Multivariate Generalization of the Confluent Hypergeometric Function Kind 1 Distribution

Abstract

The confluent hypergeometric function kind 1 distribution with the probability density function (pdf) proportional to 𝑥^_₁⁻¹𝐹₁(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.

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 variable 𝑋 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],Γ(𝛼)Γ(𝛽−𝜈)𝑥Γ(𝜈)Γ(𝛽)Γ(𝛼−𝜈)^1𝜈−1𝐹₁(𝛼;𝛽;−𝑥),𝑥>0, (1.1)where 𝛽>𝜈>0,𝛼>𝜈>0, and ₁𝐹₁ 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 by1𝑥Γ(𝜈)^𝜈−1exp(−𝑥),𝑥>0. (1.2)

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 (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_𝑝𝐹_𝑞(𝑎₁,…,𝑎_𝑝;𝑏₁,…,𝑏_𝑞;𝑧)=^∞_𝑘=0(𝑎₁)_𝑘⋯(𝑎_𝑝)_𝑘(𝑏₁)_𝑘⋯(𝑏_𝑞)_𝑘𝑧^𝑘𝑘!, (2.1)where 𝑎_𝑖,𝑖=1,…,𝑝;𝑏_𝑗,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 that₀𝐹₀(𝑧)=^∞_𝑘=0𝑧^𝑘𝑘!=exp(𝑧),₁𝐹₀(𝑎;𝑧)=^∞_𝑘=0(𝑎)_𝑘𝑧^𝑘𝑘!=(1−𝑧)^−𝑎,|𝑧|<1,₁𝐹₁(𝑎;𝑐;𝑧)=^∞_𝑘=0(𝑎)_𝑘(𝑐)_𝑘𝑧^𝑘,𝑘!²𝐹₁(𝑎,𝑏;𝑐;𝑧)=^∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘(𝑐)_𝑘𝑧^𝑘𝑘!,|𝑧|<1. (2.2)

Also, under suitable conditions,^∞₀exp(−𝛿𝑧)𝑧_𝑝^𝛼−1𝐹_𝑞(𝑎₁,…,𝑎_𝑝;𝑏₁,…,𝑏_𝑞;𝑧𝑦)𝑑𝑧=Γ(𝛼)𝛿^−𝛼_𝑝+1𝐹_𝑞(𝑎₁,…,𝑎_𝑝,𝛼;𝑏₁,…,𝑏_𝑞;𝛿⁻¹𝑦). (2.3)

The integral representations of the confluent hypergeometric function and the Gauss hypergeometric function are given as₁𝐹₁(𝑎;𝑐;𝑧)=Γ(𝑐)Γ(𝑎)Γ(𝑐−𝑎)¹₀𝑡^𝑎−1(1−𝑡)^{𝑐−𝑎−1}exp(𝑧𝑡)𝑑𝑡Re(𝑐)>Re(𝑎)>0,₂𝐹₁(𝑎,𝑏;𝑐;𝑧)=Γ(𝑐)Γ(𝑎)Γ(𝑐−𝑎)¹₀𝑡^𝑎−1(1−𝑡)^{𝑐−𝑎−1}(1−𝑧𝑡)^−𝑏𝑑𝑡,Re(𝑐)>Re(𝑎)>0,|arg(1−𝑧)|<𝜋, (2.4)(2.5)respectively. Note that the series expansions for ₁𝐹₁ and ₂𝐹₁ 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 obtain₂𝐹₁(𝑎,𝑏;𝑐;1)=Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ(𝑐−𝑎)Γ(𝑐−𝑏),Re(𝑐−𝑎−𝑏)>0,𝑐≠−1,−2,…. (2.6)The confluent hypergeometric function ₁𝐹₁(𝑎;𝑐;𝑧) satisfies Kummer's relation₁𝐹₁(𝑎;𝑐;−𝑧)=exp(−𝑧)₁𝐹₁(𝑐−𝑎;𝑐;𝑧). (2.7)The Lauricella hypergeometric function 𝐹_𝐷^(𝑚) in 𝑚 variables 𝑧₁,…,𝑧_𝑚 is defined by𝐹_𝐷^(𝑚)[𝑎,𝑏₁,…,𝑏_𝑚;𝑐;𝑧₁,…,𝑧_𝑚]=^∞_{𝑗1,…,𝑗𝑚=0}(𝑎)_{𝑗1+⋯+𝑗𝑚}(𝑏₁)_𝑗1⋯(𝑏_𝑚)_𝑗𝑚𝑧^𝑗1₁⋯𝑧^𝑗𝑚_𝑚(𝑐)_{𝑗1+⋯+𝑗𝑚}𝑗₁!⋯𝑗_𝑚!, (2.8) where max{|𝑧₁|,…,|𝑧_𝑚|}<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(𝑎)_𝑗(𝑐)_𝑗=Γ(𝑐)Γ(𝑎)Γ(𝑐−𝑎)¹₀𝑡^{𝑎+𝑗−1}(1−𝑡)^{𝑐−𝑎−1}𝑑𝑡,Re(𝑐)>Re(𝑎)>0, (2.9)for 𝑗=0,1,2,…, and^∞_𝑗𝑖=0(𝑏_𝑖)_𝑗𝑖(𝑡𝑧_𝑖)^𝑗𝑖𝑗_𝑖!=₁𝐹₀(𝑏_𝑖;𝑡𝑧_𝑖)=(1−𝑡𝑧_𝑖)^−𝑏𝑖,|𝑧_𝑖|<1,𝑖=1,…,𝑚, (2.10)in (2.8), an integral representation of 𝐹_𝐷^(𝑚) is given by𝐹_𝐷^(𝑚)[𝑎,𝑏₁,…,𝑏_𝑚;𝑐;𝑧₁,…,𝑧_𝑚]=Γ(𝑐)Γ(𝑎)Γ(𝑐−𝑎)¹₀𝑡^𝑎−1(1−𝑡)^{𝑐−𝑎−1}(1−𝑧₁𝑡)^−𝑏1⋯(1−𝑧_𝑚𝑡)^−𝑏𝑚𝑑𝑡. (2.11)Further, replacing (1−𝑧_𝑖𝑡)^−𝑏𝑖 by its equivalent gamma integral, namely,(1−𝑧_𝑖𝑡)^−𝑏𝑖=1Γ(𝑏_𝑖)^∞₀exp[−(1−𝑧_𝑖𝑡)𝑣_𝑖]𝑣^𝑏𝑖_𝑖⁻¹𝑑𝑣_𝑖,Re(𝑏_𝑖)>0, (2.12)for 𝑖=1,…,𝑚 and integrating out 𝑡, one obtains^𝑚_𝑖=1Γ(𝑏_𝑖)𝐹_𝐷^(𝑚)[𝑎,𝑏₁,…,𝑏_𝑚;𝑐;𝑧₁,…,𝑧_𝑚]=^∞₀⋯^∞₀exp(−^𝑚_𝑖=1𝑣_𝑖)^𝑚_𝑖=1𝑣^𝑏𝑖_𝑖1⁻¹𝐹₁(𝑎;𝑐;−^𝑚_𝑖=1𝑧_𝑖𝑣_𝑖)𝑑𝑣₁⋯𝑑𝑣_𝑚. (2.13)For further results and properties of this function the reader is referred to [10, 11].

