Multivariate Generalization of the Confluent Hypergeometric Function Kind 1 Distribution

Volume 2008, Article ID 152808,13pages doi:10.1155/2008/152808

Research Article

Multivariate Generalization of the Confluent

Hypergeometric Function Kind 1 Distribution

Daya K. Nagar and Fabio Humberto Sep ´ulveda-Murillo

Departamento de Matem´aticas, Universidad de Antioquia, Calle 67, 53–108 Medell´ın, Colombia

Correspondence should be addressed to Daya K. Nagar,[email protected]

Received 27 June 2008; Accepted 4 October 2008

Recommended by Andrei Volodin

The confluent hypergeometric function kind 1 distribution with the probability density function pdfproportional tox−1

1F1α;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.

Copyrightq2008 D. K. Nagar and F. H. Sep ´ulveda-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.

1. Introduction

The multivariate Liouville family of distributions was proposed by Marshall and Olkin1. Sivazlian 2introduced 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 in1–5.

In this article, we propose a multivariate generalization of the confluent hypergeomet-ric 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∼CHν, α, β, kind 1, if its probability density function pdfis given by Gupta and Nagar6,

ΓαΓβ−ν

ΓνΓβΓα−νx

ν−1

whereβ > ν >0, α > ν >0, and1F1is the confluent hypergeometric function kind 1Luke 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 in8. For α β, the density1.1 reduces to a gamma density given by

1

Γνx

ν−1_{exp−x, x >0. 1.2}

Recently, Nagar and Sep ´ulveda-Murillo9studied 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 inSection 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 symbola_ndefined by

a_n aa1· · ·an−1 a_n−1an−1forn1,2, . . . ,anda₀1. The generalized hypergeometric function of scalar argument is defined by

pFqa1, . . . , ap;b1, . . . , bq;z

∞

k0

a1k· · ·ap_k

b1k· · ·bqk zk

k!, 2.1

whereai, i1, . . . , p; bj, j 1, . . . , qare complex numbers with suitable restrictions andzis a complex variable. Conditions for the convergence of the series in2.1are available in the literature, see Luke7. From2.1it is easy to see that

0F0z

∞

k0

zk

k! expz, 1F0a;z

∞

k0 a_kz

k

k! 1−z

−a_{, |z|<1,}

1F1a;c;z

∞

k0 a_k

c_k zk

k!, 2F1a, b;c;z

∞

k0

a_kb_k

c_k zk

k!, |z|<1.

2.2

Also, under suitable conditions,

_∞

0

exp−δzzα−1pF_qa1, . . . , ap;b1, . . . , bq;zydz

Γαδ−αp1F_qa1, . . . , ap, α;b1, . . . , bq;δ−1y.

The integral representations of the confluent hypergeometric function and the Gauss hypergeometric function are given as

1F1a;c;z Γ

c

ΓaΓc−a ₁

0

ta−11−tc−a−1expztdt

Rec>Rea>0,

2.4

2F1a, b;c;z Γ

c

ΓaΓc−a 1

0

ta−11−tc−a−11−zt−bdt,

Rec>Rea>0, |arg1−z|< π,

2.5

respectively. Note that the series expansions for1F1and2F1given in2.2can be obtained by

expanding expztand1−zt−b, |zt| < 1, in2.4and2.5and integratingt. Substituting

z1 in2.5and integrating, we obtain

2F1a, b;c; 1 Γ

cΓc−a−b

Γc−aΓc−b, Rec−a−b>0, c / −1,−2, . . . . 2.6

The confluent hypergeometric function1F1a;c;zsatisfies Kummer’s relation

1F1a;c;−z exp−z1F1c−a;c;z. 2.7

The Lauricella hypergeometric functionF_Dminmvariablesz1, . . . , zmis defined by

F_Dma, b1, . . . , bm;c;z1, . . . , zm ∞

j1,...,jm0

a_j1···jmb1j1· · ·bmjmz

j1

1 · · ·z

jm

m

c_j1···jmj1!· · ·jm!

