λ1 , . . . λn ), where P is an aﬃne polynomial with respect to θ = (θ1 , . . . , θn ) and
λi > λ, for i ∈ [n] . We use the following notations ui = (1 + pi θi )
−(λi −λ)

(1 + pi θi )

= vi /ui and (1 + pi θi )

Lγ(P,Λ) (θ) = [P (θ)]

−λ

−1

Y

1/λ

= ui

1/λi

= vi

(1 + pi θi )

−λ

and vi = (1 + pi θi )

−λi

. According to equality 20, we have

−(λi −λ)

i∈[n]

X

= u[n] [1 +

T ⊂[n],|T |>1

X

= v[n] [1 +

Y
Y vi
1
1
(1 − utλ )]−λ
(−1)|T | P (− 1T )
p
ui
t∈T

|T |

(−1)

t∈T

T ⊂[n],|T |>1

= v[n] [1 +

X

i∈[n]

Y
1
1
P (− 1T )
(1 − utλ )]−λ
p

|T |

(−1)

1
Y
1
P (− 1T )
(1 − vtλt )]−λ
p

t∈T

T ⊂[n],|T |>1

= CLγ(P,Λ) (v) .

6.5
6.5.1

Proof of Proposition 18
Kendall’s tau

By injecting the given copula in formula (22) in formula (23), we obtain

1−τ
=
4

ZZ

[0,1]2

1

1

u1 u2 [1 − r12 (1 − u1λ1 )(1 − u2λ2 )]−2λ−2

{1 − r12 [1 − (1 −

1
1
1
λ
λ λ11
)u1 ](1 − u2λ2 )}{1 − r12 [1 − (1 − )u2λ2 ](1 − u1λ1 )} du1 du2 .
λ1
λ2

15

so that

1/λi

By changing the variables ti = 1 − ui
1−τ
=
4λ1 λ2

ZZ

2

, i = 1, 2, we obtain

(1 − r12 t1 t2 )−2λ (1 − t1 )2λ1 −1 (1 − t2 )2λ2 −1

[0,1]

λ
−2λ−1
2λ
2λ −1
(1 − t1 ) 1 t2 (1 − t2 ) 2
− r12 (1 − r12 t1 t2 )
λ1
λ
−2λ−1
2λ −1
2λ
− r12 (1 − r12 t1 t2 )
t1 (1 − t1 ) 1 (1 − t2 ) 2
λ2
λ2 2
+
r (1 − r12 t1 t2 )−2λ−2 t1 (1 − t1 )2λ1 t2 (1 − t2 )2λ2 dt1 dt2 .
λ1 λ2 12
By using equality (1 − r12 t1 t2 )

−2λ

=

∞

P∞

k=0

(2λ)k

k k k
t1 t2 )
(r12

k!

in the last result, we obtain,

X
1−τ
rk
=
(2λ)k 12 B (k + 1, 2λ1 ) B (k + 1, 2λ2 )
4λ1 λ2
k!
k=0

∞

−
−

X
rk
λ
r12
(2λ + 1)k 12 B (k + 1, 2λ1 + 1) B (k + 2, 2λ2 )
λ1
k!
λ
r12
λ2

k=0
∞
X

(2λ + 1)k

k=0
∞

k
r12
B (k + 2, 2λ1 ) B (k + 1, 2λ2 + 1)
k!

λ2 2 X
rk
+
r12
(2λ + 2)k 12 B (k + 2, 2λ1 + 1) B (k + 2, 2λ2 + 1) .
λ1 λ2
k!
k=0

With the equality B (α, β) = Γ (α) Γ (β) /Γ (α + β) for α, β > 0, we obtain
1−τ =

∞
X

k=0

−

k
(1)k (1)k (2λ)k
r12
(2λ1 + 1)k (2λ2 + 1)k k!
∞

X (1) (2) (2λ + 1) rk
4λ
12
k
k
k
r12
(2λ1 + 1) (2λ2 + 1)
(2λ1 + 2)k (2λ2 + 2)k k!
k=0

∞

X (2) (2) (2λ + 2) rk
λ
12
2
k
k
k
r12
.
(2λ1 + 1) (2λ2 + 1) (λ1 + 1) (λ2 + 1)
(2λ1 + 3)k (2λ2 + 3)k k!
2

+

k=0

Finally, by using equality (6), we have proved equality (25).
6.5.2

Spearman’s rho

By injecting the given copula in formula (22) in formula ((24), we obtain

̺S = 12

Z

0

1

Z

0

1

1

1

u1 u2 [1 − r12 (1 − u1λ1 )(1 − u2λ2 )]−λ du1 du2 − 3.

16

(36)

1
λ

By changing the variables vi = ui i , i = 1, 2, we obtain
Z

1
0

Z

1

0



1
1
λ1
u1 u2 [1 − r12 (1 − u1 ) (1 − u2λ2 )]−λ du1 du2
=

Z

1

0

Z

= λ1 λ2
= λ1 λ2

1

λ1 λ2 v12λ1 −1 v22λ2 −1 [1 − r12 (1 − v1 ) (1 − v2 )]

0
∞
X
k=0
∞
X

k=0
∞

(λ)k k
r
k! 12

Z

0

1

k

v12λ1 −1 (1 − v1 ) dv1

Z

1
0

−λ

dv1 dv2
k

v22λ2 −1 (1 − v2 ) dv2

(λ)k k
r B (2λ1 , k + 1) B (2λ2 , k + 1)
k! 12
2

k
(λ)k (1)k
r12
1X
=
4
(2λ1 + 1)k (2λ2 + 1)k k!
k=0

1
= F23 (1, 1, λ; 2λ1 + 1, 2λ2 + 1; r12 )
4

(37)

By injecting equality (37) in equality (36), we obtain equality (26). We remark that we have
"∞
#
X
(λ
+
1)
λ
k
F23 (λ, 1, 1; 2λ1 + 1, 2λ2 + 1; r12 ) − 1 =
r12
rk .
(k + 1)!
(2λ1 + 1) (2λ2 + 1)
(2λ1 + 2)k (2λ2 + 2)k 12
k=0

This gives equality (27).

7

Acknowledgement

I thank Gérard Letac for many helpful conversations.

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