Let 𝑓(⋅) be a continuous function and 𝛼_𝑖>0,𝑖=1,…,𝑛. The integral𝐷_𝑛(𝛼₁,…,𝛼_𝑛;𝑓)=^∞₀⋯^∞₀^𝑛_𝑖=1𝑥^𝛼𝑖_𝑖⁻¹𝑓(^𝑛_𝑖=1𝑥_𝑖)^𝑛_𝑖=1𝑑𝑥_𝑖 (2.14)is known as Liouville's integral. Substituting 𝑦_𝑖=𝑥_𝑖/𝑥,𝑖=1,…,𝑛−1 and 𝑥=∑^𝑛_𝑖=1𝑥_𝑖 with the Jacobian 𝐽(𝑥₁,…,𝑥_𝑛−1,𝑥_𝑛→𝑦₁,…,𝑦_𝑛−1,𝑥)=𝑥^𝑛−1, it is easy to see that𝐷_𝑛(𝛼₁,…,𝛼_𝑛∏;𝑓)=^𝑛𝑖=1Γ(𝛼_𝑖)Γ(∑^𝑛_𝑖=1𝛼_𝑖)^∞₀𝑥^{∑𝑛𝑖=1𝛼𝑖−1}𝑓(𝑥)𝑑𝑥. (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,…,𝑋_𝑛 are said to have a multivariate confluent hypergeometric function kind 1 distribution, denoted as (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1), if their joint pdf is given by𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)^𝑛_𝑖=1𝑥^𝜈𝑖_𝑖1⁻¹𝐹₁(𝛼;𝛽;−^𝑛_𝑖=1𝑥_𝑖),𝑥_𝑖>0,𝑖=1,…,𝑛, (3.1)where 𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽) is the normalizing constant.

The normalizing constant in (3.1) is given by{𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)}⁻¹=^∞₀⋯^∞₀^𝑛_𝑖=1𝑥^𝜈𝑖_𝑖1⁻¹𝐹₁(𝛼;𝛽;−^𝑛_𝑖=1𝑥_𝑖)^𝑛_𝑖=1𝑑𝑥_𝑖=∏^𝑛_𝑖=1Γ(𝜈_𝑖)Γ(∑^𝑛_𝑖=1𝜈_𝑖)^∞₀𝑥^{∑𝑛𝑖=1𝜈𝑖}₁⁻¹𝐹₁=∏(𝛼;𝛽;−𝑥)𝑑𝑥_𝑛𝑖=1Γ(𝜈_𝑖)Γ(∑^𝑛_𝑖=1𝜈_𝑖)^∞₀𝑥^{∑𝑛𝑖=1𝜈𝑖−1}exp(−𝑥)₁𝐹₁(𝛽−𝛼;𝛽;𝑥)𝑑𝑥, (3.2)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{𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)}⁻¹=∏^𝑛_𝑖=1Γ(𝜈_𝑖)Γ(𝛽)Γ(𝛼−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛼)Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖), (3.3)where 𝜈_𝑖>0,𝑖=1,…,𝑛,𝛼>∑^𝑛_𝑖=1𝜈_𝑖, and 𝛽>∑^𝑛_𝑖=1𝜈_𝑖. 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 result₁𝐹₁(𝑎;2𝑎;𝑧)=Γ(𝑎+1/2)exp(𝑧/2)(𝑧/4)^𝑎−1/2𝐼_𝑎−1/2(𝑧2), (3.4)where 𝐼_𝛿 is the modified Bessel function of the first kind, yield𝐶(𝜈₁,…,𝜈_𝑛;𝛼,2𝛼)4^𝛼−1/21Γ(𝛼+2)exp(−∑^𝑛_𝑖=1𝑥_𝑖/2)∏^𝑛_𝑖=1𝑥^𝜈𝑖_𝑖⁻¹(∑^𝑛_𝑖=1𝑥_𝑖)^𝛼−1/2×𝐼_𝛼−1/2(∑^𝑛_𝑖=1𝑥_𝑖2),𝑥_𝑖>0,𝑖=1,…,𝑛. (3.5)

