Natural Exponential Families and Generalized Hypergeometric Measures |
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Authors: | I-Li Lu Donald St P Richards |
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Institution: | 1. Applied Statistics, Phantom Works , The Boeing Company , Seattle , Washington , USA I-li.Lu@boeing.com;3. Department of Statistics , The Pennsylvania State University , University Park , Pennsylvania , USA |
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Abstract: | Let ν be a positive Borel measure on ?n and pFq(a1,…, ap; b1,…, bq; s) be a generalized hypergeometric series. We define a generalized hypergeometric measure, μp,q := pFq(a1,…, ap; b1,…, bq;ν), as a series of convolution powers of the measure ν, and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968
Kemp , A. W. ( 1968 ). A wide class of discrete distributions and the associated differential equations . Sankhyā, Ser. A 30 : 401 – 410 . Google Scholar]) family of distributions is an example of μp,q in which ν is a Dirac measure on ?. For the case in which ν is a Dirac measure on ?n, we relate μp,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994
Bar-Lev , S. K. ,
Bshouty , D. ,
Enis , P. ,
Letac , G. ,
Lu , I. ,
Richards , D. ( 1994 ). The diagonal natural exponential families on ? n and their classification . J. Theoret. Probab. 7 : 883 – 929 .Crossref] , Google Scholar]). For p < q, we show that certain measures μp,q can be expressed as the convolution of a sequence of independent multi-dimensional Bernoulli trials. For p = q, q + 1, we show that the measures μp,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures. |
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Keywords: | Binomial distribution Dufresne distribution Factorization theorems Moments Probability-generating function |
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