# Multivariate Generalization of the Confluent Hypergeometric Function Kind 1 Distribution

Authors: Daya K. Nagar; Fabio Humberto Sepúlveda-Murillo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

The confluent hypergeometric function kind 1 distribution with the probability density
function (pdf) proportional tox_−1F11(α;β;−x),x>0 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.


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## Body

## 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 [1–5].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 variableX is said to have a confluent hypergeometric
function kind 1 distribution, denoted by X~CH(ν,α,β,kind1),
if its probability density function (pdf) is given by Gupta and  Nagar [6],(1.1)Γ(α)Γ(β−ν)Γ(ν)Γ(β)Γ(α−ν)xν−1F11(α;β;−x),x>0,where β>ν>0,α>ν>0,
and F11 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 α=β,
the density (1.1) reduces to a gamma density given by(1.2)1Γ(ν)xν−1exp(−x),x>0.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
Section3, 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(a)n defined by (a)n=a(a+1)⋯(a+n−1)=(a)n−1(a+n−1) for n=1,2,…, and (a)0=1.The generalized hypergeometric function of scalar
argument is defined by(2.1)Fqp(a1,…,ap;b1,…,bq;z)=∑k=0∞(a1)k⋯(ap)k(b1)k⋯(bq)kzkk!,where ai,i=1,…,p;bj,j=1,…,q  are complex numbers with suitable restrictions
and z 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(2.2)F00(z)=∑k=0∞zkk!=exp(z),F10(a;z)=∑k=0∞(a)kzkk!=(1−z)−a,|z|<1,F11(a;c;z)=∑k=0∞(a)k(c)kzkk!,F21(a,b;c;z)=∑k=0∞(a)k(b)k(c)kzkk!,|z|<1.Also, under suitable  conditions,(2.3)∫0∞exp(−δz)zα−1Fpq(a1,…,ap;b1,…,bq;zy)dz=Γ(α)δ−αFp+1q(a1,…,ap,α;b1,…,bq;δ−1y).The integral representations of the confluent
hypergeometric function and the Gauss hypergeometric function are given as(2.4)F11(a;c;z)=Γ(c)Γ(a)Γ(c−a)∫01ta−1(1−t)c−a−1exp(zt)dtRe(c)>Re(a)>0,(2.5)F21(a,b;c;z)=Γ(c)Γ(a)Γ(c−a)∫01ta−1(1−t)c−a−1(1−zt)−bdt,Re(c)>Re(a)>0,|arg(1−z)|<π,respectively. Note that the
series expansions for F11 and F21 given in (2.2)  can be obtained by expanding exp(zt) and (1−zt)−b,|zt|<1,
in (2.4) and (2.5) and integrating t.
Substituting z=1 in (2.5) and integrating, we obtain(2.6)F21(a,b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b),Re(c−a−b)>0,c≠−1,−2,….The confluent hypergeometric
function F11(a;c;z) satisfies Kummer's relation(2.7)F11(a;c;−z)=exp(−z)F11(c−a;c;z).The Lauricella hypergeometric
function FD(m) in m variables z1,…,zm is defined by(2.8)FD(m)[a,b1,…,bm;c;z1,…,zm]=∑j1,…,jm=0∞(a)j1+⋯+jm(b1)j1⋯(bm)jmz1j1⋯zmjm(c)j1+⋯+jmj1!⋯jm!,
where max{|z1|,…,|zm|}<1.
For m=1,
the Lauricella hypergeometric function reduces to a Gauss hypergeometric
function and for m=2 it slides to an Appell hypergeometric function F1.
Using the result(2.9)(a)j(c)j=Γ(c)Γ(a)Γ(c−a)∫01ta+j−1(1−t)c−a−1dt,Re(c)>Re(a)>0,for j=0,1,2,…,
and(2.10)∑ji=0∞(bi)ji(tzi)jiji!=1F0(bi;tzi)=(1−tzi)−bi,|zi|<1,i=1,…,m,in (2.8), an integral
representation of FD(m) is given by(2.11)FD(m)[a,b1,…,bm;c;z1,…,zm]=Γ(c)Γ(a)Γ(c−a)∫01ta−1(1−t)c−a−1(1−z1t)−b1⋯(1−zmt)−bmdt.Further, replacing (1−zit)−bi by its equivalent gamma integral, namely,(2.12)(1−zit)−bi=1Γ(bi)∫0∞exp[−(1−zit)vi]vibi−1dvi,Re(bi)>0,for i=1,…,m and integrating out t,
one obtains(2.13)∏i=1mΓ(bi)FD(m)[a,b1,…,bm;c;z1,…,zm]=∫0∞⋯∫0∞exp(−∑i=1mvi)∏i=1mvibi−1F11(a;c;−∑i=1mzivi)dv1⋯dvm.For further results and
properties of this function the reader is referred to [10, 11].Letf(⋅) be a continuous function and αi>0,i=1,…,n.