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 𝑋 is said to have the gamma distribution with shape parameter 𝑎,𝑎>0, denoted as 𝑋∼Ga(𝑎), if its pdf is given byexp(−𝑥)𝑥^𝑎−1Γ(𝑎),𝑥>0. (3.6)

Definition 3.3. A random variable 𝑋 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 byΓ(𝑎+𝑏)𝑥Γ(𝑎)Γ(𝑏)^𝑎−1(1−𝑥)^𝑏−1,0<𝑥<1. (3.7)

Definition 3.4. A random variable 𝑋 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 byΓ(𝑎+𝑏)𝑥Γ(𝑎)Γ(𝑏)^𝑎−1(1+𝑥)^{−(𝑎+𝑏)},𝑥>0. (3.8)

If 𝑋∼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. Let 𝑌₁,…,𝑌_𝑛 and 𝑍 be independent, 𝑌_𝑖∼𝐺𝑎(𝑐_𝑖),𝑖=1,…,𝑛, and 𝑍∼𝐵1(𝑎,𝑏). Then, (𝑌₁/𝑍,…,𝑌_𝑛/𝑍)∼𝐶𝐻(𝑐₁,…,𝑐_𝑛;∑^𝑛_𝑖=1𝑐_𝑖+𝑎,∑^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏,kind1) with the pdfΓ(𝑎+𝑏)Γ(∑^𝑛_𝑖=1𝑐_𝑖+𝑎)∏^𝑛_𝑖=1Γ(𝑐_𝑖)Γ(𝑎)Γ(∑^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏)^𝑛_𝑖=1𝑥^𝑐𝑖_𝑖⁻¹×₁𝐹₁(^𝑛_𝑖=1𝑐_𝑖+𝑎;^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏;−^𝑛_𝑖=1𝑥_𝑖),𝑥_𝑖>0,𝑖=1,…,𝑛. (3.9)

Proof. The joint density of 𝑌₁,…,𝑌_𝑛 and 𝑍 is given byΓ(𝑎+𝑏)∏^𝑛_𝑖=1Γ(𝑐_𝑖)Γ(𝑎)Γ(𝑏)^𝑛_𝑖=1𝑦^𝑐𝑖_𝑖⁻¹exp(−^𝑛_𝑖=1𝑦_𝑖)𝑧^𝑎−1(1−𝑧)^𝑏−1, (3.10)where 0<𝑧<1,𝑦_𝑖>0,𝑖=1,…,𝑛. Transforming 𝑋_𝑖=𝑌_𝑖/𝑍,𝑖=1,…,𝑛 with the Jacobian 𝐽(𝑦₁,…,𝑦_𝑛,𝑧→𝑥₁,…,𝑥_𝑛,𝑧)=𝑧^𝑛 in (3.10) and integrating out 𝑧, we get the marginal density of (𝑋₁,…,𝑋_𝑛) asΓ(𝑎+𝑏)∏^𝑛_𝑖=1Γ(𝑐_𝑖)Γ(𝑎)Γ(𝑏)^𝑛_𝑖=1𝑥^𝑐𝑖_𝑖⁻¹¹₀exp(−𝑧^𝑛_𝑖=1𝑥_𝑖)𝑧^{∑𝑛𝑖=1𝑐𝑖+𝑎−1}(1−𝑧)^𝑏−1𝑑𝑧, (3.11)where 𝑥_𝑖>0,𝑖=1,…,𝑛. Now, evaluation of the above integral using (2.4) yields the desired result.

Corollary 3.6. Let 𝑌₁,…,𝑌_𝑛 and 𝑍 be independent, 𝑌_𝑖∼𝐺𝑎(𝑐_𝑖),𝑖=1,…,𝑛, and 𝑍∼𝐵1(𝑎,𝑏). Then(𝑌₁𝑌1−𝑍,…,^𝑛1−𝑍)∼CH(𝑐₁,…,𝑐_𝑛;^𝑛_𝑖=1𝑐_𝑖+𝑏,^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏,kind1). (3.12)

Corollary 3.7. Let 𝑌₁,…,𝑌_𝑛 and 𝑉 be independent, 𝑌_𝑖∼𝐺𝑎(𝑐_𝑖),𝑖=1,…,𝑛, and 𝑉∼𝐵2(𝑎,𝑏), then((1+𝑉)𝑌₁𝑉,…,(1+𝑉)𝑌_𝑛𝑉)∼CH(𝑐₁,…,𝑐_𝑛;^𝑛_𝑖=1𝑐_𝑖+𝑎,^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏,kind1),((1+𝑉)𝑌₁,…,(1+𝑉)𝑌_𝑛)∼CH(𝑐₁,…,𝑐_𝑛;^𝑛_𝑖=1𝑐_𝑖+𝑏,^𝑛_𝑖=1𝑐_𝑖+𝑎+𝑏,kind1). (3.13)

