(Received July 30, 1990; revised February 14, 1991)
Abstract. Let Vi, i = 1, .... ,k, be independent gamma random variables with shape alphai, scale beta, and location parameter gammai, 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.