Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-26T00:07:08.076Z Has data issue: false hasContentIssue false
  • English
  • Français

On Bayesian Mixture Credibility

Published online by Cambridge University Press:  17 April 2015

John W. Lau ,
Tak Kuen Siu  and
Hailiang Yang
John W. Lau
Affiliation:
Department of Mathematics, University of Bristol, Bristol, United Kingdom, Email: [email protected]
Tak Kuen Siu
Affiliation:
Department of Actuarial, Mathematics and Statistics School of Mathematical and Computer Sciences and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, United Kingdom, E-mail: [email protected]
Hailiang Yang
Affiliation:
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, Email: [email protected]
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.

We introduce a class of Bayesian infinite mixture models first introduced by Lo (1984) to determine the credibility premium for a non-homogeneous insurance portfolio. The Bayesian infinite mixture models provide us with much flexibility in the specification of the claim distribution. We employ the sampling scheme based on a weighted Chinese restaurant process introduced in Lo et al. (1996) to estimate a Bayesian infinite mixture model from the claim data. The Bayesian sampling scheme also provides a systematic way to cluster the claim data. This can provide some insights into the risk characteristics of the policyholders. The estimated credibility premium from the Bayesian infinite mixture model can be written as a linear combination of the prior estimate and the sample mean of the claim data. Estimation results for the Bayesian mixture credibility premiums will be presented.

Keywords

Credibility TheoryBayesian Mixture ModelsInfinite MixtureRisk CharacteristicsClusteringWeighted Chinese Restaurant ProcessCredibility Premium PrincipleDirichlet Process
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 36 , Issue 2 , November 2006 , pp. 573 - 588
Copyright
Copyright © ASTIN Bulletin 2006

References

Bailey, A. (1950) Credibility procedures. Proceedings of the Casualty Actuarial Society, XXXVII, 7–23 and 94–115.Google Scholar
Bühlmann, H. (1967) Experience rating and credibility. ASTIN Bulletin, 4, 199–207.CrossRefGoogle Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. New York: Springer-Verlag.Google Scholar
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and Its Applications. New York: Springer Verlag.Google Scholar
Ishwaran, H. and James, L.F. (2001) Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association, 96, 161–173.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2004) Loss Models: From Data to Decisions, 2nd Edition. New Jersey: John Wiley & Sons.Google Scholar
Lo, A.Y. (1984) On a class of Bayesian nonparametric estimates I: Density estimates. Ann. Statist, 12, 351–357.CrossRefGoogle Scholar
Lo, A.Y., Brunner, L.J. and Chan, A.T. (1996) Weighted Chinese restaurant processes and Bayesian mixture models. Research Report, Hong Kong University of Science and Technology. Available at http://www.erin.utoronto.ca/~jbrunner/papers/wcr96.pdf Google Scholar
MacEachern, S.N. (1994) Estimating normal means with a conjugate style Dirichlet Process Prior. Communications in statistics – Simulation and computation, 23, 727–741.CrossRefGoogle Scholar
Mowbray, A.H. (1914) How extensive a payroll exposure is necessary to give a dependendable pure premium? Proceedings of the Casualty Actuarial Society, I, 24–30.Google Scholar
Neal, R.M. (2000) Markov Chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9, 249–265.Google Scholar
Ramsay, C.M. (1994) Loading gross premiums of risk without using utility theory. Transactions of Society of Actuaries, XLV, 305–349.Google Scholar
Siu, T.K. and Yang, H. (1999) Subjective risk measures: Bayesian predictive scenarios analysis. Insurance: Mathematics & Economics, 25(2), 157–169.Google Scholar
Wang, S.S. and Young, V.R. (1998) Risk-adjusted credibility premiums using distorted probabilities. Scand. Actuarial Journal, 2, 143–165.CrossRefGoogle Scholar
Waters, H. (1993) Credibility Theory. Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University.Google Scholar
West, M., Müller, P. and Escobar, M.D. (1994) Hierarchical Priors and Mixture Models, With Applications in Regression and Density Estimation, in A tribute to Lindley, D.V., eds. Smith, A.F.M. and Freeman, P.R., New York: Wiley.Google Scholar
Whitney, A.W. (1918) The theory of experience rating. Proceedings of the Casualty Actuarial Society, IV, 274–292.Google Scholar
Young, V.R. (1997) Credibility using semiparametric models. ASTIN Bulletin, 27, 273–285.CrossRefGoogle Scholar
Young, V.R. (1998) Robust Bayesian credibility using semiparametric models. ASTIN Bulletin, 28, 187–203.CrossRefGoogle Scholar