- Email: [email protected]
On a generalized mixture distribution
Applied Mathematics and Computation 169 (2005) 943–952 www.elsevier.com/locate/amc
On a generalized mixture distribution Y. Ben Nakhi, S.L. Kalla
*
Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Abstract In this paper we obtain a new mixture distribution, which generalizes some other mixture distributions established earlier. Some basic functions associated with the probability density function of the mixture distribution, such as kth moments, characteristic function and factorial moments are computed. Further we obtain a three-term recurrence relation for the established mixture distribution. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Mixture distributions; x-Hyper Poisson distribution; Probability density function
1. Introduction A particular mixture distribution stems when some or all parameters of a distribution vary according to some given probability distribution, called the mixing distribution. A well known example is Poisson distribution mixture with gamma mixing distribution leading to negative binomial distribution. Such distributions have been used in a number of applications including accident proneness [5] and entomological field data [5].
*
Corresponding author. E-mail addresses: [email protected] (Y.B. Nakhi), [email protected] (S.L. Kalla).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.09.071
944
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
In a recent paper Ben Nakhi and Kalla [4] have established several mixture distributions, which are obtained by mixing discrete distributions with continuous ones. Ghitany et al. [9] have shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right and obtained a three term recurrence relation. Special functions have been used to define a variety of probability distributions [15,16]; generalized gamma-type [1,11,12], and an inverse gaussian [10], using a generalized form of KobayashiÕs [13] gamma function. Recently, Al-Saqabi and Kalla [2] have treated a probability distribution involving a confluent hypergeometric function of two variables. We consider here a new mixture distribution, which generalizes some of the mixture distributions considered in [4,9]. In Section 2, we define some special functions and give some basic results that will be used in latter sections. Our new mixture distribution is obtained in the third section by mixing a x-Hyper Poisson distribution f(x/k), defined in (3.2), with the new generalized gamma distribution g(k) defined in (3.1). We derive kth moments, characteristic function, factorial moments and three-term recurrence relation for the mixture distribution obtained. Further, in the graphical representations of probability mass function and its distribution function, the role of x is reflected. It is interesting to observe that the results obtained by Ghitany et al. [9] follow as special case of our general distributions considered in this work. Moreover, our new mixture distribution generalizes some of the mixture distributions obtained in [4,5].
2. Definitions and preliminaries Throughout this sequel, we shall use the following definitions, Laplace integrals and recurrence relations: The x-confluent hypergeometric function [19,20] x
1 U1 ða; b; zÞ
¼
1 CðbÞ X Cða þ xkÞ zk ; CðaÞ k¼0 Cðb þ xkÞ k!
jzjh1; xi0; ðb þ xkÞ 6¼ 0; 1; 2; . . .
ð2:1Þ
For x = 1, (2.1) reduces to confluent hypergeometric function [17]. 1 F 1 ða; b; zÞ
¼
1 X ðaÞn zn ; ðbÞn n! n¼0
jzj < 1; a; b > 0; b 6¼ 0; 1; 2; . . . ; ð2:2Þ
where (c)n denotes the PochhammerÕs symbol, i.e. (c)n = c(c + 1) (c + n 1).
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
The x-hypergeometric function [19] 1 x CðcÞ X ðaÞk Cðb þ xkÞ zk ; 2 R1 ða; b; c; zÞ ¼ CðbÞ k¼0 Cðc þ xkÞ k!
