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Joint modelling of the total amount and the number of claims by conditionals

  • Sarabia, José María
  • Guillén, Montserrat

In the risk theory context, let us consider the classical collective model. The aim of this paper is to obtain a flexible bivariate joint distribution for modelling the couple (S,N), where N is a count variable and S=X1+...+XN is the total claim amount. A generalization of the classical hierarchical model, where now we assume that the conditional distributions of SN and NS belong to some prescribed parametric families, is presented. A basic theorem of compatibility in conditional distributions of the type S given N and N given S is stated. Using a known theorem for exponential families and results from functional equations new models are obtained. We describe in detail the extension of two classical collective models, which now we call Poisson-Gamma and the Poisson-Binomial conditionals models. Other conditionals models are proposed, including the Poisson-Lognormal conditionals distribution, the Geometric-Gamma conditionals model and a model with inverse Gaussian conditionals. Further developments of collective risk modelling are given.

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Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 43 (2008)
Issue (Month): 3 (December)
Pages: 466-473

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Handle: RePEc:eee:insuma:v:43:y:2008:i:3:p:466-473
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  1. Dionne, G. & Vanasse, C., 1988. "A Generalization of Automobile Insurance Rating Models: the Negative Binomial Distribution with a Regression Component," Cahiers de recherche 8833, Universite de Montreal, Departement de sciences economiques.
  2. Frees, Edward W. & Wang, Ping, 2006. "Copula credibility for aggregate loss models," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 360-373, April.
  3. Spanos,Aris, 1999. "Probability Theory and Statistical Inference," Cambridge Books, Cambridge University Press, number 9780521424080.
  4. Katrien Antonio & Jan Beirlant, 2008. "Issues in Claims Reserving and Credibility: A Semiparametric Approach With Mixed Models," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 75(3), pages 643-676.
  5. J. Pinquet & M. Guillén & C. Bolancé, 2000. "Long-range contagion in automobile insurance data : estimation and implications for experience rating," THEMA Working Papers 2000-43, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  6. José Sarabia & Enrique Castillo & Marta Pascual & María Sarabia, 2007. "Bivariate income distributions with lognormal conditionals," Journal of Economic Inequality, Springer, vol. 5(3), pages 371-383, December.
  7. José María Sarabia & Enrique Castillo & Emilio Gómez-Déniz & Francisco J. Vázquez-Polo, 2005. "A Class of Conjugate Priors for Log-Normal Claims Based on Conditional Specification," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 72(3), pages 479-495.
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