The integral(2.14)Dn(α1,…,αn;f)=∫0∞⋯∫0∞∏i=1nxiαi−1f(∑i=1nxi)∏i=1ndxiis known as Liouville's
integral. Substituting yi=xi/x,i=1,…,n−1 and x=∑i=1nxi with the Jacobian J(x1,…,xn−1,xn→y1,…,yn−1,x)=xn−1,
it is easy to see that(2.15)Dn(α1,…,αn;f)=∏i=1nΓ(αi)Γ(∑i=1nαi)∫0∞x∑i=1nαi−1f(x)dx.


## 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 variablesX1,…,Xn are said to have a multivariate confluent
hypergeometric function kind 1 distribution, denoted as
(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1),
if their joint pdf is given by(3.1)C(ν1,…,νn;α,β)∏i=1nxiνi−1F11(α;β;−∑i=1nxi),xi>0,i=1,…,n,where C(ν1,…,νn;α,β) is the normalizing constant.The normalizing constant in (3.1) is given by(3.2){C(ν1,…,νn;α,β)}−1=∫0∞⋯∫0∞∏i=1nxiνi−1F11(α;β;−∑i=1nxi)∏i=1ndxi=∏i=1nΓ(νi)Γ(∑i=1nνi)∫0∞x∑i=1nνi−1F11(α;β;−x)dx=∏i=1nΓ(νi)Γ(∑i=1nνi)∫0∞x∑i=1nνi−1exp(−x)1F1(β−α;β;x)dx,where 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 get(3.3){C(ν1,…,νn;α,β)}−1=∏i=1nΓ(νi)Γ(β)Γ(α−∑i=1nνi)Γ(α)Γ(β−∑i=1nνi),where νi>0,i=1,…,n,α>∑i=1nνi, and β>∑i=1nνi.
For α=β,
multivariate confluent hypergeometric function kind 1 density simplifies to
the product of n
univariate gamma
densities. Several special cases of the density (3.1) can be obtained by
specializing α and β and using results on confluent hypergeometric
function. For example, substitution of β=2α in (3.1) and application of the result(3.4)F11(a;2a;z)=Γ(a+1/2)exp(z/2)(z/4)a−1/2Ia−1/2(z2),where Iδ is the modified Bessel function of the first
kind, yield(3.5)C(ν1,…,νn;α,2α)4α−1/2Γ(α+12)exp(−∑i=1nxi/2)∏i=1nxiνi−1(∑i=1nxi)α−1/2×Iα−1/2(∑i=1nxi2),xi>0,i=1,…,n.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 variableX is said to have the gamma distribution with
shape parameter a,a>0,
denoted as X~Ga(a),
if its pdf is given by(3.6)exp(−x)xa−1Γ(a),x>0.Definition 3.3.
A random variableX is said to have the beta type 1 distribution
with parameters (a,b),a>0,b>0,
denoted as X~B1(a,b),
if its pdf is given by(3.7)Γ(a+b)Γ(a)Γ(b)xa−1(1−x)b−1,0<x<1.Definition 3.4.
A random variableX is said to have the beta type 2 distribution
with parameters (a,b),
denoted as X~B2(a,b),a>0,b>0,
if its pdf is given by(3.8)Γ(a+b)Γ(a)Γ(b)xa−1(1+x)−(a+b),x>0.IfX~B1(a,b),
then 1−X~B1(b,a),X/(1−X)~B2(a,b),
and (1−X)/X~B2(b,a).
Further, if Y~B2(a,b),
then 1/Y~B2(b,a),Y/(1+Y)~B1(a,b),
and 1/(1+Y)~B1(b,a).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.
LetY1,…,Yn and Z be independent, Yi~Ga(ci),i=1,…,n, and Z~B1(a,b).
