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Approximating gamma distributions by normalized negative binomial distributions

Part of: Special processes Limit theorems Approximations and expansions Distribution theory Integral transforms, operational calculus

Published online by Cambridge University Press:  14 July 2016

José A. Adell  and
Jesús De La Cal
José A. Adell*
Affiliation:
Universidad de Zaragoza
Jesús De La Cal*
Affiliation:
Universidad del País Vasco
*
∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, E-50009 Zaragoza, Spain.
∗∗ Postal address: Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain.
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Abstract

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

Keywords

UNIFORM DISTANCERATE OF CONVERGENCECHARGED MULTIPLICITY DISTRIBUTIONSBASKAKOV OPERATORGAMMA OPERATOR

MSC classification

Primary: 62E20: Asymptotic distribution theory 60F05: Central limit and other weak theorems
Secondary: 44A10: Laplace transform 41A99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 2 , June 1994 , pp. 391 - 400
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

Research supported by CAI-CONAI PCB0292 and by the University of the Basque Country.

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