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Alternative modelling and inference methods for claim size distributions

Published online by Cambridge University Press:  19 June 2019

Mathias Raschke
Mathias Raschke*
Affiliation:
Independent researcher/freelancer, Stolze-Schrey Str. 1, Wiesbaden, Germany
*
E-mail: [email protected] (www.mathiasraschke.de)
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Abstract

The upper tail of a claim size distribution of a property line of business is frequently modelled by Pareto distribution. However, the upper tail does not need to be Pareto distributed, extraordinary shapes are possible. Here, the opportunities for the modelling of loss distributions are extended. The basic idea is the adjustment of a base distribution for their tails. The (generalised) Pareto distribution is used as base distribution for different reasons. The upper tail is in the focus and can be modelled well for special cases by a discrete mixture of the base distribution with a combination of the base distribution with an adapting distribution via the product of their survival functions. A kind of smoothed step is realised in this way in the original line function between logarithmic loss and logarithmic exceedance probability. The lower tail can also be adjusted. The new approaches offer the opportunity for stochastic interpretation and are applied to observed losses. For parameter estimation, a modification of the minimum Anderson Darling distance method is used. A new test is suggested to exclude that the observed upper tail is better modelled by a simple Pareto distribution. Q-Q plots are applied, and secondary results are also discussed.

Keywords

Claim size distributionminimum principlediscrete mixingextreme value indexclaim generating process
Type
Paper
Information
Annals of Actuarial Science , Volume 14 , Issue 1 , March 2020 , pp. 1 - 19
Copyright
© Institute and Faculty of Actuaries 2019 