, 2.8

where max{|z1|, . . . ,|zm|}<1. Form1, the Lauricella hypergeometric function reduces to a

Gauss hypergeometric function and form2 it slides to an Appell hypergeometric function

F1. Using the result

a_j

c_j

Γc

ΓaΓc−a ₁

0

taj−1_1−tc−a−1_{dt, Rec>Rea>0, 2.9}

forj0,1,2, . . . ,and

∞

ji0

bi_jitziji

ji! 1

in2.8, an integral representation ofFm_D is given by

Fm_D a, b1, . . . , bm;c;z1, . . . , zm

Γc

ΓaΓc−a ₁

0

ta−1_1−tc−a−1_1−z

1t−b1· · ·1−zmt−bmdt.

2.11

Further, replacing1−zit−bi by its equivalent gamma integral, namely,

1−zit−bi 1 Γbi

_∞

0

exp−1−zitvivbi−1

i dvi, Rebi>0, 2.12

fori1, . . . , mand integrating outt, one obtains

m

i1

ΓbiF_Dma, b1, . . . , bm;c;z1, . . . , zm

_∞

0

· · ·

_∞

0

exp

−m

i1

vi

_m

i1

vbi−1

i 1F1

a;c;−

m

i1

zivi

dv1· · ·dvm.

2.13

For further results and properties of this function the reader is referred to10,11. Letf·be a continuous function andαi>0, i1, . . . , n. The integral

Dnα1, . . . , αn;f _∞

0

· · ·

_∞

0

n

i1

xαi−1

i f

_n

i1

xi

_n

i1

dxi 2.14

is known as Liouville’s integral. Substitutingyi xi/x, i1, . . . , n−1 andxn_i1xiwith the JacobianJx1, . . . , xn−1, xn → y1, . . . , yn−1, x xn−1, it is easy to see that

Dnα1, . . . , αn;f

_n

i1Γαi

Γn

i1αi _∞

0

xni1αi−1_{fxdx. 2.15}

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 X1, . . . , Xn are said to have a multivariate

confluent hypergeometric function kind 1 distribution, denoted as X1, . . . , Xn∼

CHν1, . . . , νn;α, β, kind 1, if their joint pdf is given by

Cν1, . . . , νn;α, β

n

i1

xνi−1

i 1F1

α;β;−

n

i1

xi

, xi>0, i1, . . . , n, 3.1

The normalizing constant in3.1is given by

{Cν1, . . . , νn;α, β}−1 _∞

0

· · ·

_∞

0

n

i1

xνi−1

i 1F1

α;β;−

n

i1

xi

_n

i1

dxi

n

i1Γνi

Γn

i1νi _∞

0

xni1νi−1

1F1α;β;−xdx

_n

i1Γνi

Γn

i1νi _∞

0

xni1νi−1_exp−_x

1F1β−α;β;xdx,

3.2

where the last line has been obtained by using Liouville’s integral and2.7. Now, evaluating the above integral and simplifying the resulting expression using2.3and2.6, we get

{Cν1, . . . , νn;α, β}−1

n

i1ΓνiΓβΓα−n_i1νi

ΓαΓβ−n_i1νi

, 3.3

whereνi > 0, i 1, . . . , n, α > n_i1νi, and β > n_i1νi. For α β, multivariate confluent hypergeometric function kind 1 density simplifies to the product of n univariate gamma densities. Several special cases of the density3.1can be obtained by specializingαand β

and using results on confluent hypergeometric function. For example, substitution ofβ2α

in3.1and application of the result

1F1a; 2a;z

Γa1/2expz/2

z/4a−1/2 Ia−1/2

z

2

, 3.4

whereIδis the modified Bessel function of the first kind, yield

Cν1, . . . , νn;α,2α4α−1/2Γ α1

2

_exp−n i1xi/2

_n

i1xνii−1

n

i1xi α−1/2

×Iα−1/2

n i1xi

2

, xi>0, i1, . . . , n.

