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A FORM OF MULTIVARIATE GAMMA DISTRIBUTION

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A. M. MATHAI^{1} AND P. G. MOSCHOPOULOS^{2}

^{1} *Department of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6*

^{2} *Department of Mathematical Sciences, The University of Texas at El Paso,*

El Paso, TX 79968-0514, U.S.A.
(Received July 30, 1990; revised February 14, 1991)

**Abstract.**
Let *V*_{i}, *i* = 1, .... ,*k*, be independent gamma random
variables with shape *alpha*_{i}, scale *beta*, and location parameter
*gamma*_{i}, 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.

*Key words and phrases*:
Multivariate gamma model, cumulative
sums, moments, cumulants, multiple correlation, exact density, conditional
density.

**Source**
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