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On zero-truncating and mixing Poisson distributions

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

J. Valero ,
M. Pérez-Casany  and
J. Ginebra
J. Valero*
Affiliation:
Technical University of Catalonia
M. Pérez-Casany*
Affiliation:
Technical University of Catalonia
J. Ginebra*
Affiliation:
Technical University of Catalonia
*
∗ Postal address: Department of Applied Mathematics III, Technical University of Catalonia, C. Esteve Terradas 8, 08860 Castelldefels, Spain.
∗∗ Postal address: Department of Applied Mathematics II and Data Management group, Technical University of Catalonia, C. Jordi Girona 1-3, 08034 Barcelona, Spain. Email address: [email protected]
∗∗∗ Postal address: Department of Statistics and Operations Research, Technical University of Catalonia, Avda. Diagonal 647, 08028 Barcelona, Spain.
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Abstract

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The distributions that result from zero-truncating mixed Poisson (ZTMP) distributions and those obtained from mixing zero-truncated Poisson (MZTP) distributions are characterised based on their probability generating functions. One consequence is that every ZTMP distribution is an MZTP distribution, but not vice versa. These characterisations also indicate that the size-biased version of a Poisson mixture and, under certain regularity conditions, the shifted version of a Poisson mixture are neither ZTMP distributions nor MZTP distributions.

Keywords

Count variablePoisson mixturezero-truncated distributionprobability generating functionsize-biased versionshifted version

MSC classification

Primary: 60E05: Distributions
Secondary: 60E10: Characteristic functions; other transforms
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 4 , December 2010 , pp. 1013 - 1027
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Research partially supported by the Spanish Ministry of Education and Science and FEDER grants TIN2009-14560-C03-03, MTM2006-09920 and MTM2006-01477, and Generalitat de Catalunya through grant SGR-1187.

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