The real powers of the convolution of a negative binomial distribution and a Bernoulli distribution
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- by Gérard Letac, Dhafer Malouche and Stefan Maurer PDF
- Proc. Amer. Math. Soc. 130 (2002), 2107-2114 Request permission
Abstract:
For $c>0,$ this note computes essentially the set of $(x,y)$ in $[0,+\infty )^2$ such that the entire series in $z$ defined by $(1+z/c)^x(1-z)^{-y}$ has all its coefficients non-negative. If $X$ and $Y$ are independent random variables which have respectively Bernoulli and negative binomial distributions, denote by $\mu$ the distribution of $X+Y$. The above problem is equivalent to finding the set of $p>0$ such that $\mu ^{*p}$ exists; this set is a finite union of intervals and may be the first example of this type in the literature. This gives the final touch to the classification of the natural exponential families with variance functions of Babel type, i.e. of the form $aR(m)+(bm+c)\sqrt {R(m)}$, where $R$ is a polynomial with degree $\leq 2.$References
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Additional Information
- Gérard Letac
- Affiliation: Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 31062, Toulouse, France
- Email: letac@cict.fr
- Dhafer Malouche
- Affiliation: 24 Av. Mongi Slim, 1004 El Menzah V, Tunisie
- Email: dhafer_malouche@yahoo.fr
- Stefan Maurer
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22902
- Email: srm4x@virginia.edu
- Received by editor(s): May 1, 1998
- Received by editor(s) in revised form: November 4, 1998
- Published electronically: February 8, 2002
- Communicated by: Stanley Sawyer
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2107-2114
- MSC (1991): Primary 60E10; Secondary 33A65
- DOI: https://doi.org/10.1090/S0002-9939-02-05352-2
- MathSciNet review: 1896047