Applied Mathematical Sciences, Vol. 3, 2009, no. 59, 2911 - 2918
Moment Generating Function of the Bivariate Generalized Exponential Distribution Samir K. Ashour1, Essam A. Amin1 and Hiba Z. Muhammed1
Abstract Recently a new distribution, named a bivariate generalized exponential (BVGE) distribution has been introduced by Kundu and Gupta (2008). In this paper we obtain joint moments and the moment generating function for (BVGE) which is in closed form, and convenient to use in practice. Keywords: bivariate generalized exponential distribution; joint and marginal moments; moment generating function; marginal moment generating function.
1. Introduction The two parameter generalized exponential distribution was introduced by Gupta and Kundu (1999). Gupta and Kundu (2001a) observed that it can be used quite effectively to analyze positive life time data , particularly , in place of the two-parameter gamma or two-parameter Weibull distributions . Since the distribution function of the generalized exponential is in closed form, it can be used quit easily for analyzing censored data also. The frequentest and Bayesian inferences have been developed for the unknown parameters of the generalized exponential distribution. The readers are referred to the review article of Gupta and Kundu (2007) for a current account on generalized exponential distribution. Although quite bit of work has been done in the recent years on generalized exponential distribution, but not much attempt has made to extend this to the multivariate case. Recently Kundu and Gupta (2008) define a bivariate generalized exponential distribution (BVGE) distribution as : The bivariate vector (X 1 , X 2 ) has a bivariate generalized exponential distribution with the shape parameters α 1 , α 2 and α 3 , denoted by BGE (α 1 , α 2 , α 3 ) . if (X 1 , X 2 ) has the joint probability density function 1
Department of mathematical statistics, I.S.S.R., Cairo University, 5 Dr.Ahmed Zweil St. Orman, Giza, Egypt, Zip code 12613,email:
[email protected].
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S. K. Ashour, E. A. Amin and H. Z. Muhammed
⎧f 1 ( x 1 , x 2 ) ⎪ f ( x 1 , x 2 ) = ⎨f 2 ( x 1 , x 2 ) ⎪f ( x ) ⎩ 0
if
0 < x1 < x 2 < ∞
if
0 < x 2 < x1 < ∞
if
0 < x 1 = x 2 = x.
(1.1)
where f 1 ( x 1 , x 2 ) = α 2 (α 1 + α 3 )(1 − e − x1 ) α1 + α3 −1 (1 − e − x 2 ) α 2 −1 e − x1 − x 2 f 2 ( x 1 , x 2 ) = α 1 (α 2 + α 3 )(1 − e − x 2 ) α 2 + α3 −1 (1 − e − x1 ) α1 −1 e − x1 − x 2 f 0 ( x ) = α 3 (1 − e − x ) α1 + α 2 + α 3 −1 e − x
for α 1 , α 2 , α 3 > 0 .Kundu and Gupta (2008) observed that the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms, they discussed several properties of this distribution, and used the EM algorithm to compute the maximum likelihood estimators of the unknown parameters and obtained the observed and expected Fisher information matrices. One data set has been re-analyzed and it is observed that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution. The main aim of this paper is to provide joint and marginal moments of the bivariate generalized exponential distribution , and the joint moment generating function which is in closed form, and convenient to use in practice. The paper is organized as follows. The joint and marginal moments are provided in section 2. In section 3 we introduce the joint moment generating function.
2. Joint and Marginal Moments In this section we drive the rth and sth joint moments of X 1 and X 2 , as well as the marginal moments . Theorem2.1. The rth and sth joint moments of the bivariate generalized exponential distribution, denoted by μ ′r ,s is given by ∞
∞
μ ′r ,s = E(X 1r , X s2 ) = α 2 (α1 + α 3 ) r!∑ ∑ i = 0 j= 0
∞
i = 0 j= 0
∞
+ α3 ∑ ki i =0
Where
( j + 1) r +1 k ′ij
∞
+ α 1 (α 2 + α 3 ) s!∑ ∑
k ij
( j + 1) s +1
Γ(r + s + 1) . (i + 1) r +s +1
[
[
r Γ(s + 1) Γ(s + k + 1) − ] ∑ s +1 s + k +1 (i + 1) k = 0 k!(i + 2)
s Γ(r + 1) Γ(r + k + 1) ] − ∑ r +1 r + k +1 (i + 1) k = 0 k!(i + 2)
r, s = 1,2,3,...
