Annals of Studies in Science and Humanities
Volume 1 Number 2 (2015) : 60–74
http://journal.carsu.edu.ph/
Online ISSN 02408-3631
On the Distribution of the Sums, Products and Quotient of
Lomax Distributed Random Variables Based on FGM
Copula
Milburn Macalos1,∗ and Jayrold P. Arcede2
1
2
Department of Mathematics, Caraga State University, Butuan City, Philippines
Department of Mathematics, Saint Joseph Institute of Technology, Butuan City, Philippines
Received: September 15, 2014
Accepted: September 23, 2015
ABSTRACT
In this article, a Lomax distribution based on Farlie-Gumbel-Morgenstern copula is introduced. Derivations of exact distribution R = X + Y , V = XY and Z = X/(X + Y ) are
obtained in closed form. Corresponding moment properties of these distributions are also
derived. The expressions turn out to involve known special functions.
Keywords: Gauss Hypergeometric function, Lomax distribution, products of random variables, quotient of random variables, sum of random variables.
1
Introduction
Copula from the latin word copulare means to connect or to join (Sklar, 1959). Essentially, copulas’ are functions that join or "couple" multivariate distributions to their
one-dimensional marginal distribution functions (Nelsen, 1999). Its sole purpose is to describe the interdependence of several random variables (Schmidt, 2006). A copula is a
joint distribution function of the uniform marginals (Nelsen, 2003). When marginals are
uniform, they are independent. This implies a flat probability density function and any
deviation will indicate dependency (Hutchinson and Lai, 2009).
∗
Corresponding Author
Email:[email protected]
Annals of Studies in Science and Humanities
Vol. 1 No. 2 2015
To date, there has been growing interest in copula owing to its usefulness and popularity though not exempt of criticism (Mikosh, 2006). A listing of copula can be found in
Hutchinson and Lai (2009), Joe (1997, ch. 5), and Nelsen (2006: 116-119).
In this study, a Farlie-Gumbel-Morgenstern (FGM) copula is considered. Let FX (x)
and FY (y) be the distribution functions of the random variables X and Y , respectively,
and θ, −1 < θ < 1, then the probability density function of the bivariate FGM is given by
fX,Y (x, y) = fX (x)fY (y) [1 + θ (2FX (x) − 1) (2FY (y) − 1)]
(1)
where fX (x) and fY (y) are the pdf’s of random variable X and Y respectively. The
parameter θ is known as the dependence parameter of X and Y
The FGM copula was first proposed by Morgenstern (1956). According to Trivedi and
Zimmer (2007) it is a perturbation of the product copula. It is also attractive due to its
simplicity and tractability. Observe that when θ in (1) equals zero, FGM copula collapses
to independence. However, FGM copula is restrictive in the sense that dependency of
two marginals should be modest in magnitude (Mukherjee et al., 2012). An extensive
applications on FGM with varying marginals can be found in Hutchinson and Lai (2009,
ch. 2).
Nadarajah (2005) similar to their other works (Nadarajah & Espejo, 2006; Nadarajah
& Kotz, 2007) concern on obtaining exact distributions on the sum, product and quotient
of some known bivariate distributions. In this note, a bivariate Lomax distribution based
on FGM copula is introduced. As to our knowledge, there is still no research done with
this marginal.
The paper is organized as follows. Section 2 is devoted on derivations of explicit expressions for the pdfs of R = X + Y, V = XY and Z = X/(X + Y ), resp. while section 3
is devoted in derivation of raw moments of all pdfs obtained in section 2.
The calculations of this paper involve several special functions. This includes the incomplete beta function
∫ x
Bx (a, b) =
0
ta−1 (1 − t)b−1 dt,
and, the Gauss Hypergeometric function
2 F1 (a, b; c; x)
=
∞
∑
(a)k (b)k xk
k=0
(c)k
k!
,
where (e)k = e(e + 1) · · · (e + k − 1) denotes the ascending factorial. The following results which can be found in Nadarajah and Espejo (2006) are needed in the subsequent
discussions.