3.5

It may be noted here that the multivariate confluent hypergeometric function kind 1 distribution belongs to the Liouville family of distributionsSivazlian2, Gupta and Song

Definition 3.2. A random variable X is said to have the gamma distribution with shape parametera, a >0, denoted asX∼Gaa, if its pdf is given by

exp−xxa−1

Γa , x >0. 3.6

Definition 3.3. A random variable X is said to have the beta type 1 distribution with parametersa, b, a >0, b >0, denoted asX∼B1a, b, if its pdf is given by

Γab

ΓaΓbx

a−1_1−xb−1_{, 0< x <1. 3.7}

Definition 3.4. A random variable X is said to have the beta type 2 distribution with parametersa, b, denoted asX∼B2a, b, a >0, b >0, if its pdf is given by

Γab

ΓaΓbx

a−1_1x−ab_{, x >0. 3.8}

IfX∼B1a, b, then 1−X∼B1b, a, X/1−X∼B2a, b, and 1−X/X∼B2b, a. Further, ifY∼B2a, b, then 1/Y∼B2b, a, Y/1Y∼B1a, b, and 1/1Y∼B1b, a.

The matrix variate generalizations of gamma, beta type 1, beta type 2 distributions have been defined and studied extensively. For example, see6.

Theorem 3.5. Let Y1, . . . , Yn and Z be independent, Yi∼Gaci, i 1, . . . , n,and Z∼B1a, b. Then,Y1/Z, . . . , Yn/Z∼CHc1, . . . , cn;

_n i1cia,

_n

i1ciab, kind 1with the pdf

ΓabΓn_i1cia

_n

i1ΓciΓaΓni1ciab

n

i1

xci−1

i

×1F1

_n

i1

cia;

n

i1

ciab;−

n

i1

xi

, xi>0, i1, . . . , n.

3.9

Proof. The joint density ofY1, . . . , YnandZis given by

Γab

n

i1ΓciΓaΓb n

i1

yci−1

i exp

−n

i1

yi

za−11−zb−1, 3.10

where 0< z <1, yi >0, i1, . . . , n. TransformingXi Yi/Z, i 1, . . . , nwith the Jacobian Jy1, . . . , yn, z → x1, . . . , xn, z zn in 3.10 and integrating out z, we get the marginal

density ofX1, . . . , Xnas

Γab

n

i1ΓciΓaΓb

n

i1

xci−1

i 1

0

exp

−z n

i1

xi

zni1cia−1₁−_zb−1_{dz, 3.11}

Corollary 3.6. Let Y1, . . . , Yn and Z be independent,Yi∼Gaci, i 1, . . . , n, andZ∼B1a, b.

Then

Y1

1−Z, . . . , Yn

1−Z

∼CH

c1, . . . , cn;

n

i1

cib, n

i1

ciab, kind 1

. 3.12

Corollary 3.7. Let Y1, . . . , Yn and V be independent,Yi∼Gaci, i 1, . . . , n, and V∼B2a, b, then

1VY1

V , . . . ,

1VYn V

∼CH

c1, . . . , cn; n

i1

cia, n

i1

ciab, kind 1

,

1VY1, . . . ,1VYn∼CH

c1, . . . , cn; n

i1

cib, n

i1

ciab, kind 1

.

3.13

The Laplace transform of the multivariate confluent hypergeometric function kind 1 density3.1is given by

Cν1, . . . , νn;α, β _∞ 0 · · · _∞ 0 exp

−n

i1

sixi

_n

i1

xνi−1

i 1F1

α;β;−

n i1 xi _n i1

dxi, 3.14

where Resi>0, i1, . . . , n. Now, rewriting1F1by applying2.7and integratingx1, . . . , xm

using2.13, the above expression is evaluated as

ΓαΓβ−n_i1νi

ΓβΓα−n_i1νi F_Dn

β−α, ν1, . . . , νn;β;

1 1s1

, . . . , 1

1sn

. 3.15

The joint moments ofX1, . . . , Xnare given by

EX₁r1· · ·Xrn

n

Cν1, . . . , νn;α, β _∞ 0 · · · _∞ 0 n i1

xνiri−1

i 1F1

α;β;−

n i1 xi _n i1

dxi

Cν1, . . . , νn;α, β

Cn_i1νi;α, β

n

i1Γνiri

Γn

i1νiri

EXni1ri

,

3.16

whereX∼CHn_i1νi, α, β, kind 1. Note that ifX∼CHν, α, β, kind 1, then from9, one

gets

EXh Γβ−νΓνhΓα−ν−h

where Rehν>0, Reh< α−ν, and Reh< β−ν. Now, computingEXni1ri_{using the}

above result, substituting forCν1, . . . , νn;α, βandCn_i1νi;α, βfrom3.3and simplifying

the resulting expression, we obtain

EX₁r1· · ·Xrn

n

Γ

β−n_i1νi

Γα−n_i1νi− _n

i1rini1Γνiri

_n

i1ΓνiΓ

α−n_i1νi

Γβ−n_i1νi− _n

i1ri

, 3.18

whereα >n_i1νiriandβ >n_i1νiri. Substituting appropriately in the above expression and using definitions of variance, covariance, and correlation coefficient, it is straightforward to show that