Moment generating function
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⎛ α − 1⎞⎛ α + α 3 − 1⎞ ⎟⎟, k ij = (−1) i + j ⎜⎜ 2 ⎟⎟⎜⎜ 1 j ⎝ i ⎠⎝ ⎠ ⎛ α + α 2 + α 3 − 1⎞ ⎟⎟, k i = (−1) i ⎜⎜ 1 i ⎝ ⎠
⎛ α − 1⎞⎛ α + α 3 − 1⎞ ⎟⎟, k ′ij = (−1) i + j ⎜⎜ 1 ⎟⎟⎜⎜ 2 j ⎝ i ⎠⎝ ⎠
and ∞
Γ( x ) = ∫ u x −1 e − u du ,
x>0
is the complete gamma function.
0
Proof. Starting with ∞∞
E(X 1r , X s2 ) = ∫ ∫ x 1r x s2 f ( x 1 , x 2 )dx 1dx 2
r, s = 1,2,3,...
0 0
and substituting for f ( x 1 , x 2 ) from (1.1) ,we get ∞ x2
E(X , X ) = α 2 (α1 + α 3 ) ∫ ∫ x 1r x s2 (1 − e − x1 ) α1 + α 3 −1 (1 − e − x 2 ) α 2 −1 e − x1 − x 2 dx 1dx 2 r 1
s 2
0 0 ∞ x1
+ α 1 (α 2 + α 3 ) ∫ ∫ x 1r x s2 (1 − e − x 2 ) α 2 + α 3 −1 (1 − e − x1 ) α1 −1 e − x1 − x 2 dx 2 dx 1 0 0 ∞
+ α 3 ∫ x r +s (1 − e − x ) α1 + α 2 + α 3 −1 e − x dx 0
upon using the binomial series expansion ∞ ⎛ α − 1⎞ −ix ⎟⎟ e (1 − e − x ) α −1 = ∑ (−1) i ⎜⎜ i =1 ⎝ i ⎠
and the fact that c −1
zk γ (c, z) = ∫ u e du = Γ(c)[1 − e ∑ ]; c∈N* k = 0 k! 0 where γ (c, z) is an incomplete gamma function, z
c −1 − u
−z
(2.1)
The expression for E(X1r , X s2 ) is derived. Comment(2.1). From theorem 2.1, it easily to obtain different marginal moments of X 1 and X 2 in terms of infinite series by putting r = 0 and s = 0 respectively ,that is the rth marginal moment of X 1 will be
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S. K. Ashour, E. A. Amin and H. Z. Muhammed
∞
∞
μ ′r 0 = E(X 1r ) = α 2 (α1 + α 3 ) r!∑ ∑ i = 0 j= 0
∞
i = 0 j= 0
∞
+ α3 ∑ ki i =0
( j + 1) r +1 k ′ij
∞
+ α 1 (α 2 + α 3 ) ∑ ∑
k ij
[
( j + 1)
Γ(r + 1) . (i + 1) r +1
[
r 1 1 ] −∑ k +1 (i + 1) k = 0 (i + 2)
Γ(r + 1) Γ(r + 1) − ] (i + 1) r +1 (i + 2) r +1
r = 1,2,3,...
Similarly, the sth marginal moment of X 2 is given by ∞ ∞ k ij Γ(s + 1) Γ(s + 1) [ ] μ ′0s = E(X s2 ) = α 2 (α 1 + α 3 ) ∑∑ − (i + 1) s +1 (i + 2) s +1 i = 0 j= 0 ( j + 1) ∞
i = 0 j= 0
∞
+ α3 ∑ ki i =0
k ′ij
∞
+ α 1 (α 2 + α 3 ) s!∑ ∑
( j + 1) s +1
Γ(s + 1) . (i + 1) s +1
[
s 1 1 −∑ ] k +1 (i + 1) k = 0 (i + 2)
s = 1,2,3,...
We observe that the infinite series is summable and it has only a finite number of terms if the shape parameters are integers.