LEMMA 1 (Nadarajah and Espejo (2006)). For any ρ > α > 0,
∫ ∞
0
sα−1
ds = z α−ρ B(α, ρ − α),
(s + z)ρ
61
z ∈ R.
(2)
Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
where
∫ 1
B(a, b) =
0
xa−1 (1 − x)b−1 dx
for a > 0 and b > 0 is the beta function.
LEMMA 2. For 0 < α < ρ + λ,
∫ ∞
xα−1 (x + y)−ρ (x + z)−λ dx
0
=z
−λ α−ρ
y
(
B(α, ρ + λ − α)2 F1
)
y
.
α, λ; ρ + λ; 1 −
z
(3)
LEMMA 3. For p > 0 and q > 0,
∫ b
a
(x − a)p−1 (b − x)q−1 dx
(
= z(b − a)
2
p+q−1
r
(ac + d) B(p, q)2 F1
)
c(a − b)
p, −r; p + q;
.
ac + d
(4)
Pdfs
Let X and Y be two independent Lomax distributed random variables with probability
density functions (pdf) given by
fX (x; α, θ) =
αθα
;
(x + θ)α+1
x > 0, α > 0, θ > 0
(5)
fY (y; α, θ) =
αθα
;
(y + θ)α+1
y > 0, α > 0, θ > 0,
(6)
and
respectively.
The cumulative distribution functions (cdf) of X and Y are known to be
FX (x; α, θ) = 1 −
and
FY (y; α, θ) = 1 −
(
(
θ )α
;
x+θ
x > 0, α > 0, θ > 0
(7)
θ )α
;
y+θ
y > 0, α > 0, θ > 0,
(8)
respectively.
The following result is the joint pdf derived from FGM copula using Lomax distribution
as marginals. It will be used often in this paper as our random variables follows this joint
density.
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Annals of Studies in Science and Humanities
Vol. 1 No. 2 2015
Theorem 2.1. Let X and Y be random variables that follows Lomax distribution with
pdfs in (5) and (6) and cdfs in (7) and (8), respectively. Then the joint density function
is given by
[
(
( θ )α
αθα
αθα
1
+
ρ
2
−1
fX,Y (x, y; α, θ; ρ) =
(x + θ)α+1 (y + θ)α+1
x+θ
)(
(
θ )α
2
−1
y+θ
)]
(9)
where x, y, α, θ are all positive and |ρ| ≤ 1.
Proof. Plugging-in equations (5)–(6) in the FGM copula, we have
[
(
( θ )α
αθα
αθα
fX,Y (x, y; α, θ; ρ) =
1
+
ρ
2
−1
(x + θ)α+1 (y + θ)α+1
x+θ
)(
(
θ )α
2
−1
y+θ
)]
.
It can be shown that
fX,Y (x, y; α, θ; ρ) ≥ 0.
We are left to show that
∫ ∞ ∫ ∞
fX, Y (x, y; α, θ; ρ) dxdy = 1.
0
0
Now, consider the following
∫ ∞
0
αθα
(x + θ)α+1
Let u = 1 −
Hence,
(
2
θ
x+θ
1−
Thus,
[ (
∫ ∞
0
∫ ∞ ∫ ∞
0
0
)α
θ
x+θ
)α
]
. Then du =
αθα
dx.
(x+θ)α+1
(
αθα
2 1−
(x + θ)α+1
[ (
∫ ∞ ∫ ∞
0
Consequently, we have
0
(
) )]
(
θ
x+θ
)α
If x = 0, then u = 0. As x → ∞, u → 1.
)α )
dx = 1 −
][ (
−1
2
θ
y+θ
∫ 1
2udu = 0.
0
)α
]
− 1 dxdy = 0.
(αθα )2
dxdy = 1.
[(x + θ) (y + θ)]α+1
∫ ∞ ∫ ∞
fX, Y (x, y; α, θ; ρ) dxdy = 1.
