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Exponential families of mixed Poisson distributions


  • Ferrari, A.
  • Letac, G.
  • Tourneret, J.-Y.


If I=(I1,...,Id) is a random variable on [0,[infinity])d with distribution [mu](d[lambda]1,...,d[lambda]d), the mixed Poisson distribution MP([mu]) on 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,[infinity])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[less-than-or-equals, slant]d, for some vectors v0,...,vq of [0,[infinity])d with disjoint supports and for independent standard real gamma random variables Y1,...,Yq.

Suggested Citation

  • Ferrari, A. & Letac, G. & Tourneret, J.-Y., 2007. "Exponential families of mixed Poisson distributions," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1283-1292, July.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:6:p:1283-1292

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    References listed on IDEAS

    1. Bobecka, Konstancja & Wesolowski, Jacek, 2004. "Multivariate Lukacs theorem," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 143-160, November.
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