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A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution

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  • Rigby, R.A.
  • Stasinopoulos, D.M.
  • Akantziliotou, C.

Abstract

A variety of methods of modelling overdispersed count data are compared. The methods are classified into three main categories. The first category are ad hoc methods (i.e. pseudo-likelihood, (extended) quasi-likelihood, double exponential family distributions). The second category are discretized continuous distributions and the third category are observational level random effects models (i.e. mixture models comprising explicit and non-explicit continuous mixture models and finite mixture models). The main focus of the paper is a family of mixed Poisson distributions defined so that its mean [mu] is an explicit parameter of the distribution. This allows easier interpretation when [mu] is modelled using explanatory variables and provides a more orthogonal parameterization to ease model fitting. Specific three parameter distributions considered are the Sichel and Delaporte distributions. A new four parameter distribution, the Poisson-shifted generalized inverse Gaussian distribution is introduced, which includes the Sichel and Delaporte distributions as a special and a limiting case respectively. A general formula for the derivative of the likelihood with respect to [mu], applicable to the whole family of mixed Poisson distributions considered, is given. Within the framework introduced here all parameters of the distributions are modelled as parametric and/or nonparametric (smooth) functions of explanatory variables. This provides a very flexible way of modelling count data. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models.

Suggested Citation

  • Rigby, R.A. & Stasinopoulos, D.M. & Akantziliotou, C., 2008. "A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 381-393, December.
    • Handle: RePEc:eee:csdana:v:53:y:2008:i:2:p:381-393
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    References listed on IDEAS

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    1. Murray Aitkin, 1999. "A General Maximum Likelihood Analysis of Variance Components in Generalized Linear Models," Biometrics, The International Biometric Society, vol. 55(1), pages 117-128, March.
    2. R. A. Rigby & D. M. Stasinopoulos, 2005. "Generalized additive models for location, scale and shape," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(3), pages 507-554, June.
    3. J. K. Lindsey, 1999. "On the use of corrections for overdispersion," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 48(4), pages 553-561.
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    1. Moritz Berger & Gerhard Tutz, 2021. "Transition models for count data: a flexible alternative to fixed distribution models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(4), pages 1259-1283, October.
    2. Willmot, Gordon E. & Woo, Jae-Kyung, 2022. "Remarks on a generalized inverse Gaussian type integral with applications," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    3. Tzougas, George & Vrontos, Spyridon D. & Frangos, Nickolaos E., 2015. "Risk classification for claim counts and losses using regression models for location, scale and shape," LSE Research Online Documents on Economics 70921, London School of Economics and Political Science, LSE Library.
    4. Tzougas, George & Jeong, Himchan, 2021. "An expectation-maximization algorithm for the exponential-generalized inverse Gaussian regression model with varying dispersion and shape for modelling the aggregate claim amount," LSE Research Online Documents on Economics 108210, London School of Economics and Political Science, LSE Library.
    5. Farouk Mselmi, 2022. "Generalized linear model for subordinated Lévy processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(2), pages 772-801, June.
    6. Fatemeh Hassanzadeh & Iraj Kazemi, 2017. "Regression modeling of one-inflated positive count data," Statistical Papers, Springer, vol. 58(3), pages 791-809, September.
    7. Tzougas, George, 2020. "EM estimation for the Poisson-Inverse Gamma regression model with varying dispersion: an application to insurance ratemaking," LSE Research Online Documents on Economics 106539, London School of Economics and Political Science, LSE Library.
    8. Vilhelm Verendel, 2023. "Tracking artificial intelligence in climate inventions with patent data," Nature Climate Change, Nature, vol. 13(1), pages 40-47, January.
    9. José Rodríguez-Avi & María José Olmo-Jiménez, 2017. "A regression model for overdispersed data without too many zeros," Statistical Papers, Springer, vol. 58(3), pages 749-773, September.
    10. A. Baccini & L. Barabesi & M. Cioni & C. Pisani, 2014. "Crossing the hurdle: the determinants of individual scientific performance," Scientometrics, Springer;Akadémiai Kiadó, vol. 101(3), pages 2035-2062, December.
    11. Tzougas, George & Pignatelli di Cerchiara, Alice, 2021. "The multivariate mixed Negative Binomial regression model with an application to insurance a posteriori ratemaking," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 602-625.
    12. Liu, Yafen & He, Zhen & Shu, Lianjie & Wu, Zhang, 2009. "Statistical computation and analyses for attribute events," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3412-3425, July.
    13. Cordeiro, Gauss M. & Andrade, Marinho G. & de Castro, Mário, 2009. "Power series generalized nonlinear models," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1155-1166, February.
    14. Tzougas, George & Hong, Natalia & Ho, Ryan, 2022. "Mixed poisson regression models with varying dispersion arising from non-conjugate mixing distributions," LSE Research Online Documents on Economics 113616, London School of Economics and Political Science, LSE Library.
    15. Marcelo Bourguignon & Diego I. Gallardo & Rodrigo M. R. Medeiros, 2022. "A simple and useful regression model for underdispersed count data based on Bernoulli–Poisson convolution," Statistical Papers, Springer, vol. 63(3), pages 821-848, June.
    16. Rahma Abid & Célestin C. Kokonendji & Afif Masmoudi, 2021. "On Poisson-exponential-Tweedie models for ultra-overdispersed count data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(1), pages 1-23, March.
    17. Qiang Jia & Yong Zhang & Lei Cao, 2019. "Response of Anatidae Abundance to Environmental Factors in the Middle and Lower Yangtze River Floodplain, China," Sustainability, MDPI, vol. 11(23), pages 1-10, December.
    18. Rodríguez-Avi, J. & Conde-Sánchez, A. & Sáez-Castillo, A.J. & Olmo-Jiménez, M.J. & Martínez-Rodríguez, A.M., 2009. "A generalized Waring regression model for count data," Computational Statistics & Data Analysis, Elsevier, vol. 53(10), pages 3717-3725, August.
    19. Ben Omrane, Walid & Heinen, Andréas, 2010. "Public news announcements and quoting activity in the Euro/Dollar foreign exchange market," Computational Statistics & Data Analysis, Elsevier, vol. 54(11), pages 2419-2431, November.
    20. Wen, Li & Rogers, Kerrylee & Saintilan, Neil & Ling, Joanne, 2011. "The influences of climate and hydrology on population dynamics of waterbirds in the lower Murrumbidgee River floodplains in Southeast Australia: Implications for environmental water management," Ecological Modelling, Elsevier, vol. 222(1), pages 154-163.
    21. Baíllo, A. & Berrendero, J.R. & Cárcamo, J., 2009. "Tests for zero-inflation and overdispersion: A new approach based on the stochastic convex order," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2628-2639, May.
    22. George Tzougas & Himchan Jeong, 2021. "An Expectation-Maximization Algorithm for the Exponential-Generalized Inverse Gaussian Regression Model with Varying Dispersion and Shape for Modelling the Aggregate Claim Amount," Risks, MDPI, vol. 9(1), pages 1-17, January.

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