Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T20:36:41.309Z Has data issue: false hasContentIssue false
  • English
  • Français

Using Mixed Poisson Processes in Connection with Bonus-Malus Systems1

Published online by Cambridge University Press:  29 August 2014

J.F. Walhin  and
J. Paris
J.F. Walhin*
Affiliation:
Institut de Statistique, Université Catholique de Louvain, Belgium Le Mans Assurances, Belgique
J. Paris*
Affiliation:
Institut de Statistique, Université Catholique de Louvain, Belgium
*
Institut de Statistique, Voie du Roman Pays, 20, B-1348 Louvain-la-Neuve, Belgium
Institut de Statistique, Voie du Roman Pays, 20, B-1348 Louvain-La-Neuve, Belgium
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For the construction of bonus-malus systems, we propose to show how to apply, thanks to simple mathematics, a parametric method encompassing those encountered in the literature. We also compare this parametric method with a non-parametric one that has not yet been used in the actuarial literature and that however permits a simple formulation of the stationary and transition probabilities in a portfolio whenever we have the intention to construct a bonus-malus system with finite number of classes.

Keywords

Mixed Poisson processesnon-parametric estimationHofmann's distributionBayes theorembonus-malus systemsstationary distribution
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 29 , Issue 1 , May 1999 , pp. 81 - 99
Copyright
Copyright © International Actuarial Association 1999

Footnotes

1

This paper has been presented at the XXVIIIth ASTIN Colloquium, Cairns 10-13 August 1997

References

Bühlmann, H. (1970). Mathematical Models in Risk Theory. Springer-Verlag, Berlin.Google Scholar
Coene, G. and Doray, L.G. (1996). A Financially Balanced Bonus-Malus System. ASTIN Bulletin, 26: 107–115.Google Scholar
Dufresne, F. (1988). Distributions Stationnaires d'un Système Bonus-Malus et Probabilité de Ruine. ASTIN Bulletin, 18: 31–46.Google Scholar
Dufresne, F. (1995). The Efficiency of the Swiss Bonus-Malus System. Bulletin of the Swiss Actuaries, 1995(1): 29–41.Google Scholar
Feller, W. (1971). An introduction to Probability Theory and its Applications Vol II (3ed). Wiley, New York.Google Scholar
Geller, H. (1979). An introduction to Mathematical Risk Theory. University of Pennsylvania.Google Scholar
Hofmann, M. (1955). Uber zusammengesetzte poisson-prozesse und ihre Anwendungen in der Unfall versicherung. Bulletin of the Swiss Actuaries, 55: 499–575.Google Scholar
Hürlimann, W. (1990). On Maximum Likelihood Estimation for Count Data Models. Insurance: Mathematics and Economics, 9: 39–49.Google Scholar
Kestemont, R.M. and Paris, J. (1985). Sur l'Ajustement du Nombre de Sinistres. Bulletin of the Swiss Actuaries, 85: 157–164.Google Scholar
Lemaire, J. (1985). Automobile Insurance: Actuarial Models. Kluwer-Nijhoff, Netherlands.Google Scholar
Lindsay, B. (1995). Mixture Models: Theory, Geometry and Applications. Pennsylvania State University.CrossRefGoogle Scholar
Loimaranta, K. (1972). Some Asymptotic Properties of Bonus Systems. ASTIN Bulletin, 6: 233–245.Google Scholar
Maceda, E. (1948). On the Compound and Generalized Poisson Distributions. Annals of Mathematical Statistics, 19: 414–416.CrossRefGoogle Scholar
Panjer, H.H. (1981). Recursive Evaluation of a Family of Compound Distributions. ASTIN Bulletin, 12: 22–26.CrossRefGoogle Scholar
Simar, L. (1976). Maximum Likelihood Estimation of a Compound Poisson Process. The Annals of Statistics, 4: 1200–1209.Google Scholar
Tremblay, L. (1992). Using the Poisson Inverse Gaussian Distribution in Bonus-Malus Systems. ASTIN Bulletin, 22: 97–106.Google Scholar