EXj

νj

β−n_i1νi−1

α−n_i1νi−1 ,

EX2_j νjνj1

β−n_i1νi−1

β−n_i1νi−2

α−n_i1νi−1α−n_i1νi−2 ,

VarXj νj

β−n_i1νi−1

α−n_i1νi−1 2

α−n_i1νi−2

×

νjβ−α

β− n

i1

νi−2

α−

n

i1

νi−1

,

EXjXk νjνk

β−n_i1νi−1β−n_i1νi−2

α−n_i1νi−1

α−n_i1νi−2

,

CovXj, Xk

νjνk

β−n_i1νi−1

β−α

α−n_i1νi−12α−n_i1νi−2,

CorrXj, Xk

1

β−n_i1νi−2

α−n_i1νi−1

νjβ−α

×

1

β−n_i1νi−2

α−n_i1νi−1

νkβ−α

₋1/2

,

3.19

wherej /k, j, k1, . . . , 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, . . . , Xnis multivariate

confluent hypergeometric function kind 1, then the marginal distribution of any subset of

X1, . . . , Xnis multivariate confluent hypergeometric function kind 1.

Theorem 4.1. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Then, for 1≤s≤n,

X1, . . . , Xs∼CH

ν1, . . . , νs;α−

n

is1

νi, β− n

is1

νi, kind 1

Proof. Replacing1F1α;β;− n

i1xiin3.1by its integral representation and integrating out

x_s1, . . . , xn, the marginal density ofX1, . . . , Xsis derived as

Cν1, . . . , νn;α, β

Γβn_is1Γνi

ΓαΓβ−α s

i1

xνi−1

i ₁

0

tα−n

is1νi−1₁−_tβ−α−1_exp

−t n

i1

xi

dt, 4.2

wherexi>0, is1, . . . , n. Now, evaluating the above integral using2.4and simplifying

the resulting expression, we get the desired result.

Corollary 4.2. If X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1, then for any subset of variables

Xj1, . . . , Xji, it holds that

Xj1, . . . , Xji∼CH

νj1, . . . , νji;α

i

k1

νjk−

n

k1

νi, β i

k1

νjk−

n

k1

νi, kind 1

, 4.3

wherej1, . . . , jiare distinct integers with 1≤j1, . . . , ji≤n, 1≤i≤n.

Corollary 4.3. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Then, for k 1, . . . , n, Xk∼

CHνk, α−n_i/k1νi, β−n_i/k1νi, kind 1.

Using Theorem 4.1, the conditional density of X_s1, . . . , Xn given X1, . . . , Xs is

obtained as

Cν1, . . . , νn;α, β Cν1, . . . , νs;α−n_is1νi, β−

n is1νi

×

n

is1xνii−11F1α;β;− n

i1xi 1F1α−

_n

is1νi;β− _n

is1νi;− _n

i1xi ,

4.4

where 0< xi, is1, . . . , n.

Next, inTheorem 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

see13.

Definition 4.4. The random variablesU1, . . . , Unare said to have a Dirichlet type 1 distribution

with parameters ν1, . . . , νn;νn1, denoted byU1, . . . , Un∼D1ν1, . . . , νn;νn1, if their joint

pdf is given by

Γn1

i1νi _n1

i1Γνi

n

i1

uνi−1

i

1−

n

i1

ui νn1−1

, 0< ui, i1, . . . , n, n

i1

ui<1, 4.5

whereνi>0, i1, . . . , n1.

literature. By making the transformation Vj Uj/1−n_i1Ui, j 1, . . . , n, in 4.5, the Dirichlet type 2 density, which is a multivariate generalization of beta type 2 density, is obtained as

Γn1

i1νi _n1

i1Γνi

n

i1

uνi−1

i

1

n

i1

ui −n1

i1νi

, vi>0, i1, . . . , n. 4.6

We will write V1, . . . , Vn∼D2ν1, . . . , νn;νn1 if the joint density of V1, . . . , Vn is given by

4.6.