x > 0; jzj < 1:
For x = 1, (2.3) reduces to Gauss hypergeometric function [14,17,18], 1 X ðaÞn ðbÞn zn ; jzj < 1; c 6¼ 0; 1; 2; . . . 2 F 1 ða; b; c; zÞ ¼ ðcÞn n! n¼0
945
ð2:3Þ
ð2:4Þ
We introduce x-Appell hypergeometric function of two variables in the following form [3] 1 x CðcÞ CðcÞ X Cðb þ xnÞ Cðb þ xkÞ zn wk ; ðaÞnþk F 2 ða; b; b; c; c; z; wÞ ¼ CðbÞ CðbÞ n;k¼0 Cðc þ xnÞ Cðc þ xkÞ n! k! jzj; jwj < 1:
ð2:5Þ
For x = 1, (2.5) reduces to Appell hypergeometric function of two variables [6,7,17] 1 X ðbÞn zn ðbÞk wk F 2 ða; b; b; c; c; z; wÞ ¼ ðaÞnþk ; jzj; jwj < 1: ð2:6Þ n!ðcÞn k!ðcÞk n;k¼0 The following recurrence relation for x-hypergeometric functions are used in our analysis [8,19,20] x
x
x
axð1 zÞ2 R1 ða þ 1Þ ¼ ðp xaÞ2 R1 ða 1Þ þ ð2xa p þ ðb xaÞzÞ2 R1 ðaÞ; ð2:7Þ where x
2 R1 ðaÞ
x
¼ 2 R1 ða; b; p; zÞ:
We also make use of the following Laplace integrals [19,20]: Z 1 x CðcÞ x 1 c 1 sk k e 1 U1 ða; p; kÞdk ¼ c 2 R1 c; a; p; ; Re c; Reðs 1Þ; x > 0: s s 0 ð2:8Þ In similar way it can be shown that, Z 1 x x kc 1 e sk 1 U1 ða; b; kÞ1 U1 ða; b; kÞdk 0 CðcÞ x 1 1 ¼ c F 2 c; a; a; b; b; ; ; Re c; Reðs 2Þ > 0: s s s
ð2:9Þ
Moreover, we use in our analysis the differentiation formula for x-confluent hypergeometric functions
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Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
dr x CðbÞ Cða þ xrÞ x 1 U1 ða; b; zÞ ¼ 1 U1 ða þ xr; b þ xr; zÞ: r CðaÞ Cðb þ xrÞ dz
ð2:10Þ
We say X has x-Hyper Poisson distribution, with parameters b, q, k, if X has a probability mass function (pmf) uðxÞ ¼
kx CðqÞCðb þ xxÞ 1 ; x x! CðbÞCðq þ xxÞ U ðb; q; kÞ 1 1
b; q; k > 0; x ¼ 0; 1; 2; . . . ; ð2:11Þ
which yields for x = 1, the Hyper Poisson distribution, defined in [5], whose (pmf) is given by, hðxÞ ¼
kx ðbÞx 1 ; x! ðqÞx 1 F 1 ðb; q; kÞ
b; q; k > 0; x ¼ 0; 1; 2; . . .
ð2:12Þ
Moreover, letting b = q, we get the usual Poisson distribution whose (pmf) is given by kðxÞ ¼
kx k e ; x!
k > 0; x ¼ 0; 1; 2; . . .
ð2:13Þ
3. The mixture distribution f(x) In this section we consider a more general family of continuous mixture distribution. This distribution is obtained by mixing a Hyper Poisson distribution f(x/k) defined in [5], with a new generalized gamma distribution defined here by relation (3.1). The recent results of Ghitany et al. [9] follow as special cases of our general results derived in this section. We derive some basic functions associated with this density function, namely the kth moment, characteristic function and factorial moments. Furthermore, we establish a three-term recurrence relation for the new mixture distribution. We begin this section by defining the function g(k) as x
gðkÞ ¼
x
kc 1 1 U1 ða; p; kÞ1 U1 ðb; q; kÞ ða þ 2Þc e ðaþ2Þk x ; CðcÞ F ðc; a; b; p; q; 1 ; 1 Þ 2
k > 0:
ð3:1Þ
aþ2 aþ2
This function is non-negative since its sum of non-negative terms, and it satR1 isfies the condition 0 gðkÞdk ¼ 1 by virtue of the result (2.9). Therefore (3.1) represents a continuous distribution of the product of two confluent hypergeometric functions, involving AppellÕs function, which yields for p = c = b = q, the probability density function defined in [9]. Let X has a conditional x-Hyper Poisson distribution (2.11) with parameters b, q, k, that is, X has a conditional probability mass function (pmf)
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
f ðx=kÞ ¼
kx CðqÞCðb þ xxÞ 1 ; x x! CðbÞCðq þ xxÞ U ðb; q; kÞ 1 1
947
b; q; k > 0; x ¼ 0; 1; 2; . . . ; ð3:2Þ
whose characteristic function, for any real t, is given by 1 X CðqÞ 1 Cðb þ xxÞ ðkeit Þx /ðtÞ ¼ E½eitX =K ¼ k ¼ x CðbÞ U ðb; q; kÞ x¼0 Cðq þ xxÞ x! 1 1 x
¼
1 U1 ðb; q; ke x
it
Þ
ð3:3Þ
;
1 U1 ðb; q; kÞ
from which moments of any order could be evaluated by r
r
r
ðrÞ
E½X =K ¼ k ¼ i / ðtÞjt¼0 ¼
x
i r dtd r 1 U1 ðb; q; keit Þjt¼0 x
;
r ¼ 1; 2; . . .