3. Joint Moment Generating Function The following theorem gives the joint moment generating function of ( X 1 , X 2 ). Theorem 3.1 : The moment generating function for the BVGED is given by M (t1 , t 2 ) = α 2 B(1 − t 2 , α 1 + α 2 + α 3 ) × 3
F 2 [1 − t 2 , α 1 + α 3 , t1 ; α 1 + α 3 + 1, α 1 + α 2 + α 3 + 1 − t 2 ;1]
+ α 1 B(1 − t1 , α 1 + α 2 + α 3 ) × 3
F 2 [1 − t1 , α 2 + α 3 , t 2 ; α 2 + α 3 + 1, α 1 + α 2 + α 3 + 1 − t1 ;1]
+ α 3 B(1 − (t1 + t 2 ), α 1 + α 2 + α 3 ), where 1
B(α, β) = ∫ u α −1 (1 − u ) β−1 du , is a beta function, 0
(3.1)
Moment generating function
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∞ (b1 ) ... (b P ) u i i i = , is a hypergeometric ( b ,..., b ; c ,..., c ; u ) ∑ F P 1 q p q 1 ( c ) ... ( c ) q i i! i=0 1 i function, Γ( b + i) and (b) i = b(b + 1)...(b + i − 1) = (b ≠ 0, i = 1,2,...), , and p,q are Γ ( b) nonnegative integers.
Proof: The joint moment generating function of ( X1 , X 2 ) is given by ∞∞
M ( t 1 , t 2 ) = E[e t1x1 + t 2 x 2 ] = ∫ ∫ e t1x1 + t 2 x 2 f ( x 1 , x 2 ) dx 1dx 2 0 0
substituting for f ( x 1 , x 2 ) from (1.1) we get ∞
M (t1 , t 2 ) = α 2 (α 1 + α 3 ) ∫ (1 − e − x2 ) α 2 −1 e − x2 (1−t2 ) 0
∞
+ α 1 (α 2 + α 3 ) ∫ (1 − e − x1 ) α1 −1 e − x1 (1−t1 ) 0
x1
x2
∫ (1 − e
− x1 α1 +α 3 −1
)
e − x1 (1−t1 ) dx1 dx 2
0
∫ (1 − e
− x2 α 2 +α 3 −1
)
e − x2 (1−t2 ) dx 2 dx1
0
∞
+ α 3 ∫ e ( t1 +t2 ) x (1 − e − x ) α1 +α 2 +α 3 −1 dx.
(3.2)
0
Upon using the fact that xα 2 F1 (α,1 − β; α + 1; x ), α 0 where Β x (α, β) is an incomplete beta function and the identity [see Sarhan and Balakrishnan (2007)] x
Β x (α, β) = ∫ u α −1 (1 − u )
1
β −1 α −1 ∫ u (1 − u )
β −1
du =
(0 ≤ x ≤ 1)
F (c, d; ρ; u )du = Β(α, β) 3 F2 (α, c, d; ρ, α + β;1)
2 1
0
for α, β > 0 and d + β − α − c > 0, we can drive the expression for M ( t 1 , t 2 ) given in (3.1) . To show that M (0,0) = 1 ,set t 1 = t 2 = 0 in (3.1) Since i = 1,2 3 F 2 [1, α i + α 3 ,0; α i + α 3 + 1, α 1 + α 2 + α 3 + 1;1] = 1 , then M (0,0) = (α1 + α 2 + α 3 ) ⋅ B[1, α1 + α 2 + α 3 ] = 1 .