The Laplace transform of the multivariate confluent hypergeometric function kind 1 density (3.1) is given by𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)^∞₀⋯^∞₀exp(−^𝑛_𝑖=1𝑠_𝑖𝑥_𝑖)^𝑛_𝑖=1𝑥^𝜈𝑖_𝑖1⁻¹𝐹₁(𝛼;𝛽;−^𝑛_𝑖=1𝑥_𝑖)^𝑛_𝑖=1𝑑𝑥_𝑖, (3.14)where Re(𝑠_𝑖)>0,𝑖=1,…,𝑛. Now, rewriting ₁𝐹₁ by applying (2.7) and integrating 𝑥₁,…,𝑥_𝑚 using (2.13), the above expression is evaluated asΓ(𝛼)Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛽)Γ(𝛼−∑^𝑛_𝑖=1𝜈_𝑖)𝐹_𝐷^(𝑛)[𝛽−𝛼,𝜈₁,…,𝜈_𝑛1;𝛽;1+𝑠₁1,…,1+𝑠_𝑛]. (3.15)

The joint moments of 𝑋₁,…,𝑋_𝑛 are given by𝐸(𝑋^𝑟1₁⋯𝑋^𝑟𝑛_𝑛)=𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)^∞₀⋯^∞₀^𝑛_𝑖=1𝑥^{𝜈𝑖+𝑟𝑖}_𝑖1⁻¹𝐹₁(𝛼;𝛽;−^𝑛_𝑖=1𝑥_𝑖)^𝑛_𝑖=1𝑑𝑥_𝑖=𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)𝐶(∑^𝑛_𝑖=1𝜈_𝑖∏;𝛼,𝛽)^𝑛𝑖=1Γ(𝜈_𝑖+𝑟_𝑖)Γ[∑^𝑛_𝑖=1(𝜈_𝑖+𝑟_𝑖)]𝐸[𝑋^{∑𝑛𝑖=1𝑟𝑖}], (3.16)where 𝑋∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1). Note that if 𝑋∼CH(𝜈,𝛼,𝛽,kind1), then from [9], one gets 𝐸(𝑋^ℎ)=Γ(𝛽−𝜈)Γ(𝜈+ℎ)Γ(𝛼−𝜈−ℎ)Γ(𝜈)Γ(𝛼−𝜈)Γ(𝛽−𝜈−ℎ), (3.17)where Re(ℎ+𝜈)>0,Re(ℎ)<𝛼−𝜈, and Re(ℎ)<𝛽−𝜈. Now, computing 𝐸[𝑋^{∑𝑛𝑖=1𝑟𝑖}] using the above result, substituting for 𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽) and 𝐶(∑^𝑛_𝑖=1𝜈_𝑖;𝛼,𝛽) from (3.3) and simplifying the resulting expression, we obtain𝐸(𝑋^𝑟1₁⋯𝑋^𝑟𝑛_𝑛)=Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−∑^𝑛_𝑖=1𝑟_𝑖)∏^𝑛_𝑖=1Γ(𝜈_𝑖+𝑟_𝑖)∏^𝑛_𝑖=1Γ(𝜈_𝑖)Γ(𝛼−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−∑^𝑛_𝑖=1𝑟_𝑖), (3.18)where 𝛼>∑^𝑛_𝑖=1(𝜈_𝑖+𝑟_𝑖) and 𝛽>∑^𝑛_𝑖=1(𝜈_𝑖+𝑟_𝑖). Substituting appropriately in the above expression and using definitions of variance, covariance, and correlation coefficient, it is straightforward to show that𝐸(𝑋_𝑗𝜈)=^𝑗(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−1)𝛼−∑^𝑛_𝑖=1𝜈_𝑖,−1𝐸(𝑋²_𝑗𝜈)=^𝑗(𝜈_𝑗+1)(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−2)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖,−2)Var(𝑋_𝑗𝜈)=^𝑗(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)²(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−2)×[𝜈_𝑗(𝛽−𝛼)+(𝛽−^𝑛_𝑖=1𝜈_𝑖−2)(𝛼−^𝑛_𝑖=1𝜈_𝑖−1)],𝐸(𝑋_𝑗𝑋_𝑘𝜈)=^𝑗𝜈_𝑘(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−2)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖,−2)Cov(𝑋_𝑗,𝑋_𝑘𝜈)=^𝑗𝜈_𝑘(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−1)(𝛽−𝛼)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)²(𝛼−∑^𝑛_𝑖=1𝜈_𝑖,−2)Corr(𝑋_𝑗,𝑋_𝑘)=[(1+(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−2)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)𝜈_𝑗)(𝛽−𝛼)×(1+(𝛽−∑^𝑛_𝑖=1𝜈_𝑖−2)(𝛼−∑^𝑛_𝑖=1𝜈_𝑖−1)𝜈_𝑘(𝛽−𝛼))]^−1/2, (3.19) 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 of 𝑋₁,…,𝑋_𝑛 is multivariate confluent hypergeometric function kind 1, then the marginal distribution of any subset of 𝑋₁,…,𝑋_𝑛 is multivariate confluent hypergeometric function kind 1.