0
(
α
αθα
θ
dx
α+1 1 − 2 1 − x + θ
(x + θ)
0
(
(
)α )
∫ ∞
αθα
θ
2
1
−
dx.
=1−
x+θ
(x + θ)α+1
0
αθα
θ
α+1 ρ 2 x + θ
(x + θ)
Also
[
∫ ∞
− 1 dx =
0
63
Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
The following figure illustrates the pdf in (9) for specific values: α = .12, θ = 2, ρ = 0.5.
Fig. 1: Graph of the pdf in (9)
Theorems (2.2)–(2.5) derive the pdfs of R = X + Y , V = XY and W = X/(X + Y )
when X and Y are distributed according to (9). In the subsequent, we assume that α, θ
are positive real numbers and ρ ∈ [−1, 1].
Theorem 2.2. If X and Y are jointly distributed according to (9), then the density function
of V = XY is
[
α 2
fV (v; α, θ; ρ) = (αθ )
(
(1 + ρ)
θ2
B
(α
+
1,
α
+
1)
F
α
+
1,
α
+
1;
2α
+
2;
1
−
1
2
v α+1
v
)
(
4ρθ2α
θ2
+ 2α+1 B (2α + 1, 2α + 1)2 F1 2α + 1, 2α + 1; 4α + 2; 1 −
v
v
−
2ρ
v α+1
(
B (α + 1, 2α + 1)2 F1
θ2
α + 1, α + 1; 3α + 2; 1 −
v
(
)
)
2ρθ2α
θ2
− 2α+1 B (2α + 1, α + 1)2 F1 2α + 1, 2α + 1; 3α + 2; 1 −
v
v
)]
(10)
for 0 < v < ∞.
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Annals of Studies in Science and Humanities
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(
V
Proof. From (9), the joint pdf of (X, Y ) = X, X
(
fX,V
)
can be expressed as
{
)
4ρθ2α
v
1+ρ
+
x, ; α, θ; ρ = (αθα )2 [
]
[
]2α+1
α+1
x
(x + θ)( xv + θ)
(x + θ)( xv + θ)
−
2ρθα
(x + θ)2α+1
(v
x
+θ
)α+1 −
}
2ρθα
(x + θ)α+1
(v
x
+θ
)2α+1 .
By Rohatgi’s well-known result (1976, p. 141), the pdf of V = XY becomes
[
]
fV (v; α, θ; ρ) = (αθα )2 (1 + ρ) A(1, 1) + 4ρθ2α A(2, 2) − 2ρθα A(2, 1) − 2ρθα A(1, 2)
(11)
where
∫ ∞
A(h, k) =
0
xkα (x + θ)−(hα+1) (v + θ · x)−(kα+1) dx,
for h, k ∈ {1, 2}.
Using Lemma (2) we obtain
)
(
θ2
A(h, k) = θ
v
B (kα + 1, hα + 1) 2 F1 kα + 1, kα + 1; (h + k)α + 2; 1 −
.
v
(12)
Applying (12) to the equation (11) will result to (10).
kα−hα −(kα+1)
Figure below illustrate the shape of the pdf in (10) for θ = 2, 4. Each plot contains
three curves corresponding to selected values of α. The effect of the parameters is evident.
Theorem 2.3. If X and Y are jointly distributed according to (9), then the distribution
of W = X
Y is
[
2
fW (w; α, θ; ρ) = (α)
(
(1 + ρ)B (2, 2α) 2 F1 2, α + 1; 2α + 2; 1 − w−1
(
+ 4ρB (2, 4α) 2 F1 2, 2α + 1; 4α + 2; 1 − w−1
(
− 2ρB (2, 3α) 2 F1 2, α + 1; 3α + 2; 1 − w−1
(
− 2ρB (2, 3α) 2 F1 2, 2α + 1; 3α + 2; 1 − w
for 0 < w < ∞.
65
)
)
)
−1
)
(13)
]
.
Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
Fig. 2: Graph of the pdf in (10) with selected values of θ and α.