Letn1, . . . , n be nonnegative integers such that_i1ninand define

νi n∗_i

jn∗ i−11

νj, n∗₀0, n∗_i i

j1

nj, i1, . . . , . 4.7

Theorem 4.5. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. DefineZj Xj/Xi, j n∗i−1

1, . . . , n∗_i −1 andXin

∗ i

jn∗

i−11Xj, i1, . . . , . Then,

i X1, . . . , XandZn∗

i−11, . . . , Zn∗i−1, i1, . . . , , are independently distributed; ii Zn∗_i−11, . . . , Zn∗_i−1∼D1νn∗_i−11, . . . , νn∗_i−1;νn∗_i, i1, . . . , ;

iii X1, . . . , X∼CHν1, . . . , ν;α, β, kind 1.

Proof. Substitutingx_in∗i

jn∗

i−11xjandzj xj/xi, j n ∗

i−11, . . . , n∗i −1, i1, . . . , with

the Jacobian

Jx1, . . . , xn−→z1, . . . , zn1−1, x1, . . . , zn∗

−11, . . . , zn−1, x

i1

Jxn∗

i−11, . . . , xn∗i −→zn∗i−11, . . . , zn∗i−1, xi

i1

xni−1

i .

4.8

in the density of X1, . . . , Xn given by 3.1, we get the joint density of Zn∗_i−11,

. . . , Zn∗

i−1, Xi, i1, . . . , as

Cν1, . . . , νn;α, β

i1

xνi−1

i 1F1

α;β;−

i1

x_i

×

i1 _n∗

i−1

jn∗ i−11

zνj−1

j

1−

n∗i−1

jn∗ i−11

zj

νn∗_i−1

,

4.9

wherexi>0, i1, . . . , , zj>0, jn∗i−11, . . . , n∗i−1, n∗i−1

jn∗

i−11zj<1, i1, . . . , . From the

factorization in4.9, it is easy to see thatX1, . . . , XandZni∗−11, . . . , Zni∗−1, i1, . . . , ,

are independently distributed. Further X1, . . . , X∼CHν1, . . . , ν;α, β, kind 1 and

Corollary 4.6. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. DefineZiXi/Z, i1, . . . , n−1,

andZn_j1Xj. ThenZ1, . . . , Zn−1andZare independent,Z1, . . . , Zn−1∼D1ν1, . . . , νn−1;νn andZ∼CHn_i1νi, α, β, kind 1.

Corollary 4.7. If X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1, then _n

j1Xj and _s

i1Xi/ _n

i1Xi are independent. Further

_s i1Xi _n

i1Xi

∼B1

_s

i1

νi, n

is1

νi

, s < n. 4.10

Theorem 4.8. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. DefineWj Xj/Xn∗_i, j n∗_i−1

1, . . . , n∗_i −1 andXin

∗ i

jn∗

i−11Xj, i1, . . . , . Then,

i X1, . . . , XandWn∗

i−11, . . . , Wn∗i−1, i1, . . . , , are independently distributed; ii Wn∗_i−11, . . . , Wn∗_i−1∼D2νn∗_i−11, . . . , νn∗_i−1;νn∗_i, i1, . . . , ;

iii X1, . . . , X∼CHν1, . . . , ν;α, β, kind 1.

Corollary 4.9. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Define Wi Xi/Xn, i

1, . . . , n−1, and Z n_j1Xj. Then W1, . . . , Wn−1 and Z are independent,W1, . . . , Wn−1∼

D2ν1, . . . , νn−1;νnandZ∼CH _n

i1νi, α, β, kind 1.

Corollary 4.10. If X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1, then s

i1Xi/ n

is1Xi and n

j1Xjare independent. Further

s i1Xi n

is1Xi

∼B2

_s

i1

νi, n

is1

νi

, s < n. 4.11

In the following six theorems, we give several factorizations of the multivariate confluent hypergeometric function kind 1 density.

Theorem 4.11. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Define Yn n_j1Xj and Yi _i

j1Xj/ _i1

j1Xj, i 1, . . . , n−1. Then, Y1, . . . , Yn are independent, Yi∼B1 _i

j1νj, νi1, i

1, . . . , n−1, andYn∼CHn_i1νi, α, β, kind 1.

