Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T17:11:07.388Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Baayen, R. H. (2001). Word Frequency Distributions. Kluwer, Dordretch.CrossRefGoogle Scholar
Brian, M. V. (1953). Species frequencies in random samples from animal populations. J. Animal Ecol. 22, 57–64.CrossRefGoogle Scholar
Böhning, D. and Kuhnert, R. (2006). Equivalence of truncated count mixture distributions and mixtures of truncated count distributions. Biometrics 62, 1207–1215.CrossRefGoogle ScholarPubMed
Courant, R. and John, F. (1965). Introduction to Calculus and Analysis, Vol. I. Springer, New York.Google Scholar
Engen, S. (1974). On species frequency models. Biometrika 61, 263–270.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol II, 2nd edn. John Wiley, New York.Google Scholar
Grandell, J. (1997). Mixed Poisson Processes (Monogr. Statist. Appl. Prob. 77). Chapman and Hall, London.CrossRefGoogle Scholar
Holmes, D. I. (1992). A stylometric analysis of Mormon scripture and related texts. J. R. Statist. Soc. A 155, 91–120.CrossRefGoogle Scholar
Johnson, N. L., Kemp, A. W. and Kotz, S. (2005). Univariate Discrete Distributions, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Mir, K. A. (2009). On size-biased negative binomial distribution and its use in zero-truncated cases. Measurement Sci. Rev. 9, 33–35.CrossRefGoogle Scholar
Ord, J. K. and Whitmore, G. A. (1986). The Poisson-inverse Gaussian distribution as a model for species abundance. Commun. Statist. Theory Meth. 15, 853–871.CrossRefGoogle Scholar
Patil, G. P. and Rao, C. R. (1978). Weighted distributions and size-biased sampling with applications to wildlife populations and human families. Biometrics 34, 179–189.CrossRefGoogle Scholar
Puig, P. and Valero, J. (2006). Count data distributions: some characterizations with applications. J. Amer. Statist. Assoc. 101, 332–340.CrossRefGoogle Scholar
Puig, X., Ginebra, J. and Pérez-Casany, M. (2009). Extended truncated inverse Gaussian–Poisson model. Statist. Modelling 9, 151–171.CrossRefGoogle Scholar
Puri, P. S. and Goldie, C. M. (1979). Poisson mixtures and quasi-infinite divisibility of distributions. J. Appl. Prob. 16, 138–153.CrossRefGoogle Scholar
Rao, C. R. (1965). On discrete distributions arising out of methods of ascertaiment. In Classical and Contagious Discrete Distributions, ed. Patil, G. P., Statistical Publishing Society, Calcutta, pp. 320–332.Google Scholar
Sichel, H. S. (1975). On a distribution law for words frequencies. J. Amer. Statist. Assoc. 70, 542–547.Google Scholar
Sichel, H. S. (1992). Anatomy of the generalized inverse Gaussian–Poisson distribution with special applications to bibliometric studies. Inf. Processing Manag. 28, 5–17.CrossRefGoogle Scholar
Sichel, H. S. (1997). Modelling species-abundance frequencies and species-individual functions with the generalized inverse Gaussian–Poisson distribution. S. Afr. Statist. J. 31, 13–37.Google Scholar
Valero, J., Pérez-Casany, M. and Ginebra, J. (2010). Extended truncated Tweedie–Poisson model. Submitted.Google Scholar