(
X
Proof. From (9), the joint pdf of (X, Y ) = X, W
(
fX,W
)
can be expressed as
{
)
x
4ρθ2α
1+ρ
x, ; α, θ; ρ = (αθα )2 [
+
]α+1
[
]2α+1
w
(x + θ)( wx + θ)
(x + θ)( wx + θ)
−
2ρθα
(x + θ)2α+1
By Rohatgi’s result, the pdf of W =
X
Y
(x
w
+θ
)α+1 −
2ρθα
(x + θ)α+1
(x
w
+θ
}
)2α+1 .
can be expressed as
[
]
fW (w; α, θ; ρ) = (αθα )2 (1 + ρ) C(1, 1) + 4ρθ2α C(2, 2) − 2ρθα C(2, 1) − 2ρθα C(1, 2)
(14)
where
∫ ∞
wkα+1 x (x + θ)−(hα+1) (x + θ · w)−(kα+1) dx.
C(h, k) =
(15)
0
for h, k ∈ {1, 2}.
Using Lemma (2) one can get
(
)
C(h, k) = θ−(h+k)α B (2, (h + k)α) 2 F1 2, kα + 1; (h + k)α + 2; 1 − w−1 .
By (16), the following terms in (14) are obvious.
(
)
(1) (1 + ρ) C(1, 1) = (1 + ρ)θ−2α B (2, 2α) 2 F1 2, α + 1; 2α + 2; 1 − w−1 ;
(
)
(2) 4ρθ2α C(2, 2) = 4ρθ−2α B (2, 4α) 2 F1 2, 2α + 1; 4α + 2; 1 − w−1 ;
66
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Annals of Studies in Science and Humanities
Vol. 1 No. 2 2015
(
)
(3) −2ρθα C(2, 1) = −2ρθ−2α B (2, 3α) 2 F1 2, α + 1; 3α + 2; 1 − w−1 ;
)
(
(4) −2ρθα C(1, 2) = −2ρθ−2α B (2, 3α) 2 F1 2, 2α + 1; 3α + 2; 1 − w−1 ;
The result follows by using items (1)–(4) in (14).
Theorem 2.4. If X and Y are jointly distributed according to (9), then the distribution
X
of Z = X+Y
is
[
(
2
fZ (z; α, θ; ρ) = α (1 + ρ)B (2, 2α) 2 F1 2, α + 1; 2α + 2;
(
2z − 1
z
)
)
2z − 1
z
(
)
2z − 1
− 2ρB (2, 3α) 2 F1 2, α + 1; 3α + 2;
z
+ 4ρB (2, 4α) 2 F1 2, 2α + 1; 4α + 2;
(
− 2ρB (2, 3α) 2 F1
2z − 1
2, 2α + 1; 3α + 2;
z
(17)
)]
for 0 < z < 1.
(
X
Proof. Consider the transformation: (X, Y ) −→ (R, Z) = X + Y, X+Y
{
α 2
fR,Z (r, z; α, θ; ρ) = (αθ )
)
so that
1+ρ
4ρθ2α
+
[(rz + θ) (r − rz + θ)]2α+1 [(rz + θ) (r − rz + θ)]2α+1
2ρθα
2ρθα
−
−
(rz + θ)2α+1 (r − rz + θ)α+1 (rz + θ)α+1 (r − rz + θ)2α+1
}
Note that the jacobian of transformation is r, thus
{
}
fZ (z; α, θ; ρ) = (αθα )2 (1 + ρ) D (1, 1) + 4ρθ2α D (2, 2) − 2ρθα D (2, 1) − 2ρθα D (1, 2)
(18)
where
∫ ∞
D (h, k) =
0
for h, k ∈ {1, 2}.
Let u = (1 − z)r. Then dr =
∫ ∞
D (h, k) =
0
r (rz + θ)−(hα+1) (r − rz + θ)−(kα+1) dr
1
1−z du.
