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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.

MSC:

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