1 U1 ðb; q; kÞ
ð3:4Þ In particular, the mean is x
1
ð1Þ
E½X =K ¼ k ¼ i / ðtÞjt¼0
kðbÞx 1 U1 ðb þ 1; q þ 1; kÞ : ¼ x ðqÞx U ðb; q; kÞ 1
ð3:5Þ
1
From (3.4) and (2.10), we obtain the factorial moments of the x-Hyper Poisson distribution E½X ðX 1Þ ðX r þ 1Þ=K ¼ k x
CðqÞ Cðb þ xrÞ 1 U1 ðb þ r; q þ r; kÞ ¼k : x CðbÞ Cðq þ xrÞ 1 U1 ðb; q; kÞ r
ð3:6Þ
Now we state and prove our first theorem: Theorem 1. The unconditional pmf of X is given by x
1 Þ ðcÞx CðqÞCðb þ xxÞ 2 R1 ða; x þ c; p; aþ2 f ðxÞ ¼ ; x x x!ða þ 2Þ CðbÞCðq þ xxÞ F ðc; a; b; p; q; 1 ; 1 Þ 2 aþ2 aþ2
x ¼ 0; 1; 2; . . . ð3:7Þ
whose characteristic function, for any real t, is given by x
it
1 e F 2 ðc; a; b; p; q; aþ2 ; aþ2 Þ WX ðtÞ ¼ E½e ¼ x ; 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 itX
ð3:8Þ
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its factorial moment is E½X ðX 1Þ ðX r þ 1Þ ¼
ðcÞr CðqÞCðb þ xrÞ ða þ 2Þr CðbÞCðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 x 1 1 F 2 ðc; a; b; p; q; aþ2 ; aþ2 Þ ð3:9Þ
and its rth-moment is given by r CðqÞ X ðcÞr Cðb þ xrÞ E½X r ¼ Sðr; nÞ r CðbÞ n¼0 ða þ 2Þ Cðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 ; x 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 where Sðr; nÞ ¼
n X n ð 1Þk ðn kÞr
k
k¼0
n!
:
Proof. From (3.1) and (3.2) the unconditional pmf of X is Z 1 f ðx=kÞgðkÞdk f ðxÞ ¼ 0 c
ða þ 2Þ CðqÞCðb þ xxÞ 1 ¼ x!CðcÞ CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2 Z 1 x kxþc 1 e ðaþ2Þk 1 U1 ða; p; kÞdk 0 x 1 ðcÞx CðqÞCðb þ xxÞ 2 R1 a; x þ c; p; aþ2 ¼ ; x x!ða þ 2Þ CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2 x ¼ 0; 1; 2; . . . Using (3.3) the characteristic function of X is 2 x 3 it 1 U1 ðb; q; Ke Þ5 WX ðtÞ ¼ E½eitX ¼ E½E½eitX =K
¼ E4 x 1 U1 ðb; q; KÞ x 1 eit F 2 c; a; b; p; q; aþ2 ; aþ2 ¼x : 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2
ð3:10Þ
ð3:11Þ
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
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From (3.5) the factorial moment of X is E½X ðX 1Þ ðX r þ 1Þ ¼ E½E½X ðX 1Þ ðX r þ 1Þ=K
2 3 x r CðqÞ 4K Cðb þ xrÞ 1 U1 ðb þ r; q þ r; KÞ5 E ¼ x CðbÞ Cðq þ xrÞ U ðb; q; KÞ 1
1
ðcÞr CðqÞCðb þ xrÞ ¼ ða þ 2Þr CðbÞCðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 : x 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 Finally, the rth-moment of X is obtained using the relation E½X r ¼
r X
Sðr; nÞE½X ðX 1Þ ðX r þ 1Þ ;
n¼0
which completes the proof.