Then, (Y1/Z,…,Yn/Z)~CH(c1,…,cn;∑i=1nci+a,∑i=1nci+a+b,kind1) with the pdf(3.9)Γ(a+b)Γ(∑i=1nci+a)∏i=1nΓ(ci)Γ(a)Γ(∑i=1nci+a+b)∏i=1nxici−1×F11(∑i=1nci+a;∑i=1nci+a+b;−∑i=1nxi),xi>0,i=1,…,n.Proof.
The
joint density ofY1,…,Yn and Z is given by(3.10)Γ(a+b)∏i=1nΓ(ci)Γ(a)Γ(b)∏i=1nyici−1exp(−∑i=1nyi)za−1(1−z)b−1,where 0<z<1,yi>0,i=1,…,n.
Transforming Xi=Yi/Z,i=1,…,n  with the Jacobian J(y1,…,yn,z→x1,…,xn,z)=zn in (3.10) and integrating out z,
we get the marginal density of (X1,…,Xn) as(3.11)Γ(a+b)∏i=1nΓ(ci)Γ(a)Γ(b)∏i=1nxici−1∫01exp(−z∑i=1nxi)z∑i=1nci+a−1(1−z)b−1dz,where xi>0,i=1,…,n.
Now, evaluation of the above integral using (2.4) yields the desired result.Corollary 3.6.
LetY1,…,Yn and Z be independent, Yi~Ga(ci),i=1,…,n,  and Z~B1(a,b).
Then(3.12)(Y11−Z,…,Yn1−Z)~CH(c1,…,cn;∑i=1nci+b,∑i=1nci+a+b,kind1).Corollary 3.7.
LetY1,…,Yn and V be independent, Yi~Ga(ci),i=1,…,n,  and V~B2(a,b), then(3.13)((1+V)Y1V,…,(1+V)YnV)~CH(c1,…,cn;∑i=1nci+a,∑i=1nci+a+b,kind1),((1+V)Y1,…,(1+V)Yn)~CH(c1,…,cn;∑i=1nci+b,∑i=1nci+a+b,kind1).The Laplace transform of the multivariate confluent
hypergeometric function kind 1 density (3.1) is given by(3.14)C(ν1,…,νn;α,β)∫0∞⋯∫0∞exp(−∑i=1nsixi)∏i=1nxiνi−1F11(α;β;−∑i=1nxi)∏i=1ndxi,where Re(si)>0,i=1,…,n.
Now, rewriting F11 by applying (2.7) and integrating x1,…,xm using (2.13), the above expression is evaluated
as(3.15)Γ(α)Γ(β−∑i=1nνi)Γ(β)Γ(α−∑i=1nνi)FD(n)[β−α,ν1,…,νn;β;11+s1,…,11+sn].The joint moments ofX1,…,Xn are  given by(3.16)E(X1r1⋯Xnrn)=C(ν1,…,νn;α,β)∫0∞⋯∫0∞∏i=1nxiνi+ri−1F11(α;β;−∑i=1nxi)∏i=1ndxi=C(ν1,…,νn;α,β)C(∑i=1nνi;α,β)∏i=1nΓ(νi+ri)Γ[∑i=1n(νi+ri)]E[X∑i=1nri],where X~CH(∑i=1nνi,α,β,kind1).
Note that if X~CH(ν,α,β,kind1),
then from [9], one gets (3.17)E(Xh)=Γ(β−ν)Γ(ν+h)Γ(α−ν−h)Γ(ν)Γ(α−ν)Γ(β−ν−h),where Re(h+ν)>0,Re(h)<α−ν,  and Re(h)<β−ν.
Now, computing E[X∑i=1nri] using the above result, substituting for C(ν1,…,νn;α,β) and C(∑i=1nνi;α,β) from (3.3) and simplifying the resulting
expression, we obtain(3.18)E(X1r1⋯Xnrn)=Γ(β−∑i=1nνi)Γ(α−∑i=1nνi−∑i=1nri)∏i=1nΓ(νi+ri)∏i=1nΓ(νi)Γ(α−∑i=1nνi)Γ(β−∑i=1nνi−∑i=1nri),where α>∑i=1n(νi+ri) and β>∑i=1n(νi+ri).
