A Generalized Multivariate Gamma Distribution

Abstract

In this chapter, we introduce a multivariate gamma distribution whose marginals are finite mixtures of gamma distributions and correlation between any pair of variables is negative. Several of its properties such as joint moments, correlation coefficients, moment generating function, Rényi and Shannon entropies have been derived. Simulation study have been conducted to evaluate the performance of the maximum likelihood method.

Appendix

In this section, we give definitions and results that will be used in subsequent sections. Throughout this work we will use the Pochhammer symbol $(a)_{n}$ defined by $(a)_{n}=a(a+1)\cdots (a+n-1)=(a)_{n-1}(a+n-1)$ for $n=1,2,\ldots ,$ and $(a)_{0}=1$.

The fourth hypergeometric function of Lauricella, denoted by $F_{D}^{(n)}$, in n variables $z_{1},\ldots , z_{n}$ is defined by

$$\begin{aligned} F_{D}^{(n)} (a, b_{1},\ldots , b_{n};c; z_{1},\ldots , z_{n}) =\sum _{j_1, \ldots ,j_n =0}^{\infty }\frac{(a)_{j_1+\cdots +j_n}(b_{ 1})_{ j_{1}} \cdots (b_{n})_{ j_{n}} z_{1}^{j_{1}} \cdots z_{n}^{j_{n}}}{ (c )_{j_{1}+\cdots + j_{n}} j_{1}!\cdots j_{n}!}, \end{aligned}$$

where $|z_i|<1$, $i=1, \ldots , n$. An integral representation of $F_{D}^{(n)}$ in Exton [7, p. 49, Eq. (2.3.5)] is given as

$$\begin{aligned}{} & {} F_{D}^{(m)} (a, b_{1},\ldots , b_{m};c; z_{1},\ldots , z_{m})\nonumber \\{} & {} \quad = \frac{\Gamma (c)}{\prod _{i=1}^{n}\Gamma (b_i)\Gamma (c-\sum _{i=1}^{n}b_i) }\nonumber \\{} & {} \qquad \times \!\!\! \mathop {\int \!\cdots \!\int }_{\textstyle {\sum _{i=1}^{n} x_{i}<1\atop 0<x_{i},\, i=1, \ldots ,n}} \!\!\!\frac{\prod _{i=1}^{n}x_i^{b_i -1} \!\left( 1- \sum _{i=1}^{n}x_i\right) ^{c-\sum _{i=1}^{n} b_i -1}}{\left( 1- \sum _{i=1}^{n}z_i t_i\right) ^{a} } \textrm{d}x_1 \cdots \textrm{d}x_n. \end{aligned}$$

(15)

For further results and properties of this function the reader is referred to Exton [7] and Srivastava and Karlsson [32].

Let $f(\cdot )$ be a continuous function and $\alpha _{i} >0$, $i=1, \ldots ,n$. The integral

$$\begin{aligned} D_n (\alpha _1,\ldots ,\alpha _{n};f) = \int _{0}^{\infty }\cdots \int _{0}^{\infty } \prod _{i=1}^{n}x_i^{\alpha _{i} -1} f\left( \sum _{i=1}^{n}x_i\right) \prod _{i=1}^{n} \textrm{d}x_{i} \end{aligned}$$

is known as the Liouville-Dirichlet integral. Substituting $y_i=x_i/x,\ i=1,\ldots ,n-1$ and $x=\sum _{i=1}^n x_i$ with the Jacobian $J(x_{1},\ldots ,x_{n-1}, x_{n}\rightarrow y_{1},\ldots ,y_{n-1}, x ) =x^{n-1}$ it is easy to see that

$$\begin{aligned} D_n (\alpha _1,\ldots ,\alpha _{n};f) = \frac{\prod _{i=1}^{n}\Gamma (\alpha _{i})}{\Gamma (\sum _{i=1}^{n}\alpha _{i})}\int _{0}^{\infty } x^{\sum _{i=1}^{n}\alpha _{i} -1} f\left( x\right) \textrm{d}x. \end{aligned}$$

(16)

Finally, we define the beta type 1, beta type 2 and Dirichlet type 1 distributions. These definitions can be found in Wilks [34], Fang, Kotz and Ng [8], Johnson, Kotz and Balakrishnan [15], and Kotz, Balakrishnan and Johnson [16].

