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Exponential families of mixed Poisson distributions

By A. Ferrari, G. Letac and J.-Y. Tourneret


AbstractIf I=(I1,…,Id) is a random variable on [0,∞)d with distribution μ(dλ1,…,dλd), the mixed Poisson distribution MP(μ) on Nd is the distribution of (N1(I1),…,Nd(Id)) where N1,…,Nd are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v0+v1Y1+⋯+vqYq for some q⩽d, for some vectors v0,…,vq of [0,∞)d with disjoint supports and for independent standard real gamma random variables Y1,…,Yq

Publisher: Elsevier Inc.
Year: 2007
DOI identifier: 10.1016/j.jmva.2006.03.005
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