Skip to main content

A Generalized Multivariate Gamma Distribution

  • Chapter
  • First Online:
Innovations in Multivariate Statistical Modeling


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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. Aksoy, H. (2000). Use of gamma distribution in hydrological analysis. Turkish Journal of Engineering and Environmental Sciences, 24, 419–428.

    Google Scholar 

  2. Arnold, B. C., Castillo, E., & Sarabia, J. M. (1999). Conditional specification of statistical models. New York: Springer.

    MATH  Google Scholar 

  3. Balakrishnan, N., & C.-D, L. (2009). Continuous bivariate distributions (2nd ed.). Dordrecht: Springer.

    Google Scholar 

  4. Balakrishnan, N., & Ristić, M. M. (2016). Multivariate families of gamma-generated distributions with finite or infinite support above or below the diagonal. Journal of Multivariate Analysis, 143, 194–207.

    Article  MathSciNet  MATH  Google Scholar 

  5. Carpenter, M., & Diawara, N. (2007). A multivariate gamma distribution and its characterizations. American Journal of Mathematical and Management Sciences, 27(3–4), 499–507.

    Article  MathSciNet  MATH  Google Scholar 

  6. Dussauchoy, A., & Berland, R. (1975). A multivariate gamma type distribution whose marginal laws are gamma, and which has a property similar to a characteristic property of the normal case. In G. P. Patil, S. Kotz & J. K. Ord (Eds.), Statistical Distributions in Scientific Work, Vol. 1: Models and Structures, (pp. 319–328).

    Google Scholar 

  7. Exton, H. (1976). Multiple hypergeometric functions and applications. Chichester: Ellis Horwood.

    MATH  Google Scholar 

  8. Fang, K. T. D., Kotz, S., & Ng, K. W., Symmetric multivariate and realated distributions. London, New York: Chapman and Hall.

    Google Scholar 

  9. Furman, E. (2008). On a multivariate gamma distribution. Statistics and Probability Letters, 78(15), 2353–2360.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gaver, D. P., Jr. (1970). Multivariate gamma distributions generated by mixture. Sankhya: The Indian Journal of Statistics Series A, 32(1), 123–126.

    MATH  Google Scholar 

  11. Gupta, A. K., & Nagar, D. K. (2000). Matrix variate distributions. Boca Raton: Chapman & Hall/CRC.

    MATH  Google Scholar 

  12. Gupta, A. K., & Song, D. (1996). Generalized Liouville distribution. Computters & Mathematics with Applications, 32(2), 103–109.

    Article  MathSciNet  MATH  Google Scholar 

  13. Gupta, R. D., & Richards, D. S. P. (2001). The history of Dirichlet and Liouville distributions. International Statistical Review, 69(3), 433–446.

    Google Scholar 

  14. Hutchinson, T. P., & Lai, C. D. (1991). The engineering statistician’s guide to continuous bivariate distributions. Adelaide: Rumsby Scientific Publishing.

    Google Scholar 

  15. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (Vol. 2, 2nd Edn.). New York: Wiley.

    Google Scholar 

  16. Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Continuous multivariate distributions-1 (2nd ed.). New York: Wiley.

    Book  MATH  Google Scholar 

  17. Kowalczyk, T., & Tyrcha, J. (1989). Multivariate gamma distributions-properties and shape estimation. Statistics: A Journal of Theoretical and Applied Statistics, 20(3) , 465–474.

    Google Scholar 

  18. Krishnaiah, P. R., & Rao, M. M. (1961). Remarks on a multivariate gamma distribution. The American Mathematical Monthly, 68(4), 342–346.

    Article  MathSciNet  MATH  Google Scholar 

  19. Marcus, M. (2014). Multivariate gamma distributions. Electronic Communications in Probability, 19, 1–10.

    Article  MathSciNet  MATH  Google Scholar 

  20. Mathai, A. M., & Moschopoulos, P. G. (1991). On a multivariate gamma. Journal of Multivariate Analysis, 39(1), 135–153.

