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
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Natural exponential families and generalized hypergeometric measures. / Lu, I. Li; Richards, Donald.
In: Communications in Statistics - Theory and Methods, Vol. 37, No. 16, 01.01.2008, p. 2472-2487.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 -