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On the Analysis of the Truncated Generalized Poisson Distribution Using a Bayesian Method

Published online by Cambridge University Press:  29 August 2014

David P.M. Scollnik
David P.M. Scollnik*
Affiliation:
Department of Mathematics and Statistics, University of Calgary
*
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada[email protected]
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Abstract

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The generalized Poisson distribution with parameters θ and λ was introduced by Consul and Jain (1973) and has recently found several instances of application in the actuarial literature. The most frequently used version of the distribution assumes that θ > 0 and 0 ≤ λ < 1, in which case the mean and variance are θ/(1 − λ) and θ/(1 − λ)3, respectively. These simple moment expressions, along with nearly all of the other theoretical results available for this distribution, fail when λ < 0 or λ > 1 (e.g., Johnson, Kotz, and Kemp, 1992, page 397). In these cases, even the definition of the probability mass function usually given in the literature is not properly normalized so that its values do not sum to unity. For this reason, it is common to truncate the support of the distribution and explicitly normalize the probability mass function. In this paper we discuss the estimation of the parameters of this truncated generalized Poisson distribution using a Bayesian method.

Keywords

Bayesianbivariategeneralized PoissonLangrangian PoissontruncatedMarkov chain Monte Carlo
Type
Workshops
Information
ASTIN Bulletin: The Journal of the IAA , Volume 28 , Issue 1 , May 1998 , pp. 135 - 152
Copyright
Copyright © International Actuarial Association 1998

References

REFERENCES

Adelstein, A.M. (1949). Accident Proneness: A Criticism of the Concept Based Upon an Analysis of Shunters' Accidents. Journal of the Royal Statistical Society, A 115, 354–410.CrossRefGoogle Scholar
Alzaid, A.A., and Al-Osh, M.A. (1993). Some Autoregressive Moving Average Processes with Generalized Poisson Marginal Distributions. Ann. Inst. Statist. Math. 45, 223–232.CrossRefGoogle Scholar
Ambagaspitiya, R.S. and Balakrishnan, N. (1994). On the Compound Generalized Poisson Distributions. ASTIN Bulletin 24 (2), 255–263.CrossRefGoogle Scholar
Carlin, B.P., and Louis, T.A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall, New York.Google Scholar
Consul, P.C. (1994). Some Bivariate Families of Lagrangian Probability Distributions. Communications in Statistics: Theory and Method 23, 2895–2906.CrossRefGoogle Scholar
Consul, P.C. (1990). A Model for Distribution of Injuries in Auto-Accidents. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 1, 161–168.Google Scholar
Consul, P.C. (1989). Generalized Poisson Distributions: Properties and Applications. Marcel Dekker Inc., New York.Google Scholar
Consul, P.C. and Famoye, F. (1989). The Truncated Generalized Poisson Distribution and its Estimation. Communications in Statistics: Theory and Method 18 (10), 3635–3648.CrossRefGoogle Scholar
Consul, P.C., and Shoukri, M.M. (1985). The Generalized Poisson Distribution when the Sample Mean is Larger than the Sample Variance. Communications in Statistics: Simulation and Computation 14 (3), 667–681.CrossRefGoogle Scholar
Consul, P.C., and Jain, G.C. (1973). A Generalization of the Poisson Distribution. Technometrics 15, 791–799.CrossRefGoogle Scholar
Famoye, F. and Consul, P.C. (1995). Bivariate Generalized Poisson Distribution with Some Applications. Metrika 42, 127–138.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (1995). Bayesian Data Analysis. Chapman and Hall, New York.CrossRefGoogle Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996). Introducing Markov Chain Monte Carlo. In Markov Chain Monte Carlo in Practice. Edited by Gilks, W.R., Richardson, S., and Spiegelhalter, D.J.Chapman and Hall, New York.Google Scholar
Goovaerts, M.J., and Kaas, R. (1991). Evaluating Compound Generalized Poisson Distributions Recursively. ASTIN Bulletin 21, 193–197.CrossRefGoogle Scholar
Haastrup, S., and Arjas, E. (1996). Claims Reserving in Continuous Time: a Nonparametric Bayesian Approach. ASTIN Bulletin 26 (2), 139–164.CrossRefGoogle Scholar
Henze, N. and Klar, B. (1995). Bootstrap Based Goodness of Fit Tests for the Generalized Poisson Model. Communications in Statistics: Theory and Method 24 (7), 1875–1896.CrossRefGoogle Scholar
Hesselager, O. (1997). Recursions for a class of compound Lagrangian distributions. Swiss Association of Actuaries: Bulletin 1, 95–101.Google Scholar
Johnson, N.L., Kotz, S., and Kemp, A.W. (1992). Univariate Discrete Distributions. Second Edition. John Wiley & Sons Inc, New York.Google Scholar
Klugman, S.A., Panjer, H.H., and Wilmot, G.E. (1997). Loss Models: From Data to Decisions. In press.Google Scholar
Nelson, D.L. (1975). Some Remarks on Generalizations of the Negative Binomial and Poisson Distributions. Technometrics 17, 135–136.CrossRefGoogle Scholar
Pai, J.S. (1997). Bayesian Analysis of Compound Loss Distributions. Journal of Econometrics 79, 129–146.CrossRefGoogle Scholar
Scollnik, D.P.M. (1995a). The Bayesian Analysis of Generalized Poisson Models for Claim Frequency Data Utilising Markov Chain Monte Carlo Methods. Actuarial Research Clearing House 1995.1, 339–356.Google Scholar
Scollnik, D.P.M. (1995b). Bayesian Analysis of Two Overdispersed Poisson Models. Biometrics 51, 1117–1126.CrossRefGoogle Scholar
Scollnik, D.P.M. (1995c). Bayesian Analysis of Two Overdispersed Poisson Regression Models. Communications in Statistics: Theory and Method 24 (11), 2901–2918.CrossRefGoogle Scholar
Scollnik, D.P.M. (1995d). Simulating Random Variates from Makeham's Distribution and from Others with Exact or Nearly Log-concave Densities (with discussion). Transactions of the Society of Actuaries 1995 XLVII, 409–454.Google Scholar
Tanner, M.A. (1996). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions. Third Edition. Springer-Verlag, New York.CrossRefGoogle Scholar
Tierney, L. (1994). Markov Chains for Exploring Posterior Distributions (with discussion). Annals of Statistics 22, 1701–1762.Google Scholar
Vernic, R. (1997). On the Bivariate Generalized Poisson Distribution. Astin Bulletin 27 (1), 23–31.CrossRefGoogle Scholar