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Extended Truncated Tweedie-Poisson Model

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Abstract

It has been argued that by truncating the sample space of the negative binomial and of the inverse Gaussian-Poisson mixture models at zero, one is allowed to extend the parameter space of the model. Here that is proved to be the case for the more general three parameter Tweedie-Poisson mixture model. It is also proved that the distributions in the extended part of the parameter space are not the zero truncation of mixed Poisson distributions and that, other than under the negative binomial, they are not mixtures of zero truncated Poisson distributions either. By extending the parameter space one can improve the fit when the frequency of one is larger and the right tail is heavier than is allowed by the unextended model. Extending the model also allows one to use the maximum likelihood based inference tools when the m.l.e. does not exist under the unextended model. The extended model is proved useful in the analysis of frequency count data.

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References

  • Aalen OO (1992) Modelling heterogeneity in survival analysis by the compound Poisson distribution. Ann Appl Probab 4:951–972

    Article  MathSciNet  Google Scholar 

  • Baayen H (2001) Word frequency distributions. Kluwer, Dordretch

    Book  MATH  Google Scholar 

  • Bar-Lev SK, Enis P (1986) Reproducibility and natural exponential families with power variance functions. Ann Stat 4:1507–1522

    Article  MathSciNet  Google Scholar 

  • Böhning D, Kuhnert R (2006) Equivalence of truncated count mixture distributions and mixtures of truncated count distributions. Biometrics 62:1207–1215

    Article  MathSciNet  MATH  Google Scholar 

  • Dunn PK, Smyth GK (2008) Evaluation of Tweedie exponential dispersion model densities by Fourier inversion. Stat Comput 18:73–86

    Article  MathSciNet  Google Scholar 

  • El-Shaarawi AH, Zhu R, Joe H (2010) Modelling species abundance using the Poisson-Tweedie family. Environmetrics. doi:10.1002/env.1036

    Google Scholar 

  • Engen S (1974) On species frequency models. Biometrika 61:263–270

    Article  MathSciNet  MATH  Google Scholar 

  • Engen S (1978) Stochastic abundance models: with emphasis on biological communities and species diversity. Chapman Hall, London

    MATH  Google Scholar 

  • Fisher RJ, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample from an animal population. J Anim Ecol 12:42–58

    Article  Google Scholar 

  • Gerber HU (1991) From the generalized gamma to the generalized negative binomial distribution. Insurance: Math Econ 10:303–309

    Article  Google Scholar 

  • Ginebra J, Puig X (2010) On the measure and the estimation of evenness and diversity. Comput Stat Data Anal 54:2187–2201

    Article  MathSciNet  Google Scholar 

  • Good IJ (1953) The population frequencies of species and the estimation of population parameters. Biometrika 40:237–264

    MathSciNet  MATH  Google Scholar 

  • Griffiths DA (1973) Maximum likelihood estimation for the beta-binomial distribuiton and an application to the household distribution of the total of cases of a disease. Biometrics 29:637–648

    Article  Google Scholar 

  • Holmes DI (1992) A stylometric analysis of mormon scripture and related texts. J R Stat Soc Ser A 155:91–120

    Article  Google Scholar 

  • Hougaard P (1986a) Survival models for heterogeneous populations derived from stable distributions. Biometrika 73:387–396

    Article  MathSciNet  MATH  Google Scholar 

  • Hougaard P (1986b) A class of multivariate failure time distributions. Biometrika 73:671–678

    MathSciNet  MATH  Google Scholar 

  • Hougaard P (1987) Modelling multivariate survival. Scand J Statist 14:291–304

    MathSciNet  MATH  Google Scholar 

  • Hougaard P, Lee M-LT, Whitmore GA (1997) Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes. Biomerics 53:1225–1238

    Article  MathSciNet  MATH  Google Scholar 

  • Janzen DH (1973) Sweep samples of tropical foliage insects: description of study sites, with data on species abundances and size distributions. Ecology 54:659–686

    Article  Google Scholar 

  • Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions, 3rd edn. Wiley, New York

    Book  MATH  Google Scholar 

  • Jorgensen B (1987) Exponential dispersion models. J R Stat Soc Ser B 49:127–162

    Google Scholar 

  • Jorgensen B (1997) The theory of dispersion models. Chapman Hall, London

    Google Scholar 

  • Katti SK, Gurland J (1961) The Poisson Pascal distribution. Biometrics 17:527–538

    Article  MATH  Google Scholar 

  • Kokonendji CC, Dossou-Gbete S, Demétrio CGB (2004) Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demetrio classes. SORT 28:201–214

    MathSciNet  MATH  Google Scholar 

  • Ord JK, Whitmore G (1986) The Poisson-inverse Gaussian distribution as a model for species abundance. Commun Stat Theory Methods 15:853–871

    Article  MathSciNet  MATH  Google Scholar 

  • Panjer H (1981) Recursive evaluation of a family of compound distributions. ASTIN Bull 12:22–26

    MathSciNet  Google Scholar 

  • Puig P, Valero J (2006) Count data distributions: Some characterizations with applications. J Am Stat Assoc 101:332–340

    Article  MathSciNet  MATH  Google Scholar 

  • Puig X, Ginebra J, Pérez-Casany M (2009) Extended truncated inverse Gaussian-Poisson model. Stat Model 9:151–171

    Article  MathSciNet  Google Scholar 

  • Puig X, Ginebra J, Font M (2010) The Sichel model and the mixing and truncation order. J Appl Stat 37:1585–1603

    Article  MathSciNet  Google Scholar 

  • Puri PS, Goldie CM (1979) Poisson mixtures and quasi-infinite divisibility of distributions. J App Prob 16:138–153

    Article  MathSciNet  MATH  Google Scholar 

  • Riba A, Ginebra J (2006) Diversity of vocabulary and homogeneity of literary style. J Appl Stat 33:729–741

    Article  MathSciNet  MATH  Google Scholar 

  • Sichel HS (1975) On a distribution law for words frequencies. J Am Stat Assoc 70:542–547

    Google Scholar 

  • Sichel HS (1997) Modelling species-abundance frequencies and species-individual functions with the generalized inverse Gaussian-Poisson distribution. S Afr Stat J 31:13–37

    MATH  Google Scholar 

  • Tweedie MCK (1984) An index which distinguishes between some important exponential families. In: Ghosh JK, Roy J (eds) Statistics: applications and new directions, proceedings of the Indian Statistical Institute Golden Jubilee International Conference, pp 579–604. Indian Statistical Institute, Calcutta

    Google Scholar 

  • Valero J, Pérez-Casany M, Ginebra J (2010a) On zero truncating and mixing Poisson distributions. Adv Appl Probab 42:1013–1027

    Article  MATH  Google Scholar 

  • Valero J, Ginebra J, Pérez-Casany M (2010b) Extended truncated Tweedie-Poisson model. Technical Report # 965, Centre de Recerca Matemàtica, Universitat Autonoma de Barcelona

  • Willmot GE (1988) A remark on the Poisson-Pascal and some other contagious distributions. Stat Prob lett 7:217–220

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu R, Joe H (2009) Modelling heavy-tailed count data using a generalized Poisson-inverse Gaussian family. Stat Prob lett 70:1695–1703

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics III, Technical University of Catalonia, Barcelona, Spain

    Jordi Valero

  2. Dep. d’Estadística, E.T.S.E.I.B., Universitat Politècnica de Catalunya, Avgda. Diagonal 647, 6a Planta, 08028, Barcelona, Spain

    Josep Ginebra

  3. Department of Statistics and O.R., Technical University of Catalonia, Barcelona, Spain

    Josep Ginebra

  4. Department of Applied Mathematics II and Dama-UPC, Technical University of Catalonia, Barcelona, Spain

    Marta Pérez-Casany

Authors
  1. Jordi Valero
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  2. Josep Ginebra
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  3. Marta Pérez-Casany
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Correspondence to Josep Ginebra.

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Valero, J., Ginebra, J. & Pérez-Casany, M. Extended Truncated Tweedie-Poisson Model. Methodol Comput Appl Probab 14, 811–829 (2012). https://doi.org/10.1007/s11009-012-9277-8

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  • DOI: https://doi.org/10.1007/s11009-012-9277-8

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