Substituting appropriately in the above expression and using definitions of
variance, covariance, and correlation coefficient, it is straightforward to
show that(3.19)E(Xj)=νj(β−∑i=1nνi−1)α−∑i=1nνi−1,E(Xj2)=νj(νj+1)(β−∑i=1nνi−1)(β−∑i=1nνi−2)(α−∑i=1nνi−1)(α−∑i=1nνi−2),Var(Xj)=νj(β−∑i=1nνi−1)(α−∑i=1nνi−1)2(α−∑i=1nνi−2)×[νj(β−α)+(β−∑i=1nνi−2)(α−∑i=1nνi−1)],E(XjXk)=νjνk(β−∑i=1nνi−1)(β−∑i=1nνi−2)(α−∑i=1nνi−1)(α−∑i=1nνi−2),Cov(Xj,Xk)=νjνk(β−∑i=1nνi−1)(β−α)(α−∑i=1nνi−1)2(α−∑i=1nνi−2),Corr(Xj,Xk)=[(1+(β−∑i=1nνi−2)(α−∑i=1nνi−1)νj(β−α))×(1+(β−∑i=1nνi−2)(α−∑i=1nνi−1)νk(β−α))]−1/2, where j≠k,j,k=1,…,n.


## 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 ofX1,…,Xn is multivariate confluent hypergeometric
function kind 1, then the marginal distribution of
any subset of X1,…,Xn is multivariate confluent hypergeometric
function kind 1.Theorem 4.1.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Then, for 1≤s≤n,(4.1)(X1,…,Xs)~CH(ν1,…,νs;α−∑i=s+1nνi,β−∑i=s+1nνi,kind1).Proof.
ReplacingF11(α;β;−∑i=1nxi) in (3.1) by its integral representation and integrating out xs+1,…,xn,
the marginal density of X1,…,Xs is derived as(4.2)C(ν1,…,νn;α,β)Γ(β)∏i=s+1nΓ(νi)Γ(α)Γ(β−α)∏i=1sxiνi−1∫01tα−∑i=s+1nνi−1(1−t)β−α−1exp(−t∑i=1nxi)dt,where xi>0,i=s+1,…,n.
Now, evaluating the above integral using (2.4) and simplifying the resulting
expression, we get the desired result.Corollary 4.2.
If(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1),
then for any subset of variables (Xj1,…,Xji),
it holds that
(4.3)(Xj1,…,Xji)~CH(νj1,…,νji;α+∑k=1iνjk−∑k=1nνi,β+∑k=1iνjk−∑k=1nνi,kind1),where j1,…,ji are distinct integers with 1≤j1,…,ji≤n,1≤i≤n.Corollary 4.3.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Then, for k=1,…,n,   Xk~CH(νk,α−∑i(≠k)=1nνi,β−∑i(≠k)=1nνi,kind1).Using Theorem4.1, the conditional density of (Xs+1,…,Xn) given (X1,…,Xs) is obtained as(4.4)C(ν1,…,νn;α,β)C(ν1,…,νs;α−∑i=s+1nνi,β−∑i=s+1nνi)×∏i=s+1nxiνi−1F11(α;β;−∑i=1nxi)F11(α−∑i=s+1nνi;β−∑i=s+1nνi;−∑i=1nxi),where 0<xi,i=s+1,…,n.Next, in Theorem4.5, 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 variablesU1,…,Un are said to have a Dirichlet type 1
distribution with parameters ν1,…,νn;νn+1,
denoted by (U1,…,Un)~D1(ν1,…,νn;νn+1),
if their joint pdf is given by(4.5)Γ(∑i=1n+1νi)∏i=1n+1Γ(νi)∏i=1nuiνi−1(1−∑i=1nui)νn+1−1,0<ui,i=1,…,n,∑i=1nui<1,where νi>0,i=1,…,n+1.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 transformationVj=Uj/(1−∑i=1nUi),j=1,…,n,
in (4.5), the Dirichlet type 2 density, which is a multivariate generalization
of beta type 2 density, is obtained as(4.6)Γ(∑i=1n+1νi)∏i=1n+1Γ(νi)∏i=1nuiνi−1(1+∑i=1nui)−∑i=1n+1νi,vi>0,i=1,…,n.We will write (V1,…,Vn)~D2(ν1,…,νn;νn+1) if the joint density of V1,…,Vn is given by (4.6).Letn1,…,nℓ be nonnegative integers such that ∑i=1ℓni=n and define(4.7)ν(i)=∑j=ni−1*+1ni*νj,n0*=0,ni*=∑j=1inj,i=1,…,ℓ.Theorem 4.5.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Zj=Xj/X(i),j=ni−1*+1,…,ni*−1 and X(i)=∑j=ni−1*+1ni*Xj,i=1,…,ℓ.