h
Theorem 2. The distribution of X satisfies a three-term recurrence relation ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x f ðx 1Þ xxða þ 1Þ2 ðq þ ðx 1ÞxÞ2x
ðb þ xxÞx a ðc þ xÞx þ 2ðc þ xÞx p f ðxÞ: þ aþ2 xða þ 1Þðq þ xxÞx
ðx þ 1Þf ðx þ 1Þ ¼
ð3:12Þ 1 Proof. Using the recurrence relation (2.7), for a = x + c, z ¼ aþ2 , we get x x aþ2 ðc þ xÞ2 R1 ðx þ c þ 1Þ ¼ ½p ðc þ xÞx 2 R1 ðx þ c 1Þ ða þ 1Þx
x a ðc þ xÞx þ 2ðc þ xÞx p 2 R1 ðx þ cÞ ; ð3:13Þ þ aþ2
where x x R ðx þ cÞ ¼ R 2 1 2 1 a; x þ c; p;
1 : aþ2
Rewrite f(x), given by (3.7), as x 1 f ðxÞ ¼ vðxÞ2 R1 a; x þ c; p; ; aþ2
x ¼ 0; 1; 2; . . . ;
ð3:14Þ
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Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
where vðxÞ ¼
ðcÞx CðqÞCðb þ xxÞ 1 ; x!ða þ 2Þx CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2
x ¼ 0; 1; 2; . . . ; from which we get cþx ðb þ xxÞx ; ðx þ 1Þða þ 2Þ ðq þ xxÞx ðc þ x 1Þ2 ðb þ ðx 1ÞxÞ2x vðxÞ ¼ vðx 1Þ: 2 ðxÞ2 ða þ 2Þ ðq þ ðx 1ÞxÞ2x
vðx þ 1Þ ¼
ð3:15Þ
Using (3.13)–(3.15), we obtain, for x = 1, 2, . . . x 1 ðx þ 1Þf ðx þ 1Þ ¼ ðx þ 1Þvðx þ 1Þ2 R1 a; x þ c; p; aþ2 x ðx þ 1Þða þ 2Þ ½p ðc þ xÞx vðx þ 1Þ2 R1 ðx þ c 1Þ ¼ ðc þ xÞða þ 1Þx
x a ðc þ xÞx þ þ 2ðc þ xÞx p vðx þ 1Þ2 R1 ðx þ cÞ aþ2 x ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x vðx 1Þ2 R1 ðx þ c 1Þ ðq þ ðx 1ÞxÞ2x xxða þ 1Þða þ 2Þ
x ðb þ xxÞx a ðc þ xÞx þ þ 2ðc þ xÞx p vðxÞ2 R1 ðx þ cÞ ð3:16Þ xða þ 1Þðq þ xxÞx aþ2
¼
ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x f ðx 1Þ xxða þ 1Þ2 ðq þ ðx 1ÞxÞ2x
ðb þ xxÞx a ðc þ xÞx þ þ 2ðc þ xÞx p f ðxÞ; xða þ 1Þðq þ xxÞx aþ2
¼
and the proof is complete.
h
Special cases: (1) if we let x = 1, then we get the first mixture distribution obtained earlier in [4], whereas if we put b = q, then we get the second mixture distribution. (2) if x = 1, b = q and c = a then we get the mixture distribution studied/obtained by Ghitany et al. [9]. Figs. 1 and 2 respectively, show the probability mass function f(x) and its distribution function F(x) for the selected specific values of c, a, b, p, q, a.
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
951
Fig. 1. The probability mass function f(x) when c = 7, a = 3, b = 5, p = 6, q = 2 and a = 9. The symbol * represents f(x) when x = 1, whereas the symbol represents f(x) when x = 4.
Fig. 2. The distribution function F(x) when c = 7, a = 3, b = 5, p = 6, q = 2 and a = 9. The lower graph represents F(x) when x = 1, whereas the upper graph represents F(x) when x = 4.
References [1] A. Al-Zamel, On a generalized gamma-type distribution with s-confluent hypergeometric function, Kuwait J. Sci. Eng. 28 (12) (2000) 175–188. [2] B.N. Al-Saqabi, S.L. Kalla, Unified probability density function involving a confluent hypergeometric functions of two variables, Appl. Math. Comput. 146 (2003) 135–152. [3] A.H. Al-Shammery, S.L. Kalla, An extension of some hypergeometric functions of two variables, Revista de la Acad. Canaria de Ciancias (12) (2000) 175–188. [4] Y. Ben Nakhi, S.L. Kalla, On some mixture distributions, Fract. Calc. Appl. Anal. 7 (iii) (2004). [5] S.K. Bhattacharya, Confluent hypergeometric distributions of discrete and continuous type with applications to accident proneness, Calcutta Stat. Assoc. Bull. 15 (1966) 20–31. [6] A. Erdely (Ed.), Higher Trascendental Functions, vol. 1, McGraw-Hill, New York, 1953. [7] H. Exton, Handbook of Hypergeometric Integrals (Ellis Horwood Ltd.), John Wiley & Sons, New York, 1978. [8] L. Galue, S.L. Kalla, Some recurrence relations for generalized hypergeometric functions of a Gauss type, Fract. Calc. Appl. Anal. 1 (1) (1998) 233–244.