Theorem 4.1. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Then, for 1≤𝑠≤𝑛,(𝑋₁,…,𝑋_𝑠)∼CH(𝜈₁,…,𝜈_𝑠;𝛼−^𝑛_𝑖=𝑠+1𝜈_𝑖,𝛽−^𝑛_𝑖=𝑠+1𝜈_𝑖,kind1). (4.1)

Proof. Replacing ₁𝐹₁(𝛼;𝛽;−∑^𝑛_𝑖=1𝑥_𝑖) in (3.1) by its integral representation and integrating out 𝑥_𝑠+1,…,𝑥_𝑛, the marginal density of 𝑋₁,…,𝑋_𝑠 is derived as𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)Γ(𝛽)∏^𝑛_𝑖=𝑠+1Γ(𝜈_𝑖)Γ(𝛼)Γ(𝛽−𝛼)^𝑠∏_𝑖=1𝑥^𝜈𝑖_𝑖⁻¹∫¹₀𝑡^{𝛼−∑𝑛𝑖=𝑠+1𝜈𝑖−1}(1−𝑡)^{𝛽−𝛼−1}exp(−𝑡∑^𝑛_𝑖=1𝑥_𝑖)𝑑𝑡, (4.2)where 𝑥_𝑖>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,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1), then for any subset of variables (𝑋_𝑗1,…,𝑋_𝑗𝑖), it holds that (𝑋_𝑗1,…,𝑋_𝑗𝑖)∼CH(𝜈_𝑗1,…,𝜈_𝑗𝑖;𝛼+^𝑖_𝑘=1𝜈_𝑗𝑘−^𝑛_𝑘=1𝜈_𝑖,𝛽+^𝑖_𝑘=1𝜈_𝑗𝑘−^𝑛_𝑘=1𝜈_𝑖,kind1), (4.3)where 𝑗₁,…,𝑗_𝑖 are distinct integers with 1≤𝑗₁,…,𝑗_𝑖≤𝑛,1≤𝑖≤𝑛.

Corollary 4.3. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Then, for 𝑘=1,…,𝑛, 𝑋_𝑘∼CH(𝜈_𝑘,𝛼−∑^𝑛_{𝑖(≠𝑘)=1}𝜈_𝑖,𝛽−∑^𝑛_{𝑖(≠𝑘)=1}𝜈_𝑖,kind1).

Using Theorem 4.1, the conditional density of (𝑋_𝑠+1,…,𝑋_𝑛) given (𝑋₁,…,𝑋_𝑠) is obtained as𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)𝐶(𝜈₁,…,𝜈_𝑠;𝛼−∑^𝑛_𝑖=𝑠+1𝜈_𝑖,𝛽−∑^𝑛_𝑖=𝑠+1𝜈_𝑖)×∏^𝑛_𝑖=𝑠+1𝑥^𝜈𝑖_𝑖1⁻¹𝐹₁(𝛼;𝛽;−∑^𝑛_𝑖=1𝑥_𝑖)₁𝐹₁(𝛼−∑^𝑛_𝑖=𝑠+1𝜈_𝑖;𝛽−∑^𝑛_𝑖=𝑠+1𝜈_𝑖;−∑^𝑛_𝑖=1𝑥_𝑖), (4.4)where 0<𝑥_𝑖,𝑖=𝑠+1,…,𝑛.

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 𝑈₁,…,𝑈_𝑛 are said to have a Dirichlet type 1 distribution with parameters 𝜈₁,…,𝜈_𝑛;𝜈_𝑛+1, denoted by (𝑈₁,…,𝑈_𝑛)∼D1(𝜈₁,…,𝜈_𝑛;𝜈_𝑛+1), if their joint pdf is given byΓ(∑^𝑛+1_𝑖=1𝜈_𝑖)∏^𝑛+1_𝑖=1Γ(𝜈_𝑖)^𝑛_𝑖=1𝑢^𝜈𝑖_𝑖⁻¹(1−^𝑛_𝑖=1𝑢_𝑖)^{𝜈𝑛+1−1},0<𝑢_𝑖,𝑖=1,…,𝑛,^𝑛_𝑖=1𝑢_𝑖<1, (4.5)where 𝜈_𝑖>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 transformation 𝑉_𝑗=𝑈_𝑗/(1−∑^𝑛_𝑖=1𝑈_𝑖),𝑗=1,…,𝑛, in (4.5), the Dirichlet type 2 density, which is a multivariate generalization of beta type 2 density, is obtained asΓ(∑^𝑛+1_𝑖=1𝜈_𝑖)∏^𝑛+1_𝑖=1Γ(𝜈_𝑖)^𝑛_𝑖=1𝑢^𝜈𝑖_𝑖⁻¹(1+^𝑛_𝑖=1𝑢_𝑖)^{−∑𝑛+1𝑖=1𝜈𝑖},𝑣_𝑖>0,𝑖=1,…,𝑛. (4.6)We will write (𝑉₁,…,𝑉_𝑛)∼D2(𝜈₁,…,𝜈_𝑛;𝜈_𝑛+1) if the joint density of 𝑉₁,…,𝑉_𝑛 is given by (4.6).