[
(19)
One can obtain D (h, k) as follows
u
uz
+θ
1−z 1−z
]−(hα+1)
67
[u + θ]−(kα+1)
1
du.
1−z
(20)
Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
Using Lemma (2), we have
(
D (h, k) = θ−(k+h)α B(2, (h + k)α)2 F1 2, kα + 1; (h + k)α + 2;
2z − 1
z
)
(21)
Combining (21) and (18) the result in (17) follows.
The following figure illustrates the pdf in (17) for specific values: ρ = 0.5, α = 2, 4, and
6.
Fig. 3: Graph of the pdf in (17)
Theorem 2.5. If X and Y are jointly distributed according to (9) then the density function
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Annals of Studies in Science and Humanities
Vol. 1 No. 2 2015
of R = X + Y is given by
{
α 2
fR (r; α, θ; ρ) = r (αθ )
(1 + ρ) θ
−(α+1)
−(α+1)
(r + θ)
∞
∑
[(
j=1
)(
α+j−1
j−1
(
2 F1
2α −(2α+1)
+ 4ρθ θ
−(2α+1)
(r + θ)
∞
∑
2α + j − 1
j−1
(
2 F1
− 2ρθ θ
(r + θ)
−(α+1)
∞
∑
[(
j=1
− 2ρθ θ
(r + θ)
−(2α+1)
∞
∑
[(
j=1
for 0 < r < ∞.
)(
fR,Z (r, z; α, θ; ρ) = (αθ )
)]
)j−1
)j−1
−
r
θ
j −1
)]
j −1
)j−1
)]
j −1
r
j, 2α + 1; j + 1;
r+θ
(
{
j −1
r
j, α + 1; j + 1;
r+θ
X
Proof. Consider the transformation: (X, Y ) −→ (R, Z) = X + Y, X+Y
α 2
r
−
θ
r
−
θ
α+j−1
j−1
(
2 F1
)(
)(
2 F1
)j−1
r
j, 2α + 1; j + 1;
r+θ
2α + j − 1
j−1
(
α −(2α+1)
r
θ
r
j, α + 1; j + 1;
r+θ
[(
j=1
α −(α+1)
−
) ]}
)
so that
1+ρ
4ρθ2α
+
[(rz + θ) (r − rz + θ)]2α+1 [(rz + θ) (r − rz + θ)]2α+1
2ρθα
2ρθα
−
−
(rz + θ)2α+1 (r − rz + θ)α+1 (rz + θ)α+1 (r − rz + θ)2α+1
}
The jacobian of transformation is r, thus
{
}
fR (r; α, θ; ρ) = r (αθα )2 (1 + ρ) G (1, 1) + 4ρθ2α G (2, 2) − 2ρθα G (2, 1) − 2ρθα G (1, 2)
(22)
where
∫ 1
G (h, k) =
0
(rz + θ)−(hα+1) (r − rz + θ)−(kα+1) dz
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Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
for h, k ∈ {1, 2}.
Using Lemma (3), one can obtain G (h, k) as follows
G (h, k) = θ
−(hα+1)
∞
∑
[(
j=0
=θ
−(hα+1)
∞
∑
[(
j=1
hα + j
j
)(
r
−
θ
)j ∫ 1
z (−rz + r + θ)
dz
0
)(
hα + j − 1
j−1
]
−(kα+1)
j
−
r
θ
)j−1 ∫ 1
0
(z − 0)j−1 (1 − z)1−1
]
(−rz + r + θ)−(kα+1) dz
=θ
−(hα+1)
∞
∑
[(
j=1
(
2 F1
=θ
)(
hα + j − 1
j−1
r
j, kα + 1; j + 1;
r+θ
−(hα+1)
(r + θ)
−(kα+1)
∞
∑
2 F1
r
j, kα + 1; j + 1;
r+θ
)j−1
(r + θ)−(kα+1) B (j, 1)
(24)
)]
[(
j=1
(
r
−
θ
)(
hα + j − 1
j−1
r
−
θ
)j−1
j −1
)]
Combining (22) and (24), the result follows immediately.