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[9] M.E. Ghitany, S. Al-Awadhi, S.L. Kalla, On hypergeometric generalized negative binomial distribution, Int. J. Math. Math. Sci. 29 (12) (2002) 727–736. [10] A. Ismael, S.L. Kalla, H.G. Khajah, A generalized inverse Gaussian distribution with sconfluent hypergeometric function, Integral Transform Spec. Funct. 12 (2) (2001) 101–114. [11] S.L. Kalla, B.N. Al-Saqabi, H.G. Khajah, A unified form of gamma-type distributions, Appl. Math. Comput. 118 (2–3) (2001) 175–187. [12] S.L. Kalla, B.N. Al-Saqabi, Further results on a unified form of gamma-type distribution, Fract. Calc. Appl. Anal. 4 (1) (2001) 91–100. [13] K. Kobayashi, On generalized gamma functions occurring in diffraction theory, J. Phys. Soc. Jpn. 60 (5) (1991) 1501–1512. [14] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, third ed., Springer-Verlag, Berlin, Heidelberg, New York, 1966. [15] A.M. Mathai, A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Clarendon Press, Oxford, 1993. [16] A.M. Mathai, R.K. Saxena, The H-functions with Applications in Statistics and other Disciplines, Wiley Eastern Ltd., 1978. [17] A.P. Prudnikov, Y. Brichkov, O. MarichevIntegrals and Series, vols. 3,4, Gordon and Breach, London, 1992. [18] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, London and New York, 1966. [19] N. Virchenko, On some generalizations of the functions of hypergeometric type, Fract. Calc. Appl. Anal. 2 (3) (1999) 233–244. [20] N. Virchenko, S.L. Kalla, A. Al-Zamel, Some results on a generalized hypergeometric function, Integral Transform Spec. Funct. 12 (1) (2001) 89–100.
On a generalized mixture distribution Y. Ben Nakhi, S.L. Kalla
*
Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Abstract In this paper we obtain a new mixture distribution, which generalizes some other mixture distributions established earlier. Some basic functions associated with the probability density function of the mixture distribution, such as kth moments, characteristic function and factorial moments are computed. Further we obtain a three-term recurrence relation for the established mixture distribution. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Mixture distributions; x-Hyper Poisson distribution; Probability density function
1. Introduction A particular mixture distribution stems when some or all parameters of a distribution vary according to some given probability distribution, called the mixing distribution. A well known example is Poisson distribution mixture with gamma mixing distribution leading to negative binomial distribution. Such distributions have been used in a number of applications including accident proneness [5] and entomological field data [5].
*
Corresponding author. E-mail addresses: [email protected] (Y.B. Nakhi), [email protected] (S.L. Kalla).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.09.071
944
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
In a recent paper Ben Nakhi and Kalla [4] have established several mixture distributions, which are obtained by mixing discrete distributions with continuous ones. Ghitany et al. [9] have shown that the hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right and obtained a three term recurrence relation. Special functions have been used to define a variety of probability distributions [15,16]; generalized gamma-type [1,11,12], and an inverse gaussian [10], using a generalized form of KobayashiÕs [13] gamma function. Recently, Al-Saqabi and Kalla [2] have treated a probability distribution involving a confluent hypergeometric function of two variables. We consider here a new mixture distribution, which generalizes some of the mixture distributions considered in [4,9]. In Section 2, we define some special functions and give some basic results that will be used in latter sections. Our new mixture distribution is obtained in the third section by mixing a x-Hyper Poisson distribution f(x/k), defined in (3.2), with the new generalized gamma distribution g(k) defined in (3.1). We derive kth moments, characteristic function, factorial moments and three-term recurrence relation for the mixture distribution obtained. Further, in the graphical representations of probability mass function and its distribution function, the role of x is reflected. It is interesting to observe that the results obtained by Ghitany et al. [9] follow as special case of our general distributions considered in this work. Moreover, our new mixture distribution generalizes some of the mixture distributions obtained in [4,5].
2. Definitions and preliminaries Throughout this sequel, we shall use the following definitions, Laplace integrals and recurrence relations: The x-confluent hypergeometric function [19,20] x
1 U1 ða; b; zÞ
¼
1 CðbÞ X Cða þ xkÞ zk ; CðaÞ k¼0 Cðb þ xkÞ k!
jzjh1; xi0; ðb þ xkÞ 6¼ 0; 1; 2; . . .
ð2:1Þ
For x = 1, (2.1) reduces to confluent hypergeometric function [17]. 1 F 1 ða; b; zÞ
¼
1 X ðaÞn zn ; ðbÞn n! n¼0
jzj < 1; a; b > 0; b 6¼ 0; 1; 2; . . . ; ð2:2Þ
where (c)n denotes the PochhammerÕs symbol, i.e. (c)n = c(c + 1) (c + n 1).