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References

Anderson, T.W. & Darling, D.A. (1954). A test of goodness-of-fit. Journal of the American Statistical Association, 49, 765–769.10.1080/01621459.1954.10501232CrossRefGoogle Scholar
Bakar, S.A., Hamzah, N.A., Maghsoudi, M. & Nadarajah, S. (2015). Modeling loss data using composite models. Mathematics and Economics, 61, 146–154.10.1016/j.insmatheco.2014.08.008CrossRefGoogle Scholar
Balakrishnan, N. & Clifford Cohen, A. (1991). Order Statistics and Inference – Estimation Methods. Volume in Statistical Modeling and Decision Science, Academic Press, Boston.Google Scholar
Boos, D. (1982). Minimum Anderson-Darling estimation. Communications in Statistics - Theory and Methods, 11, 2747–2774.10.1080/03610928208828420CrossRefGoogle Scholar
Beirlant, J., Alves, I.F. & Gomes, I. (2016). Tail fitting for truncated and non-truncated Pareto-type distributions. Extremes, 19, 429–462.10.1007/s10687-016-0247-3CrossRefGoogle Scholar
Beirlant, J., Alves, I.F. & Reynkens, T. (2017). Fitting tails affected by truncation. Electronic Journal of Statistics, 11, 2026–2065.10.1214/17-EJS1286CrossRefGoogle Scholar
Beirlant, J., De Waal, D.J. & Teugels, J.L. (2000). The generalized Burr-gamma family of distributions with applications in extreme value analysis. To appear in: Proceedings of the 4th Conference on Limit Theorems in Probability and Statistics of the J. Bolyai Soc. Chichester, UK.Google Scholar
Beirlant, J., Goegebeur, Y., Teugels, J. & Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics, Wiley & Sons, Chichester.10.1002/0470012382CrossRefGoogle Scholar
Beirlant, J., Matthys, G. & Dierckx, G. (2001). Heavy-tailed distributions and rating. ASTIN Bulletin: The Journal of the IAA, 31, 37–58.10.2143/AST.31.1.993CrossRefGoogle Scholar
Charpentier, A. (ed.) (2014). Computational Actuarial Science with R. CRC - The R Series. Chapman and Hall, Boca Raton, FL.10.1201/b17230CrossRefGoogle Scholar
Clemente, G.P., Saveli, N. & Zappa, D. (2012). Modelling and calibration for non-life underwriting risk: from empirical data to risk capital evaluation. Presentation, ASTIN Colloquium, Mexico City, 1–4 October 2013.Google Scholar
Davidson, A.C. & Hinkley, D.V. (1997). Bootstrap methods and their application. Cambridge Series in Statistical and Probabilistic Mathematics, Band 1.Google Scholar
Drossos, C.A. & Philippou, N. (1980). A note on minimum distance estimates. Annals of the Institute of Statistical Mathematics, 32(Part A), 121–123.10.1007/BF02480318CrossRefGoogle Scholar
Dutang, C. (2016). CASdatasets. R-package. Available online at the address http://dutangc.free.fr/pub/RRepos/web/CASdatasets-index.html.Google Scholar
Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics, 7, 1–26.10.1214/aos/1176344552CrossRefGoogle Scholar
EmcienScan (2018). Sample data sets. Available online at the address http://www.scan-support.com/help/sample-data-sets [accessed January 2018].Google Scholar
Finan, M.A. (2017). An introductory guide in the construction of actuarial models: a preparation for the actuarial exam C/4. Arkansas Tech University. Available online at the address http://faculty.atu.edu/mfinan/actuarieshall/CGUIDE.pdf [accessed 11-Feb-2018].Google Scholar
Frigessi, A., Haug, O. & Rue, H. (2002). A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes, 5, 219–235.10.1023/A:1024072610684CrossRefGoogle Scholar
Gian Paolo, C. & Nino, S. (2014). Modeling premium risk for Solvency II: from empirical data to risk capital evaluation. Proceedings of IAC 2014, Washington D.C., 30 March-4 April. Available online: http://www.ordineattuari.it/media/185634/ica2014_clemente_savelli_zappa_finale_seminarioroma.pdf.Google Scholar
Hill, B.M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163–1174.10.1214/aos/1176343247CrossRefGoogle Scholar
Hüsler, J., Li, D. & Raschke, M. (2016). Extreme value index estimator using maximum likelihood and moment estimation. Communication in Statistics – Theory and Methods, 45(12), 3625–3636.10.1080/03610926.2013.861495CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 2, Wiley Series in probability and mathematical statistics, Wiley, New York.Google Scholar
McNeil (1997). Estimating the Tails of Loss Severity Distributions using Extreme Value Theory. Preprint Dept Mathematics, ETH Zentrum, CH-8092 Zurich.10.2143/AST.27.1.563210CrossRefGoogle Scholar
Meerschaert, M.M, Roy, P. & Shaoz, Q. (2012). Parameter estimation for exponentially tempered power law distributions. Communications in Statistics - Theory and Methods, 41(10), 1839–1856.10.1080/03610926.2011.552828CrossRefGoogle Scholar
Parr, W.C. & Schucany, W.R. (1980). Minimum distance and robust estimation. Journal of the American Statistical Association. 75(371), 616–624.10.1080/01621459.1980.10477522CrossRefGoogle Scholar
Raschke, M. (2009). The biased transformation and its application in goodness-of-fit tests for the beta and gamma distribution. Communication in Statistics – Simulation and Computation, 38, 1870–1890.10.1080/03610910903152631CrossRefGoogle Scholar
Raschke, M. (2014). Modeling of magnitude distributions by the generalized truncated exponential distribution. Journal of Seismology, 19, 265–271.10.1007/s10950-014-9460-1CrossRefGoogle Scholar
Raschke, M. (2017). Opportunities of the minimum Anderson-Darling estimator as a variant of the maximum likelihood method. Communication in Statistics – Simulation and Computation, 46, 6879–6888.CrossRefGoogle Scholar
Resnick, S. (1997). Discussion of the Danish data on large fire insurance losses. ASTIN Bulletin: The Journal of the IAA, 27, 139–151.10.2143/AST.27.1.563211CrossRefGoogle Scholar
Reynkensa, T., Verbelen, R., Beirlanta, J. & Antonio, K. (2017). Modelling censored losses using splicing: a global fit strategy with mixed Erlang and extreme value distributions. Insurance: Mathematics and Economics, 77, 65–77.Google Scholar
Scarrott, C. & McDonald, A. (2012). A review of extreme value threshold estimation and uncertainty quantification. REVSTAT – Statistical Journal, 10, 33–60.Google Scholar
Schlather, M. (2002). Models for stationary max-stable random fields. Extremes, 5(1), 33–4410.1023/A:1020977924878CrossRefGoogle Scholar
Skřivánková, V. & Juhás, M. (2012). EVT methods as risk management tools. Conference Paper, 6th International Scientific Conference Managing and Modelling of Financial Risks, Ostrava, 10–11 September 2012.Google Scholar
Stephens, M.A. (1986). Test based on EDF statistics. In D’Augustino, R.B. & Stephens, M.A. (Eds.), Goodness-of-Fit Techniques. Statistics: Textbooks and Monographs (vol. 68). Marcel Dekker, New York.Google Scholar
Sukhatme, P.V. (1936). On the analysis of k samples from exponential population with special reference to the problem of random intervals. Stat Res Mem, 2, 94–112.Google Scholar
Wolfowitz, J. (1957). The minimum distance method. The Annals of Mathematical Statistics, 28, 75–88.10.1214/aoms/1177707038CrossRefGoogle Scholar