Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T17:04:03.902Z Has data issue: false hasContentIssue false
  • English
  • Français

The Credible Distribution

Published online by Cambridge University Press:  29 August 2014

William S. Jewell
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.

Credibility theory is concerned with the problem of forecasting the mean performance (claim frequency, total losses, etc.) of an individual risk, selected from a collective of heterogeneous risks, based upon the statistics of the collective, and upon the individual's experience data. The classic results, derived by American actuaries in the 1920's, were further strengthened by Bailey and Mayerson in 1950 and 1965, who showed that these results were exact Bayesian for certain risk distributions. Bühlmann, in 1967, then showed that the credibility formulae were the best least-squares linearized approximation to the exact Bayesian forecast, for general risk distributions.

This paper extends credibility theory to the problem of forecasting the distribution of individual risk, based upon collective statistics and individual experience data. Although the problem is, in principle, solved by finding a Bayesian conditional distribution, this approach requires a detailed knowledge of collective structure. The credible distribution, on the other hand, requires fewer prior statistics, and is also a best least-squares linearized approximation to the exact Bayesian distribution.

Following the theoretical development, detailed computational results are given.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 7 , Issue 3 , March 1974 , pp. 237 - 269
Copyright
Copyright © International Actuarial Association 1974

References

[1]Bailey, A. L. (1945) “A Generalized Theory of Credibility”, PCAS), Vol. 32, pp. 13–20. 1) Proceedings of the Casualty Actuarial Society (and predecessors).Google Scholar
[2]Bailey, A. L. (1950) “Credibility Procedures, Laplace's Generalization of Bayes' Rule, and the Combination of Collateral Knowledge with Observed Data”, PCAS, Vol. 37, pp. 7–23, (1950); Discussion, PCAS, Vol. 37, pp. 94-115.Google Scholar
[3]Bühlmann, H. (July 1967) “Experience Rating and Credibility”, ASTIN-Bulletin, Vol. 4 Part 3, pp. 199–207.CrossRefGoogle Scholar
[4]Bühlmann, H. (1970) Mathematical Methods in Risk Theory, Springer-Verlag, New York.Google Scholar
[5]Bühlmann, H. (1971) “Credibility Procedures”, Sixth Berkeley Symposium, pp. 515–525.Google Scholar
[6]Bühlmann, H. and Straub, E. (1970) “Glaubwürdigkeit für Schadensätze”, Mitteilungen der Vereinigung Schweizer Versicherungsmathematiker, Vol. 70, pp. 111–133. Translation by C. E. Brooks, “Credibility for Loss Ratios”, (unpublished).Google Scholar
[7]De Groot, M. H. (1970) Optimal Statistical Decisions, McGraw Hill, New York.Google Scholar
[8]Ericson, W. A. (1969) “A Note on the Posterior Mean of a Population Mean”, Journal of the Royal Statistical Society, (B), Vol. 31, No. 2, PP. 332–334.Google Scholar
[9]Ferguson, T. S. (1967) Mathematical Statistics: A Decision Theoretic Approach, Academic, New York.Google Scholar
[10]Hewitt, C. C. Jr. (1971) “Credibility for Severity”, PCAS, Vol. 57, pp. 148–171, (1970); Discussion, PCAS, Vol. 58, pp. 25-31.Google Scholar
[11]Longley-CooK, L. H. (1962) “An Introduction to Credibility Theory”, PCAS, Vol. 49, pp. 194–221; Available as a separate report from Casualty Actuarial Society, 200 East 42nd St., New York, N.Y. 10017.Google Scholar
[12]Mayerson, A. L. (1965) “A Bayesian View of Credibility”, PCAS, Vol. 51, pp. 85–104, (1964); Discussion, PCAS, Vol. 52, pp. 121-127.Google Scholar
[13]Mayerson, A. L., Jones, D. A. and Bowers, N. L. Jr. (1969) “On the Credibility of the Pure Premium”, PCAS, Vol. 55, pp. 175–185, (1968); Discussion, PCAS, Vol. 56, pp. 63-82.Google Scholar
[14]Perryman, F. A. (1932) “Some Notes on Credibility Theory”, PCAS, Vol. 19, pp. 65–84.Google Scholar
[15]Salmond, D. (February, 1973) “Premium Calculation in Casualty Insurance”, ORC Report 73-2, Operations Research Center, University of California, Berkeley.Google Scholar
[16]Seal, H. L. (1969) Stochastic Theory of Risk Business, Wiley, New York.Google Scholar
[17]Straub, E. (1972) “On the Calculation of IBNR-Reserves”, IBNR-The Prize-Winning Papers in the Boleslaw Monte Fund Competition Held in 1971, Nederlandse Reassurantie Groep, N.V., Amsterdam, pp. 132.Google Scholar
[18]Whitney, A. W. (1918) “The Theory of Experience Rating”, PCAS, Vol. 4, pp. 274–292.Google Scholar
[19]Jewell, W. S. (April, 1973) “Multi-Dimensional Credibility”, ORC Report 73-7, Operations Research Center, University of California, Berkeley.Google Scholar
[20]Jewell, W. S., “Models of Multi-Dimensional Credibility”, (in preparation).Google Scholar
[21]Jewell, W. S., “Credible Mean is Exact Bayesian for Simple Exponential Families”, (in preparation).Google Scholar