    Article  MathSciNet  MATH  Google Scholar 

  21. Mathai, A. M., & Moschopoulos, P. G. (1992). A form of multivariate gamma distribution. Annals of the Institute of Statistical Mathematics, 44(1), 97–106.

    Article  MathSciNet  MATH  Google Scholar 

  22. Nadarajah, S., & Zografos, K. (2005). Expressions for Rényi and Shannon entropies for bivariate distributions. Information Sciences, 170(2–4), 173–189.

    Google Scholar 

  23. Peppas, K. P., Alexandropoulos, G. C., Datsikas, C. K., & Lazarakis, F. I. (2011). Multivariate gamma-gamma distribution with exponential correlation and its applications in radio frequency and optical wireless communications. IET Microwaves Antennas & Propagation, 5(3), 364–371.

    Article  Google Scholar 

  24. Rafiei, M., Iranmanesh, A., & Nagar, D. K. (2020). A bivariate gamma distribution whose marginals are finite mixtures of gamma distributions. Statistics, Optimization & Information Computing, 8(4), 950–971.

    Article  MathSciNet  Google Scholar 

  25. Rényi, A. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. I, pp. 547–561). Berkeley, CA: University of California Press.

    Google Scholar 

  26. Robson, J. G., & Troy, J. B. (1987). Nature of the maintained discharge of \(Q\), \(X\), and \(Y\) retinal ganglion cells of the cat. Journal of the Optical Society of America, A, 4, 2301–2307.

    Article  Google Scholar 

  27. Royen, T. (2007). Integral representations and approximations for multivariate gamma distributions. Annals of the Institute of Statistical Mathematics, 59(3), 499–513.

    Article  MathSciNet  MATH  Google Scholar 

  28. Semenikhine, V., Furman, E., & Su, J. (2018). On a multiplicative multivariate gamma distribution with applications in insurance. Risks, 6(3), 79.

    Google Scholar 

  29. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(379–423), 623–656.

    Article  MathSciNet  MATH  Google Scholar 

  30. Sivazlian, B. D. (1981). On a multivariate extension of the gamma and beta distributions. Siam Journal on Applied Mathematics, 41(2), 205–209.

    Google Scholar 

  31. Song, D., & Gupta, A. K. (1997). Properties of generalized Liouville distribution. Random Operators and Stochastic Equations, 5(4), 337–348.

    Google Scholar 

  32. Srivastava, H. M., & Karlsson, P. W. (1985). Multiple gaussian hypergeometric series. New York: Wiley.

    MATH  Google Scholar 

  33. Vaidyanathan, V. S., & Lakshmi, R. V. (2015). Parameter estimation in multivariate gamma distribution. Statistics, Optimization and Information Computing, 3, 147–159.

    Article  MathSciNet  Google Scholar 

  34. Wilks, S. S. (1962). Mathematical Statistics. New York: Wiley.

    MATH  Google Scholar 

  35. Zografos, K. (1999). On maximum entropy characterization of Pearson’s type II and VII multivariate distributions. Journal of Multivariate Analysis, 71(1), 67–75.

    Article  MathSciNet  MATH  Google Scholar 

  36. Zografos, K., & Nadarajah, S. (2005). Expressions for Rényi and Shannon entropies for multivariate distributions. Statistics & Probability Letters, 71(1), 71–84.

    Article  MathSciNet  MATH  Google Scholar 

Download references


Authors are grateful to the worthy reviewers for their constructive and helpful comments and suggestions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Anis Iranmanesh .

Editor information

Editors and Affiliations



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

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

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

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

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 (ab), \(a>0\), \(b>0\), 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 (ab), 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

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

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

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

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,

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


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

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

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


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

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


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

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

Lemma 2


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

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

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


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.    \(\square \)

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Iranmanesh, A., Rafiei, M., Nagar, D.K. (2022). A Generalized Multivariate Gamma Distribution. In: Bekker, A., Ferreira, J.T., Arashi, M., Chen, DG. (eds) Innovations in Multivariate Statistical Modeling. Emerging Topics in Statistics and Biostatistics . Springer, Cham.

Download citation

Publish with us

Policies and ethics