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The Mixed Bivariate Hofmann Distribution

Published online by Cambridge University Press:  29 August 2014

J.F. Walhin  and
J. Paris
J.F. Walhin*
Affiliation:
Université Catholique de Louvain Secura Belgian Re
J. Paris
Affiliation:
Université Catholique de Louvain
*
SECURA Belgian Re, Rue Montoyer 12, 1000 Bruxelles, E-mail: [email protected]
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Abstract

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In this paper we study a class of Mixed Bivariate Poisson Distributions by extending the Hofmann Distribution from the univariate case to the bivariate case.

We show how to evaluate the bivariate aggregate claims distribution and we fit some insurance portfolios given in the literature.

This study typically extends the use of the Bivariate Independent Poisson Distribution, the Mixed Bivariate Negative Binomial and the Mixed Bivariate Poisson Inverse Gaussian Distribution.

Keywords

Mixed Bivariate Independent Poisson DistributionsHofmann Distributionmaximum likelihoodaggregate claims distributionrecursive algorithmstable algorithm
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 31 , Issue 1 , May 2001 , pp. 123 - 138
Copyright
Copyright © International Actuarial Association 2001

References

Ambagaspitya, R.S. (1999) On the Distributions of Two Classes of Correlated Aggregate Claims. Insurance: Mathematics and Economics, 24, 301–308.Google Scholar
Besson, J.L. and Partrat, C. (1992) Trend et Systèmes de Bonus-Malus. Astin Bulletin, 22, 11–31.CrossRefGoogle Scholar
Grandell, J. (1997) Mixed Poisson Processes. Chapman and Hall.CrossRefGoogle Scholar
Hesselager, O. (1996) Recursions for Certain Bivariate Counting Distributions and their Compound Distributions. Astin Bulletin, 26, 35–52.CrossRefGoogle Scholar
Hofmann, M. (1955) Uber zusammengesetzte Poisson-Prozesse und ihre Anwendungen in der Unfallversicherung. 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
Kemp, A.W. (1981) Computer Sampling from Homogeneous Bivariate Discrete Distributions. ASA Proceedings of the Statistical Computing Section, pages 173–175.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
Kocherlakota, S. and Kocherlakota, K. (1992) Bivariate Discrete Distributions. Marcel Dekker, New-York.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
Panjer, H.H. and Willmot, G.E. (1992) Insurance Risk Models. Society of Actuaries.Google Scholar
Partrat, C. (1994) Compound Model for Two Dependent Kinds of Claims. Insurance: Mathematics and Economics, 15, 219–231.Google Scholar
Sundt, B. (1999a) Multivariate Compound Poisson Distributions and Infinite Divisibility. Statistical report 33, Department of Mathematics, University of Bergen.Google Scholar
Sundt, B. (1999b) On Multivariate Panjer Recursions. Astin Bulletin, 29, 29–45.CrossRefGoogle Scholar
Sundt, B. and Jewell, W.S. (1981) Further Results on Recursive Evaluation of Compound Distributions. Astin Bulletin, 12, 27–39.CrossRefGoogle Scholar
Walhin, J.F. and Paris, J. (2000a) The Effect of Excess of Loss Reinsurance with Reinstatements on the Cedent's Portfolio. Blatter Deutsche Gesellschaft für Versicherungsmathematik, 24, 615–627.Google Scholar
Walhin, J.F. and Paris, J. (2000b) A Large Family of Discrete and Overdispersed Probability Laws, submitted.Google Scholar
Walhin, J.F. and Paris, J. (2000c) Recursive Formulae for Some Bivariate Counting Distributions Obtained by the Trivariate Reduction Method. Astin Bulletin, 30, 141–155.CrossRefGoogle Scholar