Informatics and Applications
2017, Volume 11, Issue 3, pp 2-17
ANALOGS OF GLESER’S THEOREM FOR NEGATIVE BINOMIAL AND GENERALIZED GAMMA DISTRIBUTIONS AND SOME OF THEIR APPLICATIONS
Abstract
It is proved that the negative binomial distributions with the shape parameter less than one are mixed geometric distributions. The mixing distribution is written out explicitly. Thus, the similar result of L. Gleser, stating that the gamma distributions with the shape parameter less than one are mixed exponential distributions, is transferred to the discrete case. An analog of Gleser’s theorem is also proved for generalized gamma distributions.
For mixed binomial distributions related to the negative binomial laws with the shape parameter less than one, the case of a small probability of success is considered and an analog of the Poisson theorem is proved. The representation of the negative binomial distributions as mixed geometric laws is used to prove limit theorems for negative binomial random sums of independent identically distributed random variables, in particular, analogs of the law of large numbers and the central limit theorem. Both cases of light and heavy tails are considered.
The expressions for the moments of limit distributions are obtained. The obtained alternative equivalent mixture representations of the limit laws provide better understanding of how mixed probability (Bayesian) models are formed.
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[+] About this article
Title
ANALOGS OF GLESER’S THEOREM FOR NEGATIVE BINOMIAL AND GENERALIZED GAMMA DISTRIBUTIONS AND SOME OF THEIR APPLICATIONS
Journal
Informatics and Applications
2017, Volume 11, Issue 3, pp 2-17
Cover Date
2017-09-30
DOI
10.14357/19922264170301
Print ISSN
1992-2264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
negative binomial distribution; mixed geometric distribution; generalized gamma distribution; stable distribution; Laplace distribution; Mittag-Leffler distribution; Linnik distribution; mixed binomial distribution; Poisson theorem; random sum; law of large numbers; central limit theorem
Authors
V. Yu. Korolev  ,  , 
Author Affiliations
Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 1-52 Lenin- skiye Gory, Moscow 119991, GSP-1, Russian Federation
Institute of Informatics Problems, Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 44-2 Vavilov Str., Moscow 119333, Russian Federation
Hangzhou Dianzi University, Higher Education Zone, Hangzhou 310018, China
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