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An Evolutionary Credibility Model for Claim Numbers

Published online by Cambridge University Press:  29 August 2014

Peter Albrecht
Peter Albrecht*
Affiliation:
University of Mannheim, FR Germany
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This paper considers a particular credibility model for the claim numbers N1, N2, …, Nn, … of a single risk within a collective in successive periods 1, 2, …, n, … In the terminology of Jewell (1975) the model is an evolutionary credibility model, which means that the underlying risk parameter Λ is allowed to vary in successive periods (the structure function is allowed to be time dependent). Evolutionary credibility models for claim amounts have been studied by Bühlmann (1969, pp. 164–165), Gerber and Jones (1975), Jewell (1975, 1976), Taylor (1975), Sundt (1979, 1981, 1983) and Kremer (1982). Again in Jewell's terminology the considered model is on the other hand stationary, in the sense that the conditional distribution of Ni given the underlying risk parameter does not vary with i.

The computation of the credibility estimate of Nn+1 involves the considerable labor of inverting an n × n covariance matrix (n is the number of observations). The above mentioned papers have therefore typically looked for model structures for which this inversion is unnecessary and instead a recursive formula for the credibility forecast can be obtained. Typically nth order stationary a priori sequences (e.g., ARMA (p, q)-processes) lead to an nth order recursive scheme. In this paper we impose the restriction that the conditional distribution of Ni is Poisson (which by the way leads to a model identical to the so called “doubly stochastic Poisson sequences” considered in the theory of stochastic point processes). What we gain is a recursive formula for the coefficients of the credibility estimate (not for the estimate itself!) in case of an arbitrary weakly stationary a priori sequence. In addition to this central result the estimation of the structural parameters is considered in this case and some more special models are analyzed. Among them are EARMA-processes (which are positive-valued stationary sequences possessing exponentially distributed marginals and the same autocorrelation structure as ARMA-processes) as a priori sequence and models which can be considered as (discrete) generalizations of the Pólya process.

Keywords

Credibilitydoubly stochastic Poisson sequencesweakly stationary sequencesgeneralized Pólya sequence
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 15 , Issue 1 , April 1985 , pp. 1 - 17
Copyright
Copyright © International Actuarial Association 1985

References

Albrécht, P. (1984) Laplace Transforms, Mellin Transforms and Mixed Poisson Processes. Scandinavian Actuarial Journal, 58–64.CrossRefGoogle Scholar
Bates, G. E. and Neyman, J. (1952) Contributions to the Theory of Accident Proneness. University of California Publications in Statistics 1, 215–276.Google Scholar
Bühlmann, H. (1969) Experience Rating and Credibility. ASTIN Bulletin 5, 157–165.CrossRefGoogle Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley: New York.Google Scholar
Gaver, D. G. and Lewis, P. A. W. (1980) First-Order Autoregressive Gamma Sequences and Point Processes. Advances in Applied Probability 12, 727–745.CrossRefGoogle Scholar
Gerber, H. U. and Jones, D. A. (1975) Credibility Formulas of the Updating Type. In Kahn, (1975), pp. 89–105.Google Scholar
Grandell, J. (1971) On Stochastic Processes Generated by a Stochastic Intensity Function. Skandinavisk Aktuarietidskrift 54, 204–240.Google Scholar
Grandell, J. (1972) Statistical Inference for Doubly Stochastic Poisson Processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, edited by Lewis, P. A. W.. Wiley: New York, pp. 90–121.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics, No. 529. Springer: Berlin.CrossRefGoogle Scholar
Hannan, E. J. (1960) Time Series Analysis. Methuen: London.Google Scholar
Jacobs, P. A. and Lewis, P. A. W. (1977) A Mixed Autoregressive-Moving Average Exponential Sequence and Point Process, EARMA (1, 1). Advances in Applied Probability 9, 87–104.CrossRefGoogle Scholar
Jewell, W. S. (1975) Model Variations in Credibility Theory. In Kahn, (1975), pp. 193–244.Google Scholar
Jewell, W. S. (1976) Two Classes of Covariance Matrices Giving Simple Linear Forecasts. Scandinavian Actuarial Journal, 15–29.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Discrete Distributions. Wiley: New York.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics: Continuous Multivariate Distributions. Wiley: New York.Google Scholar
Kahn, P. M. (ed.) (1975) Credibility. New York: Academic Press.Google Scholar
Kremer, E. (1982) Credibility Theory for some Evolutionary Models. Scandinavian Actuarial Journal, 129–142.CrossRefGoogle Scholar
Krishnaiah, P. R. and Rao, M. M. (1961) Remarks on a Multivariate Gamma Distribution. American Mathematical Monthly 68, 342–346.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) An Exponential Moving Average Sequence and Point Process, EMA (1). Journal of Applied Probability 14, 98–113.CrossRefGoogle Scholar
Lawrance, A. J. and Lewis, P. A. W. (1980) The Exponential Autoregressive-Moving Average EARMA (p, q) Process. Journal of the Royal Statistical Society B 42, 150–161.Google Scholar
Snyder, D. L. (1975) Random point processes. Wiley, New York.Google Scholar
Sundt, B. (1979) An Insurance Model with Collective Seasonal Random Factors. Mitteilungen der Vereinigung schweizerischer Versicherungsmathematiker, 57–64.Google Scholar
Sundt, B. (1981) Recursive Credibility Estimation. Scandinavian Actuarial Journal, 3–21.CrossRefGoogle Scholar
Sundt, B. (1983) Finite Credibility Formulae in Evolutionary Models. Scandinavian Actuarial Journal, 106–116.CrossRefGoogle Scholar
Taylor, G. C. (1975) Credibility for Time-Heterogeneous Loss Ratio. In Kahn, (1975), pp. 363–389.Google Scholar