Another form for the moment generating function can also be obtained in the form of a series which is finite or infinite depending on whether the shape parameters is integers or not . By using the binomial series expansion ,(3.2) can be written as
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S. K. Ashour, E. A. Amin and H. Z. Muhammed
∞ ∞ ⎛ α − 1⎞ M ( t 1 , t 2 ) = α 2 (α 1 + α 3 ) ∫ ∑ (−1) i ⎜⎜ 2 ⎟⎟ e − x 2 (1− t 2 +i ) 0 i =0 ⎝ i ⎠ x2 ∞ ⎛ α + α 3 − 1⎞ − x1 (1− t1 + j) ⎟⎟ e dx 1dx 2 × ∫ ∑ (−1) j ⎜⎜ 1 j 0 j= 0 ⎠ ⎝ ∞ ∞ ⎛ α − 1⎞ + α 1 (α 2 + α 3 ) ∫ ∑ (−1) i ⎜⎜ 1 ⎟⎟ e − x1 (1− t1 +i ) 0 i =0 ⎝ i ⎠
⎛ α + α 3 − 1⎞ − x 2 (1− t 2 + j) ⎟⎟ e dx 2 dx 1 × ∫ ∑ (−1) j ⎜⎜ 2 j j = 0 0 ⎠ ⎝ ∞ ⎛ α + α 2 + α 3 − 1⎞ − x[1−( t1 + t 2 )] ⎟⎟ e dx + α 3 ∑ (−1) i ⎜⎜ 1 i i =0 ⎠ ⎝ x1 ∞
Since the quantity inside the summation is absolutely integrable, interchanging the summation and integration, and by using (2.1) we have ∞
∞
M (t1 , t 2 ) = α 2 (α 1 + α 3 )∑∑ k ij i =0 j =0
∞
∞
+ α 1 (α 2 + α 3 )∑∑ k ij′ i =0 j =0
∞
+ α 3 ∑ ki i =0
1 1 1 − [ ] 1 − t1 + j 1 − t 2 + i 2 − t 2 + i 1 1 1 − [ ] 1 − t 2 + j 1 − t1 + i 2 − t1 + i
1 1 − t1 − t 2 + i
(3.3) ( t 1 < 1 , t 2 < 1)
where ⎛ α − 1⎞⎛ α + α 3 − 1⎞ ⎟⎟, k ij = (−1) i + j ⎜⎜ 2 ⎟⎟⎜⎜ 1 j ⎝ i ⎠⎝ ⎠ ⎛ α + α 2 + α 3 − 1⎞ ⎟⎟, and k i = (−1) i ⎜⎜ 1 i ⎠ ⎝
⎛ α − 1⎞⎛ α + α 3 − 1⎞ ⎟⎟, k ′ij = (−1) i + j ⎜⎜ 1 ⎟⎟⎜⎜ 2 j ⎝ i ⎠⎝ ⎠
We observe that the infinite series is summable , differentiable and it has only a finite number of terms if the shape parameters are integers. Hence we can obtain different joint moments by differentiating and evaluating at t 1 = t 2 = 0 .
Comment 3.1. From equation (3.3), it easy to obtain the marginal moment generating functions of X 1 and X 2 in terms of infinite series by putting t 2 = 0 and t 1 = 0 respectively, that is the marginal moment generating function of X 1 will be
Moment generating function
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∞
∞
M ( t 1 ,0) = α 2 (α 1 + α 3 )∑ ∑ k ij i = 0 j= 0 ∞
1 1 1 [ − ] 1 − t1 + j 1 + i 2 + i
∞
1 1 1 [ − ] 1 + j 1 − t1 + i 2 − t1 + i
+ α 1 (α 2 + α 3 )∑ ∑ k ′ij i = 0 j= 0
∞
+ α3 ∑ ki i =0
1 1 − t1 + i
Similarly, the marginal moment generating function of X 2 is given by ∞
∞
M (0, t 2 ) = α 2 (α 1 + α 3 )∑∑ k ij i =0 j =0 ∞
1 1 1 − [ ] 1 + j 1 − t2 + i 2 − t2 + i
∞
+ α 1 (α 2 + α 3 )∑∑ k ij′ i =0 j =0
∞
+ α 3 ∑ ki i =0
1 1 1 − [ ] 1 − t2 + j 1 + i 2 + i
1 . 1 − t2 + i
References [1] A., Sarhan, and N., Balakrishnan, (2007). A new class of bivariate distribution and its mixture. Journal of the Multivariate Analysis, vol. 98, 1508 - 1527. [2] R.D. Gupta, D., Kundu, (1999). Generalized exponential distribution. Austr. NZ J. Statist. 41 (2), 173–188. [3] R.D. Gupta, D., Kundu, (2001a). Exponentiated exponential Family: an alternative to gamma and Weibull distributions , Biometrical Journal , 43(1),117-130 [4] R.D. Gupta, D., Kundu,., (2007). Generalized exponential distributions: existing results and some recent developments. Journal of Statistical Planning and Inference, vol. 137, 3525 - 3536. [5] R.D. Gupta, D., Kundu, (2008). Bivariate Generalized Exponential Distribution. To appear in Journal of multivariate analysis.
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Received: March, 2009
S. K. Ashour, E. A. Amin and H. Z. Muhammed
