### Abstract

Let 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, p,q:=pFq(a1, ap; b1, bq;), as a series of convolution powers of the measure , and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968) family of distributions is an example of p,q in which is a Dirac measure on . For the case in which is a Dirac measure on n, we relate p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994). For pq, we show that certain measures p,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 p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.

Original language | English (US) |
---|---|

Pages (from-to) | 2472-2487 |

Number of pages | 16 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 37 |

Issue number | 16 |

DOIs | |

State | Published - Jan 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

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*Communications in Statistics - Theory and Methods*, vol. 37, no. 16, pp. 2472-2487. https://doi.org/10.1080/03610920801968897

**Natural exponential families and generalized hypergeometric measures.** / Lu, I. Li; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Natural exponential families and generalized hypergeometric measures

AU - Lu, I. Li

AU - Richards, Donald

PY - 2008/1/1

Y1 - 2008/1/1

N2 - Let 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, p,q:=pFq(a1, ap; b1, bq;), as a series of convolution powers of the measure , and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968) family of distributions is an example of p,q in which is a Dirac measure on . For the case in which is a Dirac measure on n, we relate p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994). For pq, we show that certain measures p,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 p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.

AB - Let 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, p,q:=pFq(a1, ap; b1, bq;), as a series of convolution powers of the measure , and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968) family of distributions is an example of p,q in which is a Dirac measure on . For the case in which is a Dirac measure on n, we relate p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994). For pq, we show that certain measures p,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 p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.

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U2 - 10.1080/03610920801968897

DO - 10.1080/03610920801968897

M3 - Article

VL - 37

SP - 2472

EP - 2487

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 16

ER -