On the analysis of the truncated generalized Poisson distribution using a Bayesian method. (English) Zbl 1168.60311
Summary: The generalized Poisson distribution with parameters \(\theta, \lambda\) was introduced by P. C. Consul and G. C. Jain [Technometrics 15, 791–799 (1973; Zbl 0271.60020)] and has recently found several instances of application in the actuarial literature. The most frequently used version of the distribution assumes that \(\theta > 0\) and \(0 < \lambda < 1\), in which case the mean and variance are \(\theta / ( 1 - \lambda)\) and \(\theta / ( 1 - \lambda)^3\), respectively. These simple moment expressions, along with nearly all of the other theoretical results available for this distribution, fail when \(\lambda < 0\) or \(\lambda >1\) (e.g., N. L. Johnson, S. Kotz and A. W. Kemp [Univariate discrete distributions. 2nd ed. New York: John Wiley (1992; Zbl 0773.62007), p. 397]. In these cases, even the definition of the probability mass function usually given in the literature as 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
References:
| [1] | DOI: 10.1016/S0304-4076(97)00010-9 · Zbl 0873.62117 · doi:10.1016/S0304-4076(97)00010-9 |
| [2] | DOI: 10.2307/2533010 · Zbl 0875.62117 · doi:10.2307/2533010 |
| [3] | DOI: 10.1080/03610918508812463 · doi:10.1080/03610918508812463 |
| [4] | DOI: 10.1080/03610928908830114 · Zbl 0696.62022 · doi:10.1080/03610928908830114 |
| [5] | Generalized Poisson Distributions: Properties and Applications (1989) |
| [6] | Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker 1 pp 161– (1990) |
| [7] | DOI: 10.1080/03610929408831423 · Zbl 0850.62110 · doi:10.1080/03610929408831423 |
| [8] | Bayes and Empirical Bayes Methods for Data Analysis (1996) · Zbl 0871.62012 |
| [9] | DOI: 10.2143/AST.24.2.2005069 · doi:10.2143/AST.24.2.2005069 |
| [10] | DOI: 10.1007/BF00775809 · Zbl 0777.62085 · doi:10.1007/BF00775809 |
| [11] | Journal of the Royal Statistical Society 115 pp 354– (1949) |
| [12] | Loss Models: From Data to Decisions (1997) |
| [13] | Univariate Discrete Distributions (1992) |
| [14] | Swiss Association of Actuaries: Bulletin 1 pp 95– (1997) |
| [15] | DOI: 10.1080/03610929508831592 · Zbl 0937.62586 · doi:10.1080/03610929508831592 |
| [16] | DOI: 10.2143/AST.26.2.563216 · doi:10.2143/AST.26.2.563216 |
| [17] | DOI: 10.2143/AST.21.2.2005363 · doi:10.2143/AST.21.2.2005363 |
| [18] | Markov Chain Monte Carlo in Practice (1996) · Zbl 0832.00018 |
| [19] | Bayesian Data Analysis (1995) |
| [20] | Actuarial Research Clearing House 1 pp 339– (1995) |
| [21] | DOI: 10.1007/BF01894293 · Zbl 0820.62015 · doi:10.1007/BF01894293 |
| [22] | DOI: 10.1080/00401706.1973.10489112 · doi:10.1080/00401706.1973.10489112 |
| [23] | DOI: 10.2143/AST.27.1.542065 · doi:10.2143/AST.27.1.542065 |
| [24] | DOI: 10.1214/aos/1176325750 · Zbl 0829.62080 · doi:10.1214/aos/1176325750 |
| [25] | Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions (1996) · Zbl 0846.62001 |
| [26] | Transactions of the Society of Actuaries XLVII pp 409– (1995) |
| [27] | DOI: 10.1080/03610929508831658 · Zbl 0875.62126 · doi:10.1080/03610929508831658 |
| [28] | DOI: 10.1080/00401706.1975.10489284 · doi:10.1080/00401706.1975.10489284 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
