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EVOLUTIONARY HIERARCHICAL CREDIBILITY

Published online by Cambridge University Press:  02 November 2017

Greg Taylor
Greg Taylor*
Affiliation:
UNSW Business School, Level 6, East Lobby, UNSW Business School Building E12, 2052 Sydney, Australia. Phone: +61 (0) 421 338 448.
*
E-Mail: [email protected]
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Abstract

The hierarchical credibility model was introduced, and extended, in the 70s and early 80s. It deals with the estimation of parameters that characterize the nodes of a tree structure. That model is limited, however, by the fact that its parameters are assumed fixed over time. This causes the model's parameter estimates to track the parameters poorly when the latter are subject to variation over time. This paper seeks to remove this limitation by assuming the parameters in question to follow a process akin to a random walk over time, producing an evolutionary hierarchical model. The specific form of the model is compatible with the use of the Kalman filter for parameter estimation and forecasting. The application of the Kalman filter is conceptually straightforward, but the tree structure of the model parameters can be extensive, and some effort is required to retain organization of the updating algorithm. This is achieved by suitable manipulation of the graph associated with the tree. The graph matrix then appears in the matrix calculations inherent in the Kalman filter. A numerical example is included to illustrate the application of the filter to the model.

Keywords

Credibilitycredibility matrixevolutionary modelgraph matrixhierarchyKalman filtertree graph
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 339 - 374
Copyright
Copyright © Astin Bulletin 2017 

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References

Antonio, K., Frees, E.W. and Valdez, E.A. (2010) A multilevel analysis of intercompany claim counts. ASTIN Bulletin, 40 (1), 151–177.Google Scholar
Australian Bureau of Statistics (2011) Consumer price index: Concepts, sources and methods. Catalogue number 6461.0. Canberra, Australia. Available at http://www.abs.gov.au/ausstats/[email protected]/mf/6461.0.Google Scholar
Belhadj, H., Goulet, V. and Ouellet, T. (2009) On parameter estimation in hierarchical credibility. ASTIN Bulletin, 39 (2), 495–514.Google Scholar
Bondy, J.A. and Murty, U.S.R. (2008) Graph Theory with Applications. Vol. 244 of Graduate Texts in Mathematics series. London: Springer-Verlag.Google Scholar
Bühlmann, H. and Gisler, A. (2006) A Course in Credibility Theory and its Applications. Berlin: Springer.Google Scholar
Bühlmann, H. and Jewell, W.S. (1987) Hierarchical credibility revisited. Bulletin of the Swiss Association of Actuaries, 87, 35–54.Google Scholar
Cha, S.-H. (2012) On integer sequences derived from balanced k-ary trees. In: Proceedings of American Conference on Applied Mathematics, 377–381.Google Scholar
Chen, W.-K. (1971) Applied Graph Theory. Vol. 13 of North-Holland series in Applied Mathematics and Mechanics. Amsterdam: North-Holland Publishing Company.Google Scholar
Choudum, S. and Raman, I. (2009) Embedding height balanced trees and fibonacci trees in hypercubes. Journal of Applied Mathematics and Computing, 30 (1), 39–52.Google Scholar
Comcare (2017) Premium devolution. Available at https://www.comcare.gov.au/the_scheme/premium_paying_agencies/premium_devolution.Google Scholar
De Jong, P. (1991) The diffuse Kalman filter. The Annals of Statistics, 19 (2), 1073–1083.Google Scholar
Drmota, M. (2009) Random Trees: An Interplay between Combinatorics and Probability. Vienna: Springer Science & Business Media.Google Scholar
Durbin, J. and Koopman, S.J. (2012) Time Series Analysis by State Space Methods, 2nd Edition. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Ebrahimzadeh, M., Ibrahim, N.A., Jemain, A.A. and Kilicman, A. (2013) Claim dependence induced by common effects in hierarchical credibility models. Communications in Statistics-Theory and Methods, 42 (18), 3373–3400.Google Scholar
Goovaerts, M. J., Kaas, R., Van Heerwaarden, A.E. and Bauwelnckx, T. (1990) Effective Actuarial Methods. Amsterdam: North-Holland Publishing Company.Google Scholar
Goovaerts, M.J. and Hoogstad, W.J. (1987) Credibility Theory. Vol. 4 of Surveys of actuarial studies. The Netherlands: Nationale-Nederlanden NV.Google Scholar
Hachemeister, C.A. (1975) Credibility for regression models with application to trend. In Credibility: Theory and Applications (ed. Kahn, P.M.), pp. 308–348. New York: Wiley.Google Scholar
Harvey, A.C. (1989) Forecasting, Structural Time Series and the Kalman Filter. Cambridge, UK: Cambridge University Press.Google Scholar
Hesselager, O. (1991) Prediction of outstanding claims: A hierarchical credibility approach. Scandinavian Actuarial Journal, 1991 (1), 25–47.CrossRefGoogle Scholar
Jewell, W.S. (1975) The use of collateral data in credibility: A hierarchical model. Research Memorandum 75-24 of the International Institute for Applied Systems Analysis, Laxenburg, Austria: Institute for Applied Systems Analysis.Google Scholar
Kalman, R.E. (1960) A new approach to linear filtering and prediction systems. Journal of Basic Engineering, 82, 340–345.Google Scholar
Koopman, S.J. and Durbin, J. (2003) Filtering and smoothing of state vector for diffuse state-space models. Journal of Time Series Analysis, 24 (1), 85–98.CrossRefGoogle Scholar
Norberg, R. (1986) Hierarchical credibility: Analysis of a random effect linear model with nested classification. Scandinavian Actuarial Journal, 1986 (3–4), 204–222.CrossRefGoogle Scholar
Ohlsson, E. (2005) Simplified estimation of structure parameters in hierarchical credibility. 36th ASTIN Colloquium, Zurich.Google Scholar
Ohlsson, E. (2008) Combining generalized linear models and credibility models in practice. Scandinavian Actuarial Journal, 2008 (4), 301–314.Google Scholar
Pitselis, G. (2011) On a hierarchical credibility model for quantiles. 40th ASTIN Colloquium, Zurich, Available at http://www.actuaries.org/ASTIN/Colloquia/Madrid/Papers/Pitselis.pdf.Google Scholar
Rosenberg, B. (1973) Random coefficients models: The analysis of a cross section of time series by stochastically convergent parameter regression. In: Annals of Economic and Social Measurement, National Bureau of Economic Research, 2 (4), 399–428.Google Scholar
Sundt, B. (1979) A hierarchical credibility regression model. Scandinavian Actuarial Journal, 1979 (2–3), 107–114.Google Scholar
Sundt, B. (1980) A multi-level hierarchical credibility regression model. Scandinavian Actuarial Journal, 1980 (1), 25–32.Google Scholar
Taylor, G.C. (1979) Credibility analysis of a general hierarchical model. Scandinavian Actuarial Journal, 1979 (1), 1–12.Google Scholar