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
The x-hypergeometric function [19] 1 x CðcÞ X ðaÞk Cðb þ xkÞ zk ; 2 R1 ða; b; c; zÞ ¼ CðbÞ k¼0 Cðc þ xkÞ k!
x > 0; jzj < 1:
For x = 1, (2.3) reduces to Gauss hypergeometric function [14,17,18], 1 X ðaÞn ðbÞn zn ; jzj < 1; c 6¼ 0; 1; 2; . . . 2 F 1 ða; b; c; zÞ ¼ ðcÞn n! n¼0
945
ð2:3Þ
ð2:4Þ
We introduce x-Appell hypergeometric function of two variables in the following form [3] 1 x CðcÞ CðcÞ X Cðb þ xnÞ Cðb þ xkÞ zn wk ; ðaÞnþk F 2 ða; b; b; c; c; z; wÞ ¼ CðbÞ CðbÞ n;k¼0 Cðc þ xnÞ Cðc þ xkÞ n! k! jzj; jwj < 1:
ð2:5Þ
For x = 1, (2.5) reduces to Appell hypergeometric function of two variables [6,7,17] 1 X ðbÞn zn ðbÞk wk F 2 ða; b; b; c; c; z; wÞ ¼ ðaÞnþk ; jzj; jwj < 1: ð2:6Þ n!ðcÞn k!ðcÞk n;k¼0 The following recurrence relation for x-hypergeometric functions are used in our analysis [8,19,20] x
x
x
axð1 zÞ2 R1 ða þ 1Þ ¼ ðp xaÞ2 R1 ða 1Þ þ ð2xa p þ ðb xaÞzÞ2 R1 ðaÞ; ð2:7Þ where x
2 R1 ðaÞ
x
¼ 2 R1 ða; b; p; zÞ:
We also make use of the following Laplace integrals [19,20]: Z 1 x CðcÞ x 1 c 1 sk k e 1 U1 ða; p; kÞdk ¼ c 2 R1 c; a; p; ; Re c; Reðs 1Þ; x > 0: s s 0 ð2:8Þ In similar way it can be shown that, Z 1 x x kc 1 e sk 1 U1 ða; b; kÞ1 U1 ða; b; kÞdk 0 CðcÞ x 1 1 ¼ c F 2 c; a; a; b; b; ; ; Re c; Reðs 2Þ > 0: s s s
ð2:9Þ
Moreover, we use in our analysis the differentiation formula for x-confluent hypergeometric functions
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dr x CðbÞ Cða þ xrÞ x 1 U1 ða; b; zÞ ¼ 1 U1 ða þ xr; b þ xr; zÞ: r CðaÞ Cðb þ xrÞ dz
ð2:10Þ
We say X has x-Hyper Poisson distribution, with parameters b, q, k, if X has a probability mass function (pmf) uðxÞ ¼
kx CðqÞCðb þ xxÞ 1 ; x x! CðbÞCðq þ xxÞ U ðb; q; kÞ 1 1
b; q; k > 0; x ¼ 0; 1; 2; . . . ; ð2:11Þ
which yields for x = 1, the Hyper Poisson distribution, defined in [5], whose (pmf) is given by, hðxÞ ¼
kx ðbÞx 1 ; x! ðqÞx 1 F 1 ðb; q; kÞ
b; q; k > 0; x ¼ 0; 1; 2; . . .
ð2:12Þ
Moreover, letting b = q, we get the usual Poisson distribution whose (pmf) is given by kðxÞ ¼
kx k e ; x!
k > 0; x ¼ 0; 1; 2; . . .
ð2:13Þ
3. The mixture distribution f(x) In this section we consider a more general family of continuous mixture distribution. This distribution is obtained by mixing a Hyper Poisson distribution f(x/k) defined in [5], with a new generalized gamma distribution defined here by relation (3.1). The recent results of Ghitany et al. [9] follow as special cases of our general results derived in this section. We derive some basic functions associated with this density function, namely the kth moment, characteristic function and factorial moments. Furthermore, we establish a three-term recurrence relation for the new mixture distribution. We begin this section by defining the function g(k) as x
gðkÞ ¼
x
kc 1 1 U1 ða; p; kÞ1 U1 ðb; q; kÞ ða þ 2Þc e ðaþ2Þk x ; CðcÞ F ðc; a; b; p; q; 1 ; 1 Þ 2
k > 0:
ð3:1Þ
aþ2 aþ2
This function is non-negative since its sum of non-negative terms, and it satR1 isfies the condition 0 gðkÞdk ¼ 1 by virtue of the result (2.9). Therefore (3.1) represents a continuous distribution of the product of two confluent hypergeometric functions, involving AppellÕs function, which yields for p = c = b = q, the probability density function defined in [9]. Let X has a conditional x-Hyper Poisson distribution (2.11) with parameters b, q, k, that is, X has a conditional probability mass function (pmf)
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f ðx=kÞ ¼
kx CðqÞCðb þ xxÞ 1 ; x x! CðbÞCðq þ xxÞ U ðb; q; kÞ 1 1
947
b; q; k > 0; x ¼ 0; 1; 2; . . . ; ð3:2Þ
whose characteristic function, for any real t, is given by 1 X CðqÞ 1 Cðb þ xxÞ ðkeit Þx /ðtÞ ¼ E½eitX =K ¼ k ¼ x CðbÞ U ðb; q; kÞ x¼0 Cðq þ xxÞ x! 1 1 x
¼
1 U1 ðb; q; ke x
it
Þ
ð3:3Þ
;
1 U1 ðb; q; kÞ
from which moments of any order could be evaluated by r
r
r
ðrÞ
E½X =K ¼ k ¼ i / ðtÞjt¼0 ¼
x
i r dtd r 1 U1 ðb; q; keit Þjt¼0 x
;
r ¼ 1; 2; . . .