Definition 2

A random variable X is said to have the beta type I distribution with parameters (a, b), $a>0$, $b>0$, denoted as $X\sim \text {B1}(a,b)$, if its pdf is given by

$$\begin{aligned} \frac{ \Gamma (a+b) }{\Gamma (a) \Gamma (b)} x^{a-1} (1-x)^{b-1},\, 0<x<1. \end{aligned}$$

Definition 3

A random variable X is said to have the beta type II distribution with parameters (a, b), denoted as $X\sim \text {B2}(a,b)$, $a>0$, $b>0$, if its pdf is given by

$$\begin{aligned} \frac{ \Gamma (a+b) }{\Gamma (a) \Gamma (b)} x^{a-1}(1+x)^{-(a+b)},\, x>0. \end{aligned}$$

Definition 4

The random variables $U_1, \ldots , U_n$ are said to have a Dirichlet type 1 distribution with parameters $\alpha _1,\ldots ,\alpha _n$ and $\alpha _{n+1}$, denoted by $(U_1,\ldots ,U_n)\sim \text {D1}(\alpha _1,\ldots ,\alpha _n;\alpha _{n+1})$, if their joint pdf is given by

$$\begin{aligned}{} & {} \frac{\Gamma (\sum _{i=1}^{n+1}\alpha _i)}{\prod _{i=1}^{n+1}\Gamma (\alpha _i)} \prod _{i=1}^{n}u_i^{\alpha _i-1}\left( 1-\sum _{i=1}^n u_i\right) ^{\alpha _{n+1}-1},\nonumber \\{} & {} \qquad \qquad \qquad 0<u_i ,\, i=1,\ldots , n,\, \sum _{i=1}^n u_i <1, \end{aligned}$$

(17)

where $\alpha _i>0$, $i=1,\ldots ,n+1$.

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 $V_j= U_j/(1-\sum _{i=1}^{n}U_i)$, $j=1, \ldots , n$, in (17), the Dirichelt type 2 density, which is a multivariate generalization of beta type 2 density, is obtained as

$$\begin{aligned} \frac{\Gamma (\sum _{i=1}^{n+1}\alpha _i)}{\prod _{i=1}^{n+1}\Gamma (\alpha _i)} \prod _{i=1}^{n}u_i^{\alpha _i-1}\left( 1+\sum _{i=1}^n u_i\right) ^{-\sum _{i=1}^{n+1}\alpha _{i} }, \quad v_i >0,\, i=1,\ldots , n. \end{aligned}$$

(18)

We will write $(V_1,\ldots ,V_n)\sim \textrm{D2}(\alpha _1,\ldots ,\alpha _n;\alpha _{n+1})$ if the joint density of $V_1,\ldots ,V_n$ is given by (18).

The matrix variate generalizations of beta type 1, beta type 2 and Dirichlet type 1 distributions have been defined and studied extensively. For example, see Gupta and Nagar [11].

Definition 5

Multinomial Theorem: For a positive integer k and a non-negative integer m,

$$\begin{aligned} (z _1+ \cdots +z_m)^{k}=\sum _{k_1 +\cdots +k_m=k}\left( {\begin{array}{c}k\\ k_1, \ldots , k_m\end{array}}\right) z_{1}^{k_1} \cdots z_{m}^{k_m} , \end{aligned}$$

where

$$\begin{aligned} \left( {\begin{array}{c}k\\ k_1, \ldots , k_m\end{array}}\right) =\frac{k!}{k_1! \cdots k_m!}. \end{aligned}$$

