`International Journal of Mathematics and Mathematical SciencesVolume 2008, Article ID 152808, 13 pageshttp://dx.doi.org/10.1155/2008/152808`
Research Article

Multivariate Generalization of the Confluent Hypergeometric Function Kind 1 Distribution

Departamento de Matemáticas, Universidad de Antioquia, Calle 67, 53–108 Medellín, Colombia

Received 27 June 2008; Accepted 4 October 2008

Copyright © 2008 Daya K. Nagar and Fabio Humberto Sepúlveda-Murillo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The confluent hypergeometric function kind 1 distribution with the probability density function (pdf) proportional to occurs as the distribution of the ratio of independent gamma and beta variables. In this article, a multivariate generalization of this distribution is defined and derived. Several pertinent properties of this multivariate distribution are discussed that shed some light on the nature of the distribution.

1. Introduction

The multivariate Liouville family of distributions was proposed by Marshall and Olkin [1]. Sivazlian [2] introduced Liouville distributions as generalizations of gamma and Dirichlet distributions. The Dirichlet and Liouville distributions have applications in such diverse fields as Bayesian analysis, modeling of multivariate data, order statistics, limit laws, multivariate analysis, reliability theory, and stochastic processes. These distributions have also been applied in geology, biology, chemistry, forensic science, and statistical genetics. A comprehensive review on applications and theoretical developments of these distributions are given in [15].

In this article, we propose a multivariate generalization of the confluent hypergeometric function kind 1 distribution which is a new member of the multivariate Liouville family of distributions.

The random variable is said to have a confluent hypergeometric function kind 1 distribution, denoted by , if its probability density function (pdf) is given by Gupta and Nagar [6],where , and is the confluent hypergeometric function kind 1 (Luke [7]). The confluent hypergeometric function kind 1 distribution occurs as the distribution of the ratio of independent gamma and beta variables. Distributions of the product and ratio of independent beta and gamma variates can be found in [8]. For , the density (1.1) reduces to a gamma density given by

Recently, Nagar and Sepúlveda-Murillo [9] studied several properties and stochastic representations of the confluent hypergeometric function kind 1 distribution.

In this article we define and study multivariate generalization of the confluent hypergeometric function kind 1 distribution. In Section 3, the multivariate generalization of the confluent hypergeometric function kind 1 distribution is defined and derived as the distribution of the quotient of a set of independent gamma variables variable with an independent beta variable. Several properties of this distribution including marginal and conditional distributions, distribution of partial sums and several factorizations are studied in Section 4.

2. Some Useful Definitions

Several definitions and results on special functions and integrals used in this article are given in this section. Throughout this work we will use the Pochhammer symbol defined by for and .

The generalized hypergeometric function of scalar argument is defined bywhere are complex numbers with suitable restrictions and is a complex variable. Conditions for the convergence of the series in (2.1) are available in the literature, see Luke [7]. From (2.1) it is easy to see that

Also, under suitable conditions,

The integral representations of the confluent hypergeometric function and the Gauss hypergeometric function are given asrespectively. Note that the series expansions for and given in (2.2) can be obtained by expanding and , in (2.4) and (2.5) and integrating . Substituting in (2.5) and integrating, we obtainThe confluent hypergeometric function satisfies Kummer's relationThe Lauricella hypergeometric function in variables is defined by where . For , the Lauricella hypergeometric function reduces to a Gauss hypergeometric function and for it slides to an Appell hypergeometric function . Using the resultfor andin (2.8), an integral representation of is given byFurther, replacing by its equivalent gamma integral, namely,for and integrating out , one obtainsFor further results and properties of this function the reader is referred to [10, 11].

Let be a continuous function and . The integralis known as Liouville's integral. Substituting and with the Jacobian , it is easy to see that

3. Density Function

In this section, we present a multivariate generalization of the hypergeometric function kind 1 distribution and derive it using independent beta and gamma variables.