1 U1 ðb; q; kÞ
ð3:4Þ In particular, the mean is x
1
ð1Þ
E½X =K ¼ k ¼ i / ðtÞjt¼0
kðbÞx 1 U1 ðb þ 1; q þ 1; kÞ : ¼ x ðqÞx U ðb; q; kÞ 1
ð3:5Þ
1
From (3.4) and (2.10), we obtain the factorial moments of the x-Hyper Poisson distribution E½X ðX 1Þ ðX r þ 1Þ=K ¼ k x
CðqÞ Cðb þ xrÞ 1 U1 ðb þ r; q þ r; kÞ ¼k : x CðbÞ Cðq þ xrÞ 1 U1 ðb; q; kÞ r
ð3:6Þ
Now we state and prove our first theorem: Theorem 1. The unconditional pmf of X is given by x
1 Þ ðcÞx CðqÞCðb þ xxÞ 2 R1 ða; x þ c; p; aþ2 f ðxÞ ¼ ; x x x!ða þ 2Þ CðbÞCðq þ xxÞ F ðc; a; b; p; q; 1 ; 1 Þ 2 aþ2 aþ2
x ¼ 0; 1; 2; . . . ð3:7Þ
whose characteristic function, for any real t, is given by x
it
1 e F 2 ðc; a; b; p; q; aþ2 ; aþ2 Þ WX ðtÞ ¼ E½e ¼ x ; 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 itX
ð3:8Þ
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its factorial moment is E½X ðX 1Þ ðX r þ 1Þ ¼
ðcÞr CðqÞCðb þ xrÞ ða þ 2Þr CðbÞCðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 x 1 1 F 2 ðc; a; b; p; q; aþ2 ; aþ2 Þ ð3:9Þ
and its rth-moment is given by r CðqÞ X ðcÞr Cðb þ xrÞ E½X r ¼ Sðr; nÞ r CðbÞ n¼0 ða þ 2Þ Cðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 ; x 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 where Sðr; nÞ ¼
n X n ð 1Þk ðn kÞr
k
k¼0
n!
:
Proof. From (3.1) and (3.2) the unconditional pmf of X is Z 1 f ðx=kÞgðkÞdk f ðxÞ ¼ 0 c
ða þ 2Þ CðqÞCðb þ xxÞ 1 ¼ x!CðcÞ CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2 Z 1 x kxþc 1 e ðaþ2Þk 1 U1 ða; p; kÞdk 0 x 1 ðcÞx CðqÞCðb þ xxÞ 2 R1 a; x þ c; p; aþ2 ¼ ; x x!ða þ 2Þ CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2 x ¼ 0; 1; 2; . . . Using (3.3) the characteristic function of X is 2 x 3 it 1 U1 ðb; q; Ke Þ5 WX ðtÞ ¼ E½eitX ¼ E½E½eitX =K
¼ E4 x 1 U1 ðb; q; KÞ x 1 eit F 2 c; a; b; p; q; aþ2 ; aþ2 ¼x : 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2
ð3:10Þ
ð3:11Þ
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From (3.5) the factorial moment of X is E½X ðX 1Þ ðX r þ 1Þ ¼ E½E½X ðX 1Þ ðX r þ 1Þ=K
2 3 x r CðqÞ 4K Cðb þ xrÞ 1 U1 ðb þ r; q þ r; KÞ5 E ¼ x CðbÞ Cðq þ xrÞ U ðb; q; KÞ 1
1
ðcÞr CðqÞCðb þ xrÞ ¼ ða þ 2Þr CðbÞCðq þ xrÞ x 1 1 F 2 c þ r; a; b þ r; p; q þ r; aþ2 ; aþ2 : x 1 1 F 2 c; a; b; p; q; aþ2 ; aþ2 Finally, the rth-moment of X is obtained using the relation E½X r ¼
r X
Sðr; nÞE½X ðX 1Þ ðX r þ 1Þ ;
n¼0
which completes the proof.