Let 𝑛₁,…,𝑛_ℓ be nonnegative integers such that ∑^ℓ_𝑖=1𝑛_𝑖=𝑛 and define𝜈_(𝑖)=^𝑛∗𝑖_{𝑗=𝑛∗𝑖−1+1}𝜈_𝑗,𝑛^∗₀=0,𝑛^∗_𝑖=^𝑖_𝑗=1𝑛_𝑗,𝑖=1,…,ℓ. (4.7)

Theorem 4.5. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑍_𝑗=𝑋_𝑗/𝑋_(𝑖),𝑗=𝑛^∗_𝑖−1+1,…,𝑛^∗_𝑖−1 and 𝑋_(𝑖)=∑^𝑛∗𝑖_{𝑗=𝑛∗𝑖−1+1}𝑋_𝑗,𝑖=1,…,ℓ. Then,
(i)(𝑋₍₁₎,…,𝑋_(ℓ)) and (𝑍_𝑛^∗_𝑖−1+1,…,𝑍_𝑛^∗_𝑖−1),𝑖=1,…,ℓ, are independently distributed;(ii)(𝑍_𝑛^∗_𝑖−1+1,…,𝑍_𝑛^∗_𝑖−1)∼𝐷1(𝜈_𝑛^∗_𝑖−1+1,…,𝜈_𝑛^∗_𝑖−1;𝜈_𝑛^∗_𝑖),𝑖=1,…,ℓ;(iii)(𝑋₍₁₎,…,𝑋_(ℓ))∼CH(𝜈₍₁₎,…,𝜈_(ℓ);𝛼,𝛽,kind1).

Proof. Substituting 𝑥_(𝑖)=∑^𝑛∗𝑖_{𝑗=𝑛∗𝑖−1+1}𝑥_𝑗 and 𝑧_𝑗=𝑥_𝑗/𝑥_(𝑖),𝑗=𝑛^∗_𝑖−1+1,…,𝑛^∗_𝑖−1,𝑖=1,…,ℓ with the Jacobian𝐽(𝑥₁,…,𝑥_𝑛→𝑧₁,…,𝑧_𝑛1−1,𝑥₍₁₎,…,𝑧_𝑛^∗_ℓ−1+1,…,𝑧_𝑛−1,𝑥_(ℓ))=^ℓ_𝑖=1𝐽(𝑥_𝑛^∗_𝑖−1+1,…,𝑥_𝑛^∗_𝑖→𝑧_𝑛^∗_𝑖−1+1,…,𝑧_𝑛^∗_𝑖−1,𝑥_(𝑖))=^ℓ_𝑖=1𝑥^𝑛𝑖−1_(𝑖). (4.8)in the density of (𝑋₁,…,𝑋_𝑛) given by (3.1), we get the joint density of 𝑍_𝑛^∗_𝑖−1+1,…,𝑍_𝑛^∗_𝑖−1,𝑋_(𝑖),𝑖=1,…,ℓ as𝐶(𝜈₁,…,𝜈_𝑛;𝛼,𝛽)^ℓ_𝑖=1𝑥^𝜈(𝑖)₁⁻¹_(𝑖)𝐹₁(𝛼;𝛽;−^ℓ_𝑖=1𝑥_(𝑖))×^ℓ_𝑖=1[^{𝑛∗𝑖−1}_{𝑗=𝑛∗𝑖−1+1}𝑧^𝜈𝑗_𝑗⁻¹(1−^{𝑛∗𝑖−1}_{𝑗=𝑛∗𝑖−1+1}𝑧_𝑗)^{𝜈𝑛∗𝑖−1}], (4.9)where 𝑥_(𝑖)>0,𝑖=1,…,ℓ,𝑧_𝑗>0,𝑗=𝑛^∗_𝑖−1+1,…,𝑛^∗_𝑖−1,∑^{𝑛∗𝑖−1}_{𝑗=𝑛∗𝑖−1+1}𝑧_𝑗<1,𝑖=1,…,ℓ. From the factorization in (4.9), it is easy to see that (𝑋₍₁₎,…,𝑋_(ℓ)) and (𝑍_𝑛^∗_𝑖−1+1,…,𝑍_𝑛^∗_𝑖−1),𝑖=1,…,ℓ, are independently distributed. Further (𝑋₍₁₎,…,𝑋_(ℓ))∼CH(𝜈₍₁₎,…,𝜈_(ℓ);𝛼,𝛽,kind1) and (𝑍_𝑛^∗_𝑖−1+1,…,𝑍_𝑛^∗_𝑖−1)∼D1(𝜈_𝑛^∗_𝑖−1+1,…,𝜈_𝑛^∗_𝑖−1;𝜈_𝑛^∗_𝑖),𝑖=1,…,ℓ.

