Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T12:55:19.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Discussion of the Danish Data on Large Fire Insurance Losses

Published online by Cambridge University Press:  29 August 2014

Sidney I. Resnick
Sidney I. Resnick*
Affiliation:
Cornell University
*
Cornell University, School of Operations Research and Industrial Engineering, Rhodes Hall 223, Ithaca, NY 14853, USA E-mail: [email protected]
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.

Alexander McNeil's (1996) study of the Danish data on large fire insurance losses provides an excellent example of the use of extreme value theory in an important application context. We point out how several alternate statistical techniques and plotting devices can buttress McNeil's conclusions and provide flexible tools for other studies.

Keywords

Heavy tailsregular variationHill estimatorPoisson processeslinear programmingparameter estimation weak convergenceconsistencyestimationindependenceautocorrelations
Type
Workshop
Information
ASTIN Bulletin: The Journal of the IAA , Volume 27 , Issue 1 , May 1997 , pp. 139 - 151
Copyright
Copyright © International Actuarial Association 1997

References

Brockwell, P. and Davis, R., Time Series: Theory and Methods, 2nd edition, Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
Brockwell, P. and Davis, R.ITSM: An Interactive Time Series Modelling Package for the PC, Springer-Verlag, New York, 1991.Google Scholar
Castillo, Enrique, Extreme Value Theory in Engineering, Academic Press, San Diego, California, 1988.Google Scholar
Davis, R. and Resnick, S., Limit theory for moving averages of random variables with regularly varying tail probabilities., Ann. Probability 13 (1985a), 179–195.CrossRefGoogle Scholar
Davis, R. and Resnick, S., More limit theory for the sample correlation function of moving averages. Stochastic Processes and their Applications, 20 (1985b), 257–279.CrossRefGoogle Scholar
Davis, R. and Resnick, S., Limit theory for the sample covariance and correlation functions of moving averages, Ann. Statist. 14 (1989), 533–558.Google Scholar
Dekkers, A., Einmahl, J., and Haan, L. de, A moment estimator for the index of an extreme value distribution, Ann. Statist. 17 (1989), 1833–1855.Google Scholar
Dekkers, A. and Haan, L. de, On the estimation of the extreme value index and large quantile estimation, Ann. Statist. 17 (1989), 1795–1832.CrossRefGoogle Scholar
Embrechts, P., Kluppelbero, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance, To appear, Springer-Verlag, Heidelberg, 1997.CrossRefGoogle Scholar
Feigin, P. and Resnick, S., Limit distributions for linear programming time series estimators. Stochastic Processes and their Applications 51 (1994), 135–166.CrossRefGoogle Scholar
Feigin, P. and Resnick, S., Pitfalls of fitting autoregressive models for heavy-tailed time series, Available at http:/www.orie.cornell.edu/trlist/trlist.htlml as TR1163.ps.Z (1996).Google Scholar
Feigin, P., Resnick S., and Stӑricӑ, C., Testing for independence in heavy tailed and positive innovation time series. Stochastic Models 11 (1995), 587–612.CrossRefGoogle Scholar
Haan, L. de, On Regular Variation and its Application to the Weak Convergence of Sample Extremes, Mathematical Centre Tract 32, Mathematical Centre, Amsterdam, Holland, 1970.Google Scholar
Haan, L. de, Extreme Value Statistics, Lecture Notes, Econometric Institute, Erasmus University, Rotterdam (1991).Google Scholar
Haan, L. de and Resnick, S., On asymptotic normality of the Hill estimator, TRI 155.ps.Z available at http:/www.orie.cornell.edu/trlist/trlist.htlml (1996).Google Scholar
Hill, B., A simple approach to inference about the tail of a distribution, Ann. Statist. 3 (1975), 1163–1174.CrossRefGoogle Scholar
Hsing, T., Extreme value theory for suprema of random variables with regularly varying tail probabilities., Stoch. Proc. and their Appl. 22 (1986), 51–57.CrossRefGoogle Scholar
Kratz, M. and Resnick, S., The qq-estimator and heavy tails, Stochastic Models 12 (1996), 699–724.CrossRefGoogle Scholar
Leadbetter, M., Lindgren, G. and Rootzen, H., Extremes and Related Properties of Random Sequences and Processes, Spring Verlag, New York, 1983.CrossRefGoogle Scholar
Mason, D., Laws of large numbers for sums of extreme values, Ann. Probability 10 (1982), 754–764.CrossRefGoogle Scholar
McNeil, A., Estimating the tails of loss severity distributions using extreme value theory, Preprint: Dept. Mathematics, ETH Zentrum, CH-8092 Zürich (1966).Google Scholar
Resnick, S., Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.CrossRefGoogle Scholar
Resnick, S., Heavy tail modelling and teletraffic data. Available as TRI 134.ps.Z at http:/www.orie.cornell.edu/trlist/trlist.htlml, Ann. Statist. (1995), (to appear).Google Scholar
Resnick, S., Why non-linearities can ruin the heavy tailed modeler's day, Available as TR1157.ps.Z at http:/www.orie.cornell.edu/trlist/trlist.htlml, A PRACTICAL GUIDE TO HEAVY TAILS: Statistical Techniques for Analysing Heavy Tailed Distributions (Adler, Robert, Feldman, Raisa, Taqqu, Murad S., ed.), Birkhauser, Boston 1996, (to appear).Google Scholar
Resnick, S. and Stӑricӑ, C., Consistency of Hill's estimator for dependent data, J. Applied Probability 32 (1995), 139–167.CrossRefGoogle Scholar
Resnick, S. and Stӑricӑ, C., Smoothing the Hill estimator, To appear: J. Applied Probability (1996a).Google Scholar
Resnick, S. and Stӑricӑ, C., Smoothing the moment estimator of the extreme value parameter. Available as TRI 158.ps.Z at http:/www.orie.cornell.edu/trlist/trlist.htlml, Preprint (1996b).Google Scholar
Rootzen, H., Leadbetter, M. and De Haan, L., Tail and quantile estimation for strongly mixing stationary sequences, Technical Report 292, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill, NC27599–3260 (1990).CrossRefGoogle Scholar
Rootzen, H., The tail emprirical process for stationary sequences. Preprint 1995:9 ISSN 1100-2255, Studies in Statistical Quality Control and Reliablity, Chalmers University of Technology (1995).Google Scholar