Geometric log-Lindley distribution as an alternative to the Poisson model with application to insurance data

Document Type : Original Scientific Paper

Author

Deprtment of Statistics‎, ‎University of Hormozgan‎, ‎BandarAbbas‎, ‎Iran

Abstract

Mixed discrete distributions are primarily used for modeling over-dispersed count data. The construction of mixed models, such as the mixed Poisson model, is based on the assumption that the distribution parameter of interest is a random variable that follows a specified distribution. In this framework, the marginal distribution of a discrete random variable forms a mixed distribution. This paper introduces a novel discrete distribution derived from the geometric distribution, assuming that the model's parameter follows the log-Lindley distribution. This approach is motivated by situations where the parameter of the geometric distribution is not constant across populations, as is often the case in insurance data, where the probability of a claim varies between different portfolios. This distribution is particularly well-suited for modeling over-dispersed discrete data. The statistical properties of the proposed distribution are examined, and the parameters of the resulting model are estimated. To evaluate the accuracy of the estimates, a simulation study is conducted, and the model's performance is demonstrated using real data.

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References

Boucher, J.P., Denuit, M. and Guillen, M. (2007). Risk classification for claim count: a comparative analysis of various zero-inflated mixed Poisson and hurdle models. North American Actuarial Journal, 11(4):110–131.
Denuit, M., Marechal, X., Pitrebois, S. and Walhin, J.F. (2007). Actuarial Modeling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems. West Sussex, UK: John Wiley and Sons.
Famoye, F. and Singh, K.P. (2006). Zero-inflated generalized Poisson regression model with an application to domestic violence data. Journal of Data Science, 4(1):117–130.
Gómez-Déniz, E., Sordo, M.A. and Calderín-Ojeda, E. (2014). The Log-Lindley distribution as an alternative to the beta regression model with applications in insurance. Insurance: Mathematics and Economics, 54:49–57.
Greene, W. (2008). Functional forms for the negative binomial model for count data. Economics Letters, 99(3):585–590.
Ismail, N. and Zamani, H. (2013). Estimation of claim count data using negative binomial, generalized Poisson, zero-inflated negative binomial and zero-inflated generalized Poisson regression models. Casualty Actuarial Society E-Forum, 41(20), 1–28.
Izsák, R. (2008). Maximum likelihood fitting of the Poisson lognormal distribution. Environmental and Ecological Statistics, 15(2):143–156.
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012). Loss Models: From Data to Decisions. John Wiley & Sons.
Meng, S., Wei, Y. and Whitmore, G.A. (1999). Accounting for individual overdispersion in a bonus-malus system. ASTIN Bulletin: The Journal of the International Actuarial Association, 29(2):327–337.
Motwani, R. (1995). Randomized Algorithms. Cambridge: Cambridge University Press.
Navarro, D. and Perfors, A. (2005). An Introduction to the Beta-Binomial Model. University of Adelaide.
R Core Team (2014). R: A Language and Environment for Statistical Computing. Austria, Vienna: R Foundation for Statistical Computing.
Ridout, M.S., Hinde, J.P. and Demétrio, C.G.B. (2001). A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives. Biometrics, 57(1):219–223.
Tremblay, L. (1992). Using the Poisson inverse Gaussian in bonus-malus systems. ASTIN Bulletin: The Journal of the IAA, 22(1):97–106.
Weinberg, C.R. and Gladen, B.C. (1986). The beta-geometric distribution applied to comparative fecundability studies. Biometrics, 42:547–560.
Willmot, G.E. (1987). The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scandinavian Actuarial Journal, 1987(2):113–127.
Wongrin, W. and Bodhisuwan, W. (2017). Generalized Poisson-Lindley linear model for count data. Journal of Applied Statistics, 44(15):2659–2671.
Yip, K.C.H. and Yau, K.K.W. (2005). On modeling claim frequency data in general insurance with extra zeros. Insurance: Mathematics and Economics, 36(2):153–163.
Zakerzadeh, H. and Dolati, A. (2009). Generalized Lindley distribution. Journal of Mathematical Extension, 3(2):13–25.
Zamani, H. and Ismail, N. (2010). Negative binomial-Lindley distribution and its application. Journal of Mathematics and Statistics, 6(1):4–9.
Zamani, H. and Ismail, N. (2014). Functional form for the zero-inflated generalized Poisson regression model. Communications in Statistics-Theory and Methods, 43(3):515–529.
Journal of Statistical Modelling: Theory and Applications
Volume 6, Issue 1
January 2025
Pages 151-166

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History

  • Receive Date: 14 August 2025
  • Revise Date: 04 December 2025
  • Accept Date: 22 January 2026

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