h
Theorem 2. The distribution of X satisfies a three-term recurrence relation ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x f ðx 1Þ xxða þ 1Þ2 ðq þ ðx 1ÞxÞ2x
ðb þ xxÞx a ðc þ xÞx þ 2ðc þ xÞx p f ðxÞ: þ aþ2 xða þ 1Þðq þ xxÞx
ðx þ 1Þf ðx þ 1Þ ¼
ð3:12Þ 1 Proof. Using the recurrence relation (2.7), for a = x + c, z ¼ aþ2 , we get x x aþ2 ðc þ xÞ2 R1 ðx þ c þ 1Þ ¼ ½p ðc þ xÞx 2 R1 ðx þ c 1Þ ða þ 1Þx
x a ðc þ xÞx þ 2ðc þ xÞx p 2 R1 ðx þ cÞ ; ð3:13Þ þ aþ2
where x x R ðx þ cÞ ¼ R 2 1 2 1 a; x þ c; p;
1 : aþ2
Rewrite f(x), given by (3.7), as x 1 f ðxÞ ¼ vðxÞ2 R1 a; x þ c; p; ; aþ2
x ¼ 0; 1; 2; . . . ;
ð3:14Þ
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where vðxÞ ¼
ðcÞx CðqÞCðb þ xxÞ 1 ; x!ða þ 2Þx CðbÞCðq þ xxÞ Fx c; a; b; p; q; 1 ; 1 2 aþ2 aþ2
x ¼ 0; 1; 2; . . . ; from which we get cþx ðb þ xxÞx ; ðx þ 1Þða þ 2Þ ðq þ xxÞx ðc þ x 1Þ2 ðb þ ðx 1ÞxÞ2x vðxÞ ¼ vðx 1Þ: 2 ðxÞ2 ða þ 2Þ ðq þ ðx 1ÞxÞ2x
vðx þ 1Þ ¼
ð3:15Þ
Using (3.13)–(3.15), we obtain, for x = 1, 2, . . . x 1 ðx þ 1Þf ðx þ 1Þ ¼ ðx þ 1Þvðx þ 1Þ2 R1 a; x þ c; p; aþ2 x ðx þ 1Þða þ 2Þ ½p ðc þ xÞx vðx þ 1Þ2 R1 ðx þ c 1Þ ¼ ðc þ xÞða þ 1Þx
x a ðc þ xÞx þ þ 2ðc þ xÞx p vðx þ 1Þ2 R1 ðx þ cÞ aþ2 x ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x vðx 1Þ2 R1 ðx þ c 1Þ ðq þ ðx 1ÞxÞ2x xxða þ 1Þða þ 2Þ
x ðb þ xxÞx a ðc þ xÞx þ þ 2ðc þ xÞx p vðxÞ2 R1 ðx þ cÞ ð3:16Þ xða þ 1Þðq þ xxÞx aþ2
¼
ðc þ x 1Þ½p ðc þ xÞx ðb þ ðx 1ÞxÞ2x f ðx 1Þ xxða þ 1Þ2 ðq þ ðx 1ÞxÞ2x
ðb þ xxÞx a ðc þ xÞx þ þ 2ðc þ xÞx p f ðxÞ; xða þ 1Þðq þ xxÞx aþ2
¼
and the proof is complete.
h
Special cases: (1) if we let x = 1, then we get the first mixture distribution obtained earlier in [4], whereas if we put b = q, then we get the second mixture distribution. (2) if x = 1, b = q and c = a then we get the mixture distribution studied/obtained by Ghitany et al. [9]. Figs. 1 and 2 respectively, show the probability mass function f(x) and its distribution function F(x) for the selected specific values of c, a, b, p, q, a.
Y.B. Nakhi, S.L. Kalla / Appl. Math. Comput. 169 (2005) 943–952
951
Fig. 1. The probability mass function f(x) when c = 7, a = 3, b = 5, p = 6, q = 2 and a = 9. The symbol * represents f(x) when x = 1, whereas the symbol represents f(x) when x = 4.
Fig. 2. The distribution function F(x) when c = 7, a = 3, b = 5, p = 6, q = 2 and a = 9. The lower graph represents F(x) when x = 1, whereas the upper graph represents F(x) when x = 4.
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