On the Lagrangian Katz family of distributions as a claim frequency model
AbstractThe Panjer (Katz) family of distributions is defined by a particular first-order recursion which is built on the basis of two parameters. It is known to characterize the Poisson, negative binomial and binomial distributions. In insurance, its main usefulness is to yield a simple recursive algorithm for the aggregate claims distribution. The present paper is concerned with the more general Lagrangian Katz family of distributions. That family satisfies an extended recursion which now depends on three parameters. To begin with, this recursion is derived through a certain first-crossing problem and two applications in risk theory are described. The distributions covered by the recursion are then identified as the generalized Poisson, generalized negative binomial and binomial distributions. A few other properties of the family are pointed out, including the index of dispersion, an extended Panjer algorithm for compound sums and the asymptotic tail behaviour. Finally, the relevance of the family is illustrated with several data sets on the frequency of car accidents.
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Bibliographic InfoArticle provided by Elsevier in its journal Insurance: Mathematics and Economics.
Volume (Year): 47 (2010)
Issue (Month): 1 (August)
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Web page: http://www.elsevier.com/locate/inca/505554
Claim number distribution Lagrangian Katz family of distributions Generalized Poisson distribution Generalized negative binomial distribution Generalized Markov-Polya distribution Index of dispersion Extended Panjer algorithm Asymptotic tail behaviour;
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- Ambagaspitiya, R. S., 1995. "A family of discrete distributions," Insurance: Mathematics and Economics, Elsevier, vol. 16(2), pages 107-127, May.
- Ambagaspitiya, Rohana S., 1998. "Compound bivariate Lagrangian Poisson distributions," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 21-31, October.
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- Zhao, Xiaobing & Zhou, Xian, 2012. "Copula models for insurance claim numbers with excess zeros and time-dependence," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 191-199.
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