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Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory

Published online by Cambridge University Press:  29 August 2014

Alexander J. McNeil
Alexander J. McNeil*
Affiliation:
Departement Mathematik, ETE Zentrum, CH-8092 Zürich March 7, 1997
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Abstract

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Good estimates for the tails of loss severity distributions are essential for pricing or positioning high-excess loss layers in reinsurance. We describe parametric curve-fitting methods for modelling extreme historical losses. These methods revolve around the generalized Pareto distribution and are supported by extreme value theory. We summarize relevant theoretical results and provide an extensive example of their application to Danish data on large fire insurance losses.

Keywords

Loss severity distributionshigh excess layersextreme value theoryexcesses over high thresholdsgeneralized Pareto distribution
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 27 , Issue 1 , May 1997 , pp. 117 - 137
Copyright
Copyright © International Actuarial Association 1997

References

Balkema, A. and De Haan, L. (1974), ‘Residual life time at great age”, Annals of Probability, 2, 792–804.CrossRefGoogle Scholar
Beirlant, J. and Teugels, J. (1992), ‘Modelling large claims in non-life insurance”, Insurance: Mathematics and Economics, 11, 17–29.Google Scholar
Beirlant, J., Teugels, J. and Vynckier, P. (1996), Practical analysis of extreme values, Leuven University Press, Leuven.Google Scholar
Davison, A. (1984), Modelling excesses over high thresholds, with an application, in Oliveira, J. de, ed., “Statistical Extremes and Applications”, D. Reidel, 461–482.Google Scholar
Davison, A. and Smith, R. (1990), “Models for exceedances over high thresholds (with discussion)’, Journal of the Royal Statistical Society, Series B, 52, 393–442.Google Scholar
De Haan, L. (1990), ‘Fighting the arch-enemy with mathematics’, Statistica Neerlandica, 44, 45–68.CrossRefGoogle Scholar
Embrechts, P., Kluppelberg, C. (1983), ‘Some aspects of insurance mathematics’, Theory of Probability and its Applications 38, 262–295CrossRefGoogle Scholar
Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997), Modelling extremal events for insurance and finance, Springer Verlag, Berlin. To appear.CrossRefGoogle Scholar
Falk, M., Hüsler, J. and Reiss, R. (1994), Laws of Small numbers: extremes and rare events, Birkhäuser, Basel.Google Scholar
Fisher, R. and Tippett, L. (1928), ‘Limiting forms of the frequency distribution of the largest or smallest member of a sample’, Proceedings of the Cambridge Philosophical Society, 24, 180–190.CrossRefGoogle Scholar
Gnedenko, B. (1943), ‘Sur la distribution limite du terme maximum d'une série aléatoire’, Annals of Mathematics, 44, 423–453.CrossRefGoogle Scholar
Gumbel, E. (1958), Statistics of Extremes, Columbia University Press, New York.CrossRefGoogle Scholar
Hogg, R. and Klugman, S. (1984), Loss Distributions, Wiley, New York.CrossRefGoogle Scholar
Hosking, J. and Wallis, J. (1987), ‘Parameter and quantile estimation for the generalized Pareto distribution’, Technometrics, 29, 339–349.CrossRefGoogle Scholar
Pickands, J. (1975), ‘Statistical inference using extreme order statistics’, The Annals of Statistics, 3, 119–131.Google Scholar
Reiss, R. and Thomas, M. (1996), “Statistical analysis of extreme values’. Documentation for XTREMES software package.Google Scholar
Rootzén, H. and Tajvidi, N. (1996), ‘Extreme value statistics and wind storm losses: a case study’. To appear in Scandinavian Actuarial Journal.Google Scholar
Smith, R. (1989), ‘Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone’, Statistical Science, 4, 367–393.Google Scholar