Proof. Substitutingx1 yn _n−1

i1yi, x2 yn1−y1 _n−1

i2yi, . . . , xn−1 yn1−yn−2yn−1 and

xnyn1−yn−1with the JacobianJx1, . . . , xn → y1, . . . , yn

_n

i2yii−1in3.1, we get the

desired result.

Theorem 4.12. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. DefineZn _n

j1Xj and Zi Xi1/i_j1Xj, i 1, . . . , n − 1. Then Z1, . . . , Zn are independent, Zi∼B2νi1,i_j1νj, i

Proof. The desired result follows from Theorem 4.11 by noting that 1 − Yi/Yi∼B2νi1, i

j1νj.

Theorem 4.13. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Define Wn _n

j1Xj andWi i

j1Xj/Xi1, i 1, . . . , n −1. Then W1, . . . , Wn are independent, Wi∼B2i_j1νj, νi1, i

1, . . . , n−1, andWn∼CH _n

i1νi, α, β, kind 1.

Proof. The result follows fromTheorem 4.12by noting that 1/Zi∼B2i_j1νj, νi1.

Theorem 4.14. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Define Yn n_j1Xj and Yi

Xi/n_jiXj, i1, . . . , n−1. ThenY1, . . . , Ynare independent,Yi∼B1νi,n_ji1νj, i1, . . . , n−1,

andYn∼CHn_i1νi, α, β, kind 1.

Proof. Substitutingx1yny1, x2yny21−y1, . . . , xn−1ynyn−11−y1· · ·1−yn−2, andxn yn1−y1· · ·1−yn−1with the JacobianJx1, . . . , xn → y1, . . . , yn ynn−1

_n−2

i11−yin−i−1in

3.1, we get the desired result.

Theorem 4.15. Let X1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. DefineZn n_j1Xj and Zi Xi/n_ji1Xj, i1, . . . , n−1. ThenZ1, . . . , Znare independent,Zi∼B2νi,n_ji1νj, i1, . . . , n−

1, andZn∼CHn_i1νi, α, β, kind 1.

Proof. The result follows fromTheorem 4.14by noting thatYi/1−Yi∼B2νi,n_ji1νj.

Theorem 4.16. LetX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1. Define Wn n_j1Xj andWi n

ji1Xj/Xi, i 1, . . . , n − 1. Then W1, . . . , Wn are independent, Wi∼B2n_ji1νj, νi, i

1, . . . , n−1, andWn∼CHn_i1νi, α, β, kind 1.

Proof. The result follows fromTheorem 4.15by noting that 1/Wi∼B2n_ji1νj, νi.

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

Theorem 4.17. IfX1, . . . , Xn∼CHν1, . . . , νn;α, β, kind 1, then

X1

β , . . . , Xn

β

d

−→W1, . . . , Wn, 4.12

whereW1, . . . , Wn∼D2ν1, . . . , νn;α−n_i1νi, and “

d

→” denotes the convergence in distribution.

Proof. Substitutingxi βui, i1, . . . , nwith the JacobianJx1, . . . , xn → u1, . . . , un βnin 3.1, the joint p.d.f. ofU1, . . . , Unis given by

gu1, . . . , un Γα Γα−n_i1νi

_n

i1

uνi−1

i

Γνi

Γβ−n_i1νi

Γββ−ni1νi 1F1

α;β;−β n

i1

ui

Now, using the results

lim

β→ ∞1F1

α;β;−β

n

i1

ui

1F0

α;−

n

i1

ui

1

n

i1

ui ₋α

,

lim

β→ ∞

Γβ−n_i1νi

Γββ−ni1νi 1

4.14

it is easy to see that

lim

β→ ∞Fu1, . . . , un Gu1, . . . , un, 4.15

whereFu1, . . . , unis the joint cumulative distribution functioncdfofX1, . . . , Xn/βand

Gu1, . . . , un is the joint cdf of Dirichlet type 2 variables with parameters ν1, . . . , νn;α− _n

i1νi.

Acknowledgment

The research work of DKN was supported by the Comit´e para el Desarrollo de la Investigaci ´on, Universidad de Antioquia research Grant no. E 01252.

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