Then,
(i)
(X(1),…,X(ℓ)) and (Zni−1*+1,…,Zni*−1),i=1,…,ℓ,
are independently distributed;(ii)
(Zni−1*+1,…,Zni*−1)~D1(νni−1*+1,…,νni*−1;νni*),i=1,…,ℓ;(iii)
(X(1),…,X(ℓ))~CH(ν(1),…,ν(ℓ);α,β,kind1).Proof.
Substitutingx(i)=∑j=ni−1*+1ni*xj and zj=xj/x(i),j=ni−1*+1,…,ni*−1,i=1,…,ℓ  with the Jacobian(4.8)J(x1,…,xn→z1,…,zn1−1,x(1),…,znℓ−1*+1,…,zn−1,x(ℓ))=∏i=1ℓJ(xni−1*+1,…,xni*→zni−1*+1,…,zni*−1,x(i))=∏i=1ℓx(i)ni−1.in the density of (X1,…,Xn) given by (3.1), we get the joint density of Zni−1*+1,…,Zni*−1,X(i),i=1,…,ℓ  as(4.9)C(ν1,…,νn;α,β)∏i=1ℓx(i)ν(i)−1F11(α;β;−∑i=1ℓx(i))×∏i=1ℓ[∏j=ni−1*+1ni*−1zjνj−1(1−∑j=ni−1*+1ni*−1zj)νni*−1],where x(i)>0,i=1,…,ℓ,zj>0,j=ni−1*+1,…,ni*−1,∑j=ni−1*+1ni*−1zj<1,i=1,…,ℓ.
From the factorization in (4.9), it is easy to see that (X(1),…,X(ℓ)) and (Zni−1*+1,…,Zni*−1),i=1,…,ℓ,
are independently distributed. Further (X(1),…,X(ℓ))~CH(ν(1),…,ν(ℓ);α,β,kind1) and (Zni−1*+1,…,Zni*−1)~D1(νni−1*+1,…,νni*−1;νni*),i=1,…,ℓ.Corollary 4.6.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Zi=Xi/Z,i=1,…,n−1,
and Z=∑j=1nXj.
Then (Z1,…,Zn−1) and Z are independent, (Z1,…,Zn−1)~D1(ν1,…,νn−1;νn) and Z~CH(∑i=1nνi,α,β,kind1).Corollary 4.7.
If(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1),
then ∑j=1nXj and ∑i=1sXi/∑i=1nXi are independent. Further(4.10)∑i=1sXi∑i=1nXi~B1(∑i=1sνi,∑i=s+1nνi),s<n.Theorem 4.8.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Wj=Xj/Xni*,j=ni−1*+1,…,ni*−1 and X(i)=∑j=ni−1*+1ni*Xj,i=1,…,ℓ.
Then,
(i)
(X(1),…,X(ℓ)) and (Wni−1*+1,…,Wni*−1),i=1,…,ℓ,
are independently distributed;(ii)
(Wni−1*+1,…,Wni*−1)~D2(νni−1*+1,…,νni*−1;νni*),i=1,…,ℓ;(iii)
(X(1),…,X(ℓ))~CH(ν(1),…,ν(ℓ);α,β,kind1).Corollary 4.9.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Wi=Xi/Xn,i=1,…,n−1,
and Z=∑j=1nXj.
Then (W1,…,Wn−1) and Z are independent, (W1,…,Wn−1)~D2(ν1,…,νn−1;νn) and Z~CH(∑i=1nνi,α,β,kind1).Corollary 4.10.
If(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1),
then ∑i=1sXi/∑i=s+1nXi and ∑j=1nXj are independent. Further(4.11)∑i=1sXi∑i=s+1nXi~B2(∑i=1sνi,∑i=s+1nνi),s<n.In the following six theorems, we give several
factorizations of the multivariate confluent hypergeometric functionkind1
density.Theorem 4.11.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Yn=∑j=1nXj and Yi=∑j=1iXj/∑j=1i+1Xj,i=1,…,n−1.
Then, Y1,…,Yn are independent, Yi~B1(∑j=1iνj,νi+1),i=1,…,n−1,
and Yn~CH(∑i=1nνi,α,β,kind1).Proof.