Corollary 4.6. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑍_𝑖=𝑋_𝑖/𝑍,𝑖=1,…,𝑛−1, and 𝑍=∑^𝑛_𝑗=1𝑋_𝑗. Then (𝑍₁,…,𝑍_𝑛−1) and 𝑍 are independent, (𝑍₁,…,𝑍_𝑛−1)∼𝐷1(𝜈₁,…,𝜈_𝑛−1;𝜈_𝑛) and 𝑍∼𝐶𝐻(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Corollary 4.7. If (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1), then ∑^𝑛_𝑗=1𝑋_𝑗 and ∑^𝑠_𝑖=1𝑋_𝑖/∑^𝑛_𝑖=1𝑋_𝑖 are independent. Further∑^𝑠_𝑖=1𝑋_𝑖∑^𝑛_𝑖=1𝑋_𝑖∼𝐵1(^𝑠_𝑖=1𝜈_𝑖,^𝑛_𝑖=𝑠+1𝜈_𝑖),𝑠<𝑛. (4.10)

Theorem 4.8. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑊_𝑗=𝑋_𝑗/𝑋_𝑛^∗_𝑖,𝑗=𝑛^∗_𝑖−1+1,…,𝑛^∗_𝑖−1 and 𝑋_(𝑖)=∑^𝑛∗𝑖_{𝑗=𝑛∗𝑖−1+1}𝑋_𝑗,𝑖=1,…,ℓ. Then,
(i)(𝑋₍₁₎,…,𝑋_(ℓ)) and (𝑊_𝑛^∗_𝑖−1+1,…,𝑊_𝑛^∗_𝑖−1),𝑖=1,…,ℓ, are independently distributed;(ii)(𝑊_𝑛^∗_𝑖−1+1,…,𝑊_𝑛^∗_𝑖−1)∼𝐷2(𝜈_𝑛^∗_𝑖−1+1,…,𝜈_𝑛^∗_𝑖−1;𝜈_𝑛^∗_𝑖),𝑖=1,…,ℓ;(iii)(𝑋₍₁₎,…,𝑋_(ℓ))∼CH(𝜈₍₁₎,…,𝜈_(ℓ);𝛼,𝛽,kind1).

Corollary 4.9. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑊_𝑖=𝑋_𝑖/𝑋_𝑛,𝑖=1,…,𝑛−1, and 𝑍=∑^𝑛_𝑗=1𝑋_𝑗. Then (𝑊₁,…,𝑊_𝑛−1) and 𝑍 are independent, (𝑊₁,…,𝑊_𝑛−1)∼𝐷2(𝜈₁,…,𝜈_𝑛−1;𝜈_𝑛) and 𝑍∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Corollary 4.10. If (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1), then ∑^𝑠_𝑖=1𝑋_𝑖/∑^𝑛_𝑖=𝑠+1𝑋_𝑖 and ∑^𝑛_𝑗=1𝑋_𝑗 are independent. Further∑^𝑠_𝑖=1𝑋_𝑖∑^𝑛_𝑖=𝑠+1𝑋_𝑖∼𝐵2(^𝑠_𝑖=1𝜈_𝑖,^𝑛_𝑖=𝑠+1𝜈_𝑖),𝑠<𝑛. (4.11)

In the following six theorems, we give several factorizations of the multivariate confluent hypergeometric function kind1 density.

Theorem 4.11. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑌_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑌_𝑖=∑^𝑖_𝑗=1𝑋_𝑗/∑^𝑖+1_𝑗=1𝑋_𝑗,𝑖=1,…,𝑛−1. Then, 𝑌₁,…,𝑌_𝑛 are independent, 𝑌_𝑖∼𝐵1(∑^𝑖_𝑗=1𝜈_𝑗,𝜈_𝑖+1),𝑖=1,…,𝑛−1, and 𝑌_𝑛∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Proof. Substituting 𝑥₁=𝑦_𝑛∏^𝑛−1_𝑖=1𝑦_𝑖,𝑥₂=𝑦_𝑛(1−𝑦₁)∏^𝑛−1_𝑖=2𝑦_𝑖,…,𝑥_𝑛−1=𝑦_𝑛(1−𝑦_𝑛−2)𝑦_𝑛−1 and 𝑥_𝑛=𝑦_𝑛(1−𝑦_𝑛−1) with the Jacobian 𝐽(𝑥₁,…,𝑥_𝑛→𝑦₁,…,𝑦_𝑛)=∏^𝑛_𝑖=2𝑦_𝑖^𝑖−1 in (3.1), we get the desired result.

Theorem 4.12. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑍_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑍_𝑖=𝑋_𝑖+1/∑^𝑖_𝑗=1𝑋_𝑗,𝑖=1,…,𝑛−1. Then 𝑍₁,…,𝑍_𝑛 are independent, 𝑍_𝑖∼𝐵2(𝜈_𝑖+1,∑^𝑖_𝑗=1𝜈_𝑗),𝑖=1,…,𝑛−1, and 𝑍_𝑛∼CH(∑^𝑛_𝑗=1𝜈_𝑗,𝛼,𝛽,kind1).

Proof. The desired result follows from Theorem 4.11 by noting that (1−𝑌_𝑖)/𝑌_𝑖∼B2(𝜈_𝑖+1,∑^𝑖_𝑗=1𝜈_𝑗).

Theorem 4.13. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑊_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑊_𝑖=∑^𝑖_𝑗=1𝑋_𝑗/𝑋_𝑖+1,𝑖=1,…,𝑛−1. Then 𝑊₁,…,𝑊_𝑛 are independent, 𝑊_𝑖∼𝐵2(∑^𝑖_𝑗=1𝜈_𝑗,𝜈_𝑖+1),𝑖=1,…,𝑛−1, and 𝑊_𝑛∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Proof. The result follows from Theorem 4.12 by noting that 1/𝑍_𝑖∼B2(∑^𝑖_𝑗=1𝜈_𝑗,𝜈_𝑖+1).

