A form of multivariate gamma distribution

  • A. M. Mathal
  • P. G. Moschopoulos


Let Vi, i=1,..., k, be independent gamma random variables with shape αi, scale β, and location parameter γi, and consider the partial sums Z1=V1, Z2=V1+V2,..., Zk=V1+...+Vk. When the scale parameters are all equal, each partial sum is again distributed as gamma, and hence the joint distribution of the partial sums may be called a multivariate gamma. This distribution, whose marginals are positively correlated has several interesting properties and has potential applications in stochastic processes and reliability. In this paper we study this distribution as a multivariate extension of the three-parameter gamma and give several properties that relate to ratios and conditional distributions of partial sums. The general density, as well as special cases are considered.

Key words and phrases

Multivariate gamma model cumulative sums moments cumulants multiple correlation exact density conditional density 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Becker, P. J. and Roux, J. J. J. (1981). A bivariate extension of the gamma distribution, South African Statist. J., 15, 1–12.Google Scholar
  2. Çinlar, E. (1975). Introduction to Stochastic Processes, Prentice Hall, New Jersey.Google Scholar
  3. Dussauchoy, A. and Berland, R. (1974). A multivariate gamma type distribution whose marginal laws are gamma, and which has a property similar to a characteristic property of the normal case, Statistical Distributions in Scientific Works, Vol. 1 (eds. G. P.Patil, S.Kotz and J. K.Ord), 319–328, Reidel, Dordrecht.Google Scholar
  4. Eagleson, G. K. (1964). Polynomial expansions of bivariate distributions, Ann. Math. Statist., 35, 1208–1215.Google Scholar
  5. Freund, J. E. (1961). A bivariate extension of the exponential distribution, J. Amer. Statist. Assoc., 56, 971–977.Google Scholar
  6. Gaver, D. P.Jr. (1970). Multivariate gamma distributions generated by mixture, Sankhyā Ser. A, 32, 123–126.Google Scholar
  7. Ghirtis, G. C. (1967). Some problems of statistical inference relating to double gamma distribution, Trabajos de Estadistica, 18, 67–87.Google Scholar
  8. Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics, Vol. 4, Houghton Mifflin, Boston.Google Scholar
  9. Kibble, W. F. (1941). A two-variate gamma distribution, Sankhyā, 5, 137–150.Google Scholar
  10. Kowalczyk, T. and Tyrcha, J. (1989). Multivariate gamma distributions-properties and shape estimation, Statistics, 20, 465–474.Google Scholar
  11. Krishnaiah, P. R. and Rao, M. M. (1961). Remarks on a multivariate gamma distribution, Amer. Math. Monthly, 68, 342–346.Google Scholar
  12. Krishnaiah, P. R., Hagis, P. and Steinberg, L. (1963). A note on the bivariate chi-distribution, SIAM Rev., 5, 140–144.Google Scholar
  13. Lingappaiah, G. S. (1984). Bivariate gamma distribution as a life test model, Applikace Matematiky, 29, 182–188.Google Scholar
  14. Mathai, A. M. and Moschopoulos, P. G. (1991). On a multivariate gamma, J. Multivariate Anal., 39, 135–153.Google Scholar
  15. Miller, K. S., Bernstein, R. I. and Blumenson, L. E. (1958). Generalized Rayleigh processes, Quart. Appl Math., 16, 137–145 (Correction: ibid. (1963). 20, p. 395).Google Scholar
  16. Moran, P. A. P. (1967). Testing for correlation between non-negative variates, Biometrika, 54, 385–394.Google Scholar
  17. Moran, P. A. P. (1969). Statistical inference with bivariate gamma distributions, Biometrika, 56, 627–634.Google Scholar
  18. Moran, P. A. P. (1970). The methodology of rain making experiments, Review of the International Statistical Institute, 38, 105–115.Google Scholar
  19. Sarmanov, I. O. (1970). Gamma correlation process and its properties, Dokl. Akad. Nauk SSSR, 191, 30–32 (in Russian).Google Scholar
  20. Steel, S. J. and leRoux, N. J. (1987). A reparameterisation of a bivariate gamma extension, Comm. Statist. Theory Methods, 16, 293–305.Google Scholar
  21. Wilks, S. S. (1962). Mathematical Statistics, Wiley, New York.Google Scholar

Copyright information

© The Institute of Statistical Mathematics 1992

Authors and Affiliations

  • A. M. Mathal
    • 1
  • P. G. Moschopoulos
    • 2
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.Department of Mathematical SciencesThe University of Texas at El PasoEl PasoU.S.A.

Personalised recommendations