The numbers appearing in the theorem are the multinomial coefficients. They can be expressed in numerous ways, including as a product of binomial coefficients of factorials:

$$\begin{aligned} {\displaystyle {k \atopwithdelims ()k_{1},k_{2},\ldots ,k_{m}}={\frac{k!}{k_{1}!\,k_{2}!\cdots k_{m}!}}={k_{1} \atopwithdelims ()k_{1}}{k_{1}+k_{2} \atopwithdelims ()k_{2}}\cdots {k_{1}+k_{2}+\cdots +k_{m} \atopwithdelims ()k_{m}}} \end{aligned}$$

Lemma 1

For $a_1>0, \ldots , a_m>0$ and $k\in \mathbb {N}$, we have

$$\begin{aligned} k! \sum _{k_1+\cdots +k_m=k} \frac{\left( a_1\right) _{k_1}\ldots \left( a_m\right) _{k_m}}{k_1!\cdots k_m!}= & {} (a_1+\cdots + a_m)_k\\= & {} \frac{\Gamma (a_1+\cdots + a_m+k)}{\Gamma (a_1+\cdots + a_m)}. \end{aligned}$$

Proof

Writing $(1-\theta )^{- (a_1+\cdots + a_m)}$ as $(1-\theta )^{- a_1} \cdots (1-\theta )^{- a_m}$ and using power series expansion, for $0<\theta <1$, we get

$$\begin{aligned} (1-\theta )^{- a_1} \cdots (1-\theta )^{- a_m}= & {} \sum _{k_1=0}^{\infty }\cdots \sum _{k_m=0}^{\infty }\frac{\left( a_1\right) _{k_1}\ldots \left( a_m\right) _{k_m}}{k_1!\cdots k_m!} \theta ^{k_1+\cdots +k_n}\\= & {} \sum _{k=0}^{\infty } \theta ^k \sum _{k_1+\cdots +k_m=k} \frac{\left( a_1\right) _{k_1}\ldots \left( a_m\right) _{k_m}}{k_1!\cdots k_m!} \end{aligned}$$

and

$$\begin{aligned} (1-\theta )^{- (a_1+\cdots + a_m)} = \sum _{k=0}^{\infty } \frac{(a_1+\cdots + a_m)_k}{k!} \theta ^k. \end{aligned}$$

Now, comparing coefficients of $\theta ^k$, we get the desired result.    $\square$

Lemma 2

Let

$$\begin{aligned} g(a_1, \ldots ,a_m; \beta , k) = \int _{0}^{\infty }\!\cdots \! \int _{0}^{\infty } \prod _{i=1}^{m}z_{i}^{a_{i}-1} \left( \sum _{i=1}^{m}z_{i}\right) ^k \exp \left( -\frac{1}{\beta } \sum _{i=1}^{m}z_{i}\right) dz_1\cdots dz_m, \end{aligned}$$

(19)

where $a_1>0, \ldots , a_m>0$ and $k\in \mathbb {N}$. Then

$$\begin{aligned} g(a_1, \ldots ,a_m; \beta , k) = \beta ^{\sum _{i=1}^{m}a_{i}+ k} \left[ \prod _{i=1}^{m} \Gamma (a_{i})\right] \left( a_1+\cdots + a_m\right) _k \end{aligned}$$

Proof

Expanding $\left( \sum _{i=1}^{m}z_{i}\right) ^k$ in (19) by using multinomial theorem and integrating $z_1,\ldots , z_m$, we obtain

$$\begin{aligned} g(a_1, \ldots ,a_m; \beta , k)= & {} \sum _{k_1 + \cdots + k_m=k} \left( {\begin{array}{c}k\\ k_1, \ldots , k_m\end{array}}\right) \prod _{i=1}^{m} \int _{0}^{\infty } z_{i}^{z_{i}+ k_i-1} \exp \left( -\frac{1}{\beta } z_{i}\right) dz_i\nonumber \\= & {} \beta ^{\sum _{i=1}^{m}\alpha _{i}+ k} \sum _{k_1+\cdots +k_m=k} \left( {\begin{array}{c}k\\ k_1, \ldots , k_m\end{array}}\right) \prod _{i=1}^{m} \Gamma (a_{i}+ k_i). \end{aligned}$$

Now, using Lemma 1, we get the desired result.    $\square$