3
Moments
Theorem 3.1. Let X and Y be jointly distributed according to (9). Then the (a, b)-th
product moment of bivariate Lomax density function denoted by µ′ a,b;ρ (X, Y ) is given by
µ′ a,b;ρ (X, Y ) = Γ(a + 1)Γ(b + 1)θa+b
( Γ(2α − a)
+ρ
Γ(2α)
[ Γ(α − a)Γ(α − b)
Γ2 (α)
Γ(α − a) )( Γ(2α − b) Γ(α − b) )]
−
−
Γ(α)
Γ(2α)
Γ(α)
where x, y, α, θ, are all positive, |ρ| ≤ 1 and max{a, b} < α.
70
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Vol. 1 No. 2 2015
Proof. By definition, one can expressed the (a, b)-th moment of fX,Y (x, y; α, θ; ρ) as
∫ ∞∫ ∞
′
µ a,b;ρ (X, Y ) =
0
0
xa y b
[ (∫
αθα
αθα
dxdy
(x + θ)α+1 (y + θ)α+1
∞(
+ρ
0
(
θ
2
y+θ
)α
(∫ ∞ ( (
0
θ
2
x+θ
)
αθα
−1
y b dy
(y + θ)α+1
)α
)
)
)]
αθα
−1
xa dx
(x + θ)α+1
.
By Lemma 1, one can show the following integrals:
(1)
∫ ∞
xa
0
(2)
∫ ∞
yb
0
(3)
∫ ∞
xa
0
(3) Finally,
∫ ∞
yb
0
αθα
dx = αθa B(a + 1, α − a);
(x + θ)α+1
αθα
dy = αθb B(b + 1, α − b);
(y + θ)α+1
αθ2α
dx = αθa B(a + 1, 2α − a);
(x + θ)2α+1
αθ2α
dy = αθb B(b + 1, 2α − b);
(y + θ)2α+1
Then the result follows directly.
Theorem 3.2. If X and Y are jointly distributed according to 9, then the a-th raw moment
of the random variable V is
[
µ′a;ρ (V
(
Γ2 (α − a)
Γ (2α − a) Γ (α − a)
) = θ Γ (a + 1)
+ρ
−
2
Γ (α)
Γ (2α)
Γ (α)
2a 2
)2 ]
.
(26)
Proof. Notice that
E (V a ) = E ((X · Y )a ) = E (X a · Y a ) .
Putting b = a in (25), the result follows.
We state the next result without proof since the proof is similar to that of Theorem
3.2.
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Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
Theorem 3.3. If X and Y are jointly distributed according to (9), then a-th raw moment
of W = X
Y is
[
µ′a;ρ (W )
(
Γ (α − a) Γ (α + a)
Γ (2α − a) Γ (α − a)
= Γ (a + 1) Γ (1 − a)
+ρ
−
2
Γ (α)
Γ (2α)
Γ (α)
(
Γ (2α + a) Γ (α + a)
−
Γ (2α)
Γ (α)
)
)]
(27)
Theorem 3.4. If X and Y are jointly distributed according to (9), then the a-th raw
X
is
moment of Z = X+Y
µ′a,ρ (Z)
=
(
)
∞
∑
a−1+k
(−1)k Γ (a + k + 1) Γ (1 − a − k) ·
k
k=0
[
Γ (α − a − k) Γ (α + a + k)
+
Γ2 (α)
(
)(
Γ (2α − a − k) Γ (α − a − k)
ρ
−
Γ (2α)
Γ (α)
Γ (2α + a + k) Γ (α + a + k)
−
Γ (2α)
Γ (α)
)]
(28)
Proof. Notice that
(
−a
E (Z ) = E X · (X + Y )
a
a
((
=E
=E
X
Y
)
∞
∑
k=0
(
X
Y
)a (
a−1+k
(−1)k
k
k=0
(∞ (
)
∑ a−1+k
k=0
=
=E
(
)a ∑
∞
k
((
)
(−1)k
)
(
X
Y
(
X
+1
Y
X
Y
)−a )
)k )
)a+k )
(
)
a−1+k
(−1)k E W a+k
k
Using Theorem 3.2, the result in (28) follows.
