A form of multivariate gamma distribution

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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.