
Digital
Library of the European Council for Modelling and Simulation 
Title: 
Generalized Gamma Distributions As Mixed Exponential
Laws And Related Limit Theorems 
Authors: 
Victor Korolev, Andrey Gorshenin, Alexander Korchagin,
Alexander Zeifman 
Published in: 
(2017).ECMS 2017 Proceedings
Edited by: Zita Zoltay
Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes VidovicsDancs, János Péter Rádics European Council for Modeling and Simulation. doi:10.7148/2017 ISBN:
9780993244049/ ISBN:
9780993244056 (CD) 31st European Conference on Modelling and Simulation, Budapest, Hungary, May 23^{rd}
– May 26^{th}, 2017 
Citation
format: 
Victor
Korolev, Andrey Gorshenin, Alexander Korchagin,
Alexander Zeifman (2017). Generalized Gamma
Distributions As Mixed Exponential Laws And Related Limit Theorems, ECMS 2017
Proceedings Edited by: Zita Zoltay
Paprika, Péter Horák, Kata Váradi, Péter Tamás Zwierczyk, Ágnes VidovicsDancs, János Péter Rádics European Council
for Modeling and Simulation. doi:
10.7148/20170642 
DOI: 
https://doi.org/10.7148/20170642 
Abstract: 
A
theorem due to L. J. Gleser stating that a gamma
distribution with shape parameter no greater than one is a mixed exponential
distribution is extended to generalized gamma distributions introduced by E.
W. Stacy as a special family of lifetime distributions containing both gamma
distributions, exponential power and Weibull
distributions. It is shown that the mixing distribution is a scale mixture of
strictly stable laws concentrated on the nonnegative halfline.
As a corollary, the representation is obtained for the mixed Poisson
distribution with the generalized gamma mixing law as a mixed geometric
distribution. Limit theorems are proved establishing the convergence of the
distributions of statistics constructed from samples with random sizes
obeying the mixed Poisson distribution with the generalized gamma mixing law
including random sums to special normal mixtures. 
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