Theorem 3.5. If X and Y are jointly distributed according to (9), then the a-th raw
72
Annals of Studies in Science and Humanities
Vol. 1 No. 2 2015
moment of R = X + Y is
µ′a;ρ (R)
=θ
a
( ){
a
∑
a
i=0
Γ (i + 1) Γ (a − i + 1)
i
[
(
Γ (α − i) Γ (α − (a − i))
Γ (2α − i) Γ (α − i)
+ρ
−
Γ2 (α)
Γ (2α)
Γ (α)
(
Proof. Since Ra = (X + Y )a =
µ′a;ρ (R)
a
= E (R ) =
(a) i
i=0 i X
∑a
( )
a
∑
a
i=0
i
(
Γ (2α − (a − i)) Γ (α − (a − i))
−
Γ (2α)
Γ (α)
)
(29)
) ]}
· Y a−i , then
i
E XY
a−i
)
( )
a
(
)
∑
a ′
=
µi,a−i;ρ X i Y a−i .
i=0
i
By putting a = i and b = a − i in (25), the result follows.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of
this paper.
References
Balakrishnan, N., and Lai, C. D. (2009). Continuous bivariate distributions. New York:
Springer.
Gupta, A. K., and Nadarajah, S. (2006). Sums, products and ratios for Mckays bivariate gamma distribution. Mathematical and computer modelling, 43 (1), 185-193.
Joarder A.H. et al.(2012). A Bivariate Probability Model Based on Type II Gumbel Random Variable and its Properties. Technical Report Series 428. Department
of Mathematical Sciences. King Fahd University of Petroleum & Minerals, Saudi
Arabia.
Mikosch, T. (2006). Copulas: Tales and facts. Extremes, 9 (1), 3-20.
Morgenstern, D. (1956). Einfache Beispiele Zweidimensionaler Verteilungen. Mitteilingsblatt fur Mathematische Statistik. 8, 234-235
Mukherjee, S., Jafari, F., & Kim, J. M. (2012). Characterization of Differentiable
Copulas. arXiv preprint arXiv:1210.2953.
73
Arcede J.P. & Macalos M.O.
Vol. 1 No. 2 2015
Nadarajah, S. (2005). Sums, products, and ratios for the bivariate gumbel distribution.
Mathematical and computer modelling, 42 (5), 499-518.
Nadarajah, S., & Espejo, M. R. (2006). Sums, products, and ratios for the generalized
bivariate Pareto distribution. Kodai Mathematical Journal, 29 (1), 72-83.
Nadarajah, S., & Kotz, S. (2007). On the convolution of Pareto and gamma distributions. Computer Networks, 51 (12), 3650-3654.
Nelsen, R. B. (1999). An introduction to copulas. Springer.
Nelsen, R. B. (2003, September). Properties and applications of copulas: A brief
survey. In Proceedings of the First Brazilian Conference on Statistical Modeling in
Insurance and Finance,(Dhaene, J., Kolev, N., Morettin, PA (Eds.)), University
Press USP: Sao Paulo (pp. 10-28).
Prieger, J. E. (2002). A flexible parametric selection model for nonnormal data with
application to health care usage. Journal of Applied Econometrics, 17(4), 367-392.
Rohatgi, V.K. (1976). An Introduction to Probability Theory Mathematical Statistics.
Wiley, New York.
Sklar, M. (1959). Fonctions de ré partition à n dimensions et leurs marges. Université
Paris 8.
Schmidt, T. (2007). Coping with copulas. Copulas-From Theory to Application in
Finance, 3-34.
Trivedi, P. K., and Zimmer, D. M. (2007). Copula modeling: an introduction for
practitioners. Now Publishers Inc.
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