Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T02:38:58.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting

Published online by Cambridge University Press:  29 August 2014

Rudolf Grübel  and
Renate Hermesmeier
Rudolf Grübel*
Affiliation:
Universität Hannover
Renate Hermesmeier
Affiliation:
Universität Hannover
*
Institut für Mathematische Stochastik, Universität Hannover, Postfach 60 09, D-30060 Hannover, e-mail:rgrubel@stochastik.uni-hannover.de
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.

Numerical evaluation of compound distributions is one of the central numerical tasks in insