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A simple algorithm for computing the probabilities of count models based on pure birth processes

Author

Listed:
  • Mongkol Hunkrajok

    (Independent Researcher)

  • Wanrudee Skulpakdee

    (National Institute of Development Administration)

Abstract

Recently, non-monotonic rate sequences of pure birth processes have been the focus of much attention in the analysis of count data due to their ability to provide a combination of over-, under-, and equidispersed distributions without the need to reuse covariates (traditional methods). They also permit the modeling of excess counts, a frequent issue arising when using count models based on monotonic rate sequences such as the Poisson, gamma, Weibull, Conway-Maxwell-Poisson (CMP), Faddy (1997), etc. Matrix-exponential approaches have always been used for computing the probabilities for count models based on pure birth processes, although none have been proposed for them as a specific algorithm. It is intractable to calculate these pure birth probabilities numerically in an analytic form because severe numerical cancellations may occur. However, we circumvent this difficulty by exploiting a Taylor series expansion, and then a new analytic form is derived. We developed a simple algorithm for efficiently implementing the new formula and conducted numerical experiments to study the efficiency and accuracy of the developed algorithm. The results indicate that this new approach is faster and more accurate than the matrix-exponential methods.

Suggested Citation

  • Mongkol Hunkrajok & Wanrudee Skulpakdee, 2025. "A simple algorithm for computing the probabilities of count models based on pure birth processes," Computational Statistics, Springer, vol. 40(1), pages 249-272, January.
  • Handle: RePEc:spr:compst:v:40:y:2025:i:1:d:10.1007_s00180-024-01491-4
    DOI: 10.1007/s00180-024-01491-4
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    References listed on IDEAS

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    1. Alina Peluso & Veronica Vinciotti & Keming Yu, 2019. "Discrete Weibull generalized additive model: an application to count fertility data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 68(3), pages 565-583, April.
    2. Gordon K. Smyth & Heather M. Podlich, 2002. "An Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process," Computational Statistics, Springer, vol. 17(1), pages 17-28, March.
    3. Mabel Morales-Otero & Vicente Núñez-Antón, 2021. "Comparing Bayesian Spatial Conditional Overdispersion and the Besag–York–Mollié Models: Application to Infant Mortality Rates," Mathematics, MDPI, vol. 9(3), pages 1-33, January.
    4. Marcelo Bourguignon & Rodrigo M. R. Medeiros, 2022. "A simple and useful regression model for fitting count data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(3), pages 790-827, September.
    5. Seth D. Guikema & Jeremy P. Goffelt, 2008. "A Flexible Count Data Regression Model for Risk Analysis," Risk Analysis, John Wiley & Sons, vol. 28(1), pages 213-223, February.
    6. M. J. Faddy & D. M. Smith, 2011. "Analysis of count data with covariate dependence in both mean and variance," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(12), pages 2683-2694, February.
    7. Winkelmann, Rainer, 1995. "Duration Dependence and Dispersion in Count-Data Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(4), pages 467-474, October.
    8. M. J. Faddy & D. M. Smith, 2008. "Extended Poisson process modelling of dilution series data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 57(4), pages 461-471, September.
    9. Smith, David M. & Faddy, Malcolm J., 2016. "Mean and Variance Modeling of Under- and Overdispersed Count Data," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 69(i06).
    10. Robert Jung & Gerd Ronning & A. Tremayne, 2005. "Estimation in conditional first order autoregression with discrete support," Statistical Papers, Springer, vol. 46(2), pages 195-224, April.
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