Theorem 4.14. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑌_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑌_𝑖=𝑋_𝑖/∑^𝑛_𝑗=𝑖𝑋_𝑗,𝑖=1,…,𝑛−1. Then 𝑌₁,…,𝑌_𝑛 are independent, 𝑌_𝑖∼𝐵1(𝜈_𝑖,∑^𝑛_𝑗=𝑖+1𝜈_𝑗),𝑖=1,…,𝑛−1, and 𝑌_𝑛∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Proof. Substituting 𝑥₁=𝑦_𝑛𝑦₁,𝑥₂=𝑦_𝑛𝑦₂(1−𝑦₁),…,𝑥_𝑛−1=𝑦_𝑛𝑦_𝑛−1(1−𝑦₁)⋯(1−𝑦_𝑛−2), and 𝑥_𝑛=𝑦_𝑛(1−𝑦₁)⋯(1−𝑦_𝑛−1) with the Jacobian 𝐽(𝑥₁,…,𝑥_𝑛→𝑦₁,…,𝑦_𝑛)=𝑦_𝑛^𝑛−1∏^𝑛−2_𝑖=1(1−𝑦_𝑖)^{𝑛−𝑖−1} in (3.1), we get the desired result.

Theorem 4.15. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑍_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑍_𝑖=𝑋_𝑖/∑^𝑛_𝑗=𝑖+1𝑋_𝑗,𝑖=1,…,𝑛−1. Then 𝑍₁,…,𝑍_𝑛 are independent, 𝑍_𝑖∼𝐵2(𝜈_𝑖,∑^𝑛_𝑗=𝑖+1𝜈_𝑗),𝑖=1,…,𝑛−1, and 𝑍_𝑛∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Proof. The result follows from Theorem 4.14 by noting that 𝑌_𝑖/(1−𝑌_𝑖)∼B2(𝜈_𝑖,∑^𝑛_𝑗=𝑖+1𝜈_𝑗).

Theorem 4.16. Let (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1). Define 𝑊_𝑛=∑^𝑛_𝑗=1𝑋_𝑗 and 𝑊_𝑖=∑^𝑛_𝑗=𝑖+1𝑋_𝑗/𝑋_𝑖,𝑖=1,…,𝑛−1. Then 𝑊₁,…,𝑊_𝑛 are independent, 𝑊_𝑖∼𝐵2(∑^𝑛_𝑗=𝑖+1𝜈_𝑗,𝜈_𝑖),𝑖=1,…,𝑛−1, and 𝑊_𝑛∼CH(∑^𝑛_𝑖=1𝜈_𝑖,𝛼,𝛽,kind1).

Proof. The result follows from Theorem 4.15 by noting that 1/𝑊_𝑖∼B2(∑^𝑛_𝑗=𝑖+1𝜈_𝑗,𝜈_𝑖).

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

Theorem 4.17. If (𝑋₁,…,𝑋_𝑛)∼CH(𝜈₁,…,𝜈_𝑛;𝛼,𝛽,kind1), then(𝑋₁𝛽𝑋,…,^𝑛𝛽)→−^𝑑(𝑊₁,…,𝑊_𝑛), (4.12)where (𝑊₁,…,𝑊_𝑛)∼𝐷2(𝜈₁,…,𝜈_𝑛;𝛼−∑^𝑛_𝑖=1𝜈_𝑖), and “→−^𝑑” denotes the convergence in distribution.

Proof. Substituting 𝑥_𝑖=𝛽𝑢_𝑖,𝑖=1,…,𝑛 with the Jacobian 𝐽(𝑥₁,…,𝑥_𝑛→𝑢₁,…,𝑢_𝑛)=𝛽^𝑛 in (3.1), the joint p.d.f. of 𝑈₁,…,𝑈_𝑛 is given by𝑔(𝑢₁,…,𝑢_𝑛)=Γ(𝛼)Γ(𝛼−∑^𝑛_𝑖=1𝜈_𝑖)(^𝑛_𝑖=1𝑢^𝜈𝑖_𝑖⁻¹Γ(𝜈_𝑖))Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛽)𝛽^{−∑𝑛𝑖=1𝜈𝑖1}𝐹₁(𝛼;𝛽;−𝛽^𝑛_𝑖=1𝑢_𝑖). (4.13)Now, using the resultslim_1𝛽→∞𝐹₁(𝛼;𝛽;−𝛽∑^𝑛_𝑖=1𝑢_𝑖)=₁𝐹₀(𝛼;−∑^𝑛_𝑖=1𝑢_𝑖)=(1+∑^𝑛_𝑖=1𝑢_𝑖)^−𝛼,lim_𝛽→∞Γ(𝛽−∑^𝑛_𝑖=1𝜈_𝑖)Γ(𝛽)𝛽^{−∑𝑛𝑖=1𝜈𝑖}=1 (4.14)it is easy to see thatlim_𝛽→∞𝐹(𝑢₁,…,𝑢_𝑛)=𝐺(𝑢₁,…,𝑢_𝑛), (4.15)where 𝐹(𝑢₁,…,𝑢_𝑛) is the joint cumulative distribution function (cdf) of (𝑋₁,…,𝑋_𝑛)/𝛽 and 𝐺(𝑢₁,…,𝑢_𝑛) is the joint cdf of Dirichlet type 2 variables with parameters (𝜈₁,…,𝜈_𝑛;𝛼−∑^𝑛_𝑖=1𝜈_𝑖).

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.