Substitutingx1=yn∏i=1n−1yi,x2=yn(1−y1)∏i=2n−1yi,…,xn−1=yn(1−yn−2)yn−1 and xn=yn(1−yn−1) with the Jacobian J(x1,…,xn→y1,…,yn)=∏i=2nyii−1 in (3.1), we get the desired result.Theorem 4.12.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Zn=∑j=1nXj and Zi=Xi+1/∑j=1iXj,i=1,…,n−1.
Then Z1,…,Zn are independent, Zi~B2(νi+1,∑j=1iνj),i=1,…,n−1,
and Zn~CH(∑j=1nνj,α,β,kind1).Proof.
The desired result follows from
Theorem4.11 by noting that (1−Yi)/Yi~B2(νi+1,∑j=1iνj).Theorem 4.13.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Wn=∑j=1nXj and Wi=∑j=1iXj/Xi+1,i=1,…,n−1.
Then W1,…,Wn are independent, Wi~B2(∑j=1iνj,νi+1),i=1,…,n−1,
and Wn~CH(∑i=1nνi,α,β,kind1).Proof.
The result follows from
Theorem4.12
by noting that 1/Zi~B2(∑j=1iνj,νi+1).Theorem 4.14.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Yn=∑j=1nXj and Yi=Xi/∑j=inXj,i=1,…,n−1.
Then Y1,…,Yn are independent, Yi~B1(νi,∑j=i+1nνj),i=1,…,n−1,
and Yn~CH(∑i=1nνi,α,β,kind1).Proof.
Substitutingx1=yny1,x2=yny2(1−y1),…,xn−1=ynyn−1(1−y1)⋯(1−yn−2),
and xn=yn(1−y1)⋯(1−yn−1) with the Jacobian J(x1,…,xn→y1,…,yn)=ynn−1∏i=1n−2(1−yi)n−i−1 in (3.1), we get the desired result.Theorem 4.15.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Zn=∑j=1nXj and Zi=Xi/∑j=i+1nXj,i=1,…,n−1.
Then Z1,…,Zn are independent, Zi~B2(νi,∑j=i+1nνj),i=1,…,n−1,
and Zn~CH(∑i=1nνi,α,β,kind1).Proof.
The result follows from Theorem4.14
by noting that Yi/(1−Yi)~B2(νi,∑j=i+1nνj).Theorem 4.16.
Let(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1).
Define Wn=∑j=1nXj and Wi=∑j=i+1nXj/Xi,i=1,…,n−1.
Then W1,…,Wn are independent, Wi~B2(∑j=i+1nνj,νi),i=1,…,n−1,
and Wn~CH(∑i=1nνi,α,β,kind1).Proof.
The result follows from Theorem4.15
by noting that 1/Wi~B2(∑j=i+1nνj,νi).Now, we consider an approximation of the multivariate
confluent hypergeometric function kind 1 distribution whenβ increases.Theorem 4.17.
If(X1,…,Xn)~CH(ν1,…,νn;α,β,kind1),
then(4.12)(X1β,…,Xnβ)→d(W1,…,Wn),where (W1,…,Wn)~D2(ν1,…,νn;α−∑i=1nνi),
and “→d” denotes the convergence in distribution.Proof.
Substitutingxi=βui,i=1,…,n with the Jacobian J(x1,…,xn→u1,…,un)=βn in (3.1), the joint p.d.f. of U1,…,Un is given by(4.13)g(u1,…,un)=Γ(α)Γ(α−∑i=1nνi)(∏i=1nuiνi−1Γ(νi))Γ(β−∑i=1nνi)Γ(β)β−∑i=1nνiF11(α;β;−β∑i=1nui).Now, using the results(4.14)limβ→∞F11(α;β;−β∑i=1nui)=1F0(α;−∑i=1nui)=(1+∑i=1nui)−α,limβ→∞Γ(β−∑i=1nνi)Γ(β)β−∑i=1nνi=1it is easy to see that(4.15)limβ→∞F(u1,…,un)=G(u1,…,un),where F(u1,…,un) is the joint cumulative distribution function
(cdf) of (X1,…,Xn)/β and G(u1,…,un) is the joint cdf of Dirichlet type 2 variables
with parameters (ν1,…,νn;α−∑i=1nνi).



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*Source: 152808-2009-02-08.xml*