A form of multivariate gamma distribution. (English) Zbl 0760.62049

Summary: Let \(V_ i\), \(i=1,\dots,k\), be independent gamma random variables with shape \(\alpha_ i\), scale \(\beta\), and location parameter \(\gamma_ i\), and consider the partial sums \(Z_ 1=V_ 1,Z_ 2=V_ 1+V_ 2,\dots,Z_ k=V_ 1+\dots+V_ k\). 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. 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.


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H10 Multivariate distribution of statistics
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