A FORM OF MULTIVARIATE GAMMA DISTRIBUTION

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

1Department of Mathematics and Statistics, McGill University, Montreal, Canada H3A 2K6

2Department 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 α_{i}, scale β, and location parameter γ_{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.