Definition 3.1. The random variables are said to have a multivariate confluent hypergeometric function kind 1 distribution, denoted as , if their joint pdf is given bywhere is the normalizing constant.

The normalizing constant in (3.1) is given bywhere the last line has been obtained by using Liouville's integral and (2.7). Now, evaluating the above integral and simplifying the resulting expression using (2.3) and (2.6), we getwhere , and . For , multivariate confluent hypergeometric function kind 1 density simplifies to the product of univariate gamma densities. Several special cases of the density (3.1) can be obtained by specializing and and using results on confluent hypergeometric function. For example, substitution of in (3.1) and application of the resultwhere is the modified Bessel function of the first kind, yield

It may be noted here that the multivariate confluent hypergeometric function kind 1 distribution belongs to the Liouville family of distributions (Sivazlian [2], Gupta and Song [3]). Because of mathematical tractability of the confluent hypergeometric function and its several special cases, the multivariate confluent hypergeometric function kind 1 distribution enriches the class of multivariate Liouville distributions and may serve as an alternative to many existing distributions belonging to this class. The next theorem derives the multivariate confluent hypergeometric function kind 1 distribution using independent gamma and beta variables. First, we define the gamma, beta type 1 and beta type 2 distributions. These definitions can be found in [12].

Definition 3.2. A random variable is said to have the gamma distribution with shape parameter , denoted as , if its pdf is given by

Definition 3.3. A random variable is said to have the beta type 1 distribution with parameters , denoted as , if its pdf is given by

Definition 3.4. A random variable is said to have the beta type 2 distribution with parameters , denoted as , if its pdf is given by

If , then , and . Further, if , then , and .

The matrix variate generalizations of gamma, beta type 1, beta type 2 distributions have been defined and studied extensively. For example, see [6].

Theorem 3.5. Let and be independent, and . Then, with the pdf

Proof. The joint density of and is given bywhere . Transforming with the Jacobian in (3.10) and integrating out , we get the marginal density of aswhere . Now, evaluation of the above integral using (2.4) yields the desired result.

Corollary 3.6. Let and be independent, , and . Then

Corollary 3.7. Let and be independent, , and , then

The Laplace transform of the multivariate confluent hypergeometric function kind 1 density (3.1) is given bywhere . Now, rewriting by applying (2.7) and integrating using (2.13), the above expression is evaluated as

The joint moments of are given bywhere . Note that if , then from [9], one gets where , and . Now, computing using the above result, substituting for and from (3.3) and simplifying the resulting expression, we obtainwhere and . Substituting appropriately in the above expression and using definitions of variance, covariance, and correlation coefficient, it is straightforward to show that where .

4. Properties

In this section we derive marginal and conditional distributions, distribution of partial sums, and several factorizations of the multivariate confluent hypergeometric function kind 1 distribution.

The following theorem shows that if the joint distribution of is multivariate confluent hypergeometric function kind 1, then the marginal distribution of any subset of is multivariate confluent hypergeometric function kind 1.

Theorem 4.1. Let . Then, for ,

Proof. Replacing in (3.1) by its integral representation and integrating out , the marginal density of is derived aswhere . Now, evaluating the above integral using (2.4) and simplifying the resulting expression, we get the desired result.

Corollary 4.2. If , then for any subset of variables , it holds that where are distinct integers with .

Corollary 4.3. Let . Then, for , .

Using Theorem 4.1, the conditional density of given is obtained aswhere .

Next, in Theorem 4.5, we give the joint distribution of partial sums of random variables distributed jointly as multivariate confluent hypergeometric function kind 1. Since the theorem requires familiarity with the Dirichlet type 1 distribution, we first give its definition (see [13]).

Definition 4.4. The random variables are said to have a Dirichlet type 1 distribution with parameters , denoted by , if their joint pdf is given bywhere .

The Dirichlet type 1 distribution, which is a multivariate generalization of the beta type 1 distribution, has been considered by several authors and is well known in the scientific literature. By making the transformation , in (4.5), the Dirichlet type 2 density, which is a multivariate generalization of beta type 2 density, is obtained asWe will write if the joint density of is given by (4.6).

Let be nonnegative integers such that and define

Theorem 4.5. Let . Define and . Then,
(i) and , are independently distributed;(ii);(iii).

Proof. Substituting and with the Jacobianin the density of given by (3.1), we get the joint density of aswhere . From the factorization in (4.9), it is easy to see that and , are independently distributed. Further and .

Corollary 4.6. Let . Define , and . Then and are independent, and .

Corollary 4.7. If , then and are independent. Further

Theorem 4.8. Let . Define and . Then,
(i) and , are independently distributed;(ii);(iii).

Corollary 4.9. Let . Define , and . Then and are independent, and .

Corollary 4.10. If , then and are independent. Further

In the following six theorems, we give several factorizations of the multivariate confluent hypergeometric function density.

Theorem 4.11. Let . Define and . Then, are independent, , and .

Proof. Substituting and with the Jacobian in (3.1), we get the desired result.

Theorem 4.12. Let . Define and . Then are independent, , and .

Proof. The desired result follows from Theorem 4.11 by noting that .

Theorem 4.13. Let . Define and . Then are independent, , and .

Proof. The result follows from Theorem 4.12 by noting that .

Theorem 4.14. Let . Define and . Then are independent, , and .

Proof. Substituting , and with the Jacobian in (3.1), we get the desired result.

Theorem 4.15. Let . Define and . Then are independent, , and .

Proof. The result follows from Theorem 4.14 by noting that .

Theorem 4.16. Let . Define and . Then are independent, , and .

Proof. The result follows from Theorem 4.15 by noting that .

Now, we consider an approximation of the multivariate confluent hypergeometric function kind 1 distribution when increases.

Theorem 4.17. If , thenwhere , and “” denotes the convergence in distribution.

Proof. Substituting with the Jacobian in (3.1), the joint p.d.f. of is given byNow, using the resultsit is easy to see thatwhere is the joint cumulative distribution function (cdf) of and is the joint cdf of Dirichlet type 2 variables with parameters .

Acknowledgment

The research work of DKN was supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research Grant no. E 01252.

References

1. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.
2. B. D. Sivazlian, “On a multivariate extension of the gamma and beta distributions,” SIAM Journal on Applied Mathematics, vol. 41, no. 2, pp. 205–209, 1981.
3. A. K. Gupta and D. Song, “Generalized Liouville distribution,” Computers & Mathematics with Applications, vol. 32, no. 2, pp. 103–109, 1996.
4. R. D. Gupta and D. St. P. Richards, “The history of the Dirichlet and Liouville distributions,” International Statistical Review, vol. 69, no. 3, pp. 433–446, 2001.
5. D. Song and A. K. Gupta, “Properties of generalized Liouville distributions,” Random Operators and Stochastic Equations, vol. 5, no. 4, pp. 337–348, 1997.
6. A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, vol. 104 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000.
7. Y. L. Luke, The Special Functions and Their Approximations, Vol. 1, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York, NY, USA, 1969.
8. D. K. Nagar and E. Zarrazola, “Distributions of the product and the quotient of independent Kummer-beta variables,” Scientiae Mathematicae Japonicae, vol. 61, no. 1, pp. 109–117, 2005.
9. D. K. Nagar and F. H. Sepúlveda-Murillo, “Properties of confluent hypergeometric function kind 1 distribution,” Journal of Interdisciplinary Mathematics, vol. 11, no. 4, pp. 495–504, 2008.
10. H. Exton, Multiple Hypergeometric Functions and Applications, Mathematics & Its Application, Ellis Horwood, Chichester, UK, 1976.
11. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and Its Applications, John Wiley & Sons, New York, NY, USA, 1985.
12. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions. Vol. 2, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1995.
13. S. Kotz, N. Balakrishnan, and N. L. Johnson, Continuous Multivariate Distributions. Vol. 1, John Wiley & Sons, New York, NY, USA, 2nd edition, 2000.