Abstract
Let V i, i=1,..., k, be independent gamma random variables with shape αi, scale β, and location parameter γi, and consider the partial sums Z 1=V 1, Z 2=V 1+V 2,..., Z k=V 1+...+V k. 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Becker, P. J. and Roux, J. J. J. (1981). A bivariate extension of the gamma distribution, South African Statist. J., 15, 1–12.
Çinlar, E. (1975). Introduction to Stochastic Processes, Prentice Hall, New Jersey.
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.
Eagleson, G. K. (1964). Polynomial expansions of bivariate distributions, Ann. Math. Statist., 35, 1208–1215.
Freund, J. E. (1961). A bivariate extension of the exponential distribution, J. Amer. Statist. Assoc., 56, 971–977.
Gaver, D. P.Jr. (1970). Multivariate gamma distributions generated by mixture, Sankhyā Ser. A, 32, 123–126.
Ghirtis, G. C. (1967). Some problems of statistical inference relating to double gamma distribution, Trabajos de Estadistica, 18, 67–87.
Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics, Vol. 4, Houghton Mifflin, Boston.
Kibble, W. F. (1941). A two-variate gamma distribution, Sankhyā, 5, 137–150.
Kowalczyk, T. and Tyrcha, J. (1989). Multivariate gamma distributions-properties and shape estimation, Statistics, 20, 465–474.
Krishnaiah, P. R. and Rao, M. M. (1961). Remarks on a multivariate gamma distribution, Amer. Math. Monthly, 68, 342–346.
Krishnaiah, P. R., Hagis, P. and Steinberg, L. (1963). A note on the bivariate chi-distribution, SIAM Rev., 5, 140–144.
Lingappaiah, G. S. (1984). Bivariate gamma distribution as a life test model, Applikace Matematiky, 29, 182–188.
Mathai, A. M. and Moschopoulos, P. G. (1991). On a multivariate gamma, J. Multivariate Anal., 39, 135–153.
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).
Moran, P. A. P. (1967). Testing for correlation between non-negative variates, Biometrika, 54, 385–394.
Moran, P. A. P. (1969). Statistical inference with bivariate gamma distributions, Biometrika, 56, 627–634.
Moran, P. A. P. (1970). The methodology of rain making experiments, Review of the International Statistical Institute, 38, 105–115.
Sarmanov, I. O. (1970). Gamma correlation process and its properties, Dokl. Akad. Nauk SSSR, 191, 30–32 (in Russian).
Steel, S. J. and leRoux, N. J. (1987). A reparameterisation of a bivariate gamma extension, Comm. Statist. Theory Methods, 16, 293–305.
Wilks, S. S. (1962). Mathematical Statistics, Wiley, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Mathal, A.M., Moschopoulos, P.G. A form of multivariate gamma distribution. Ann Inst Stat Math 44, 97–106 (1992). https://doi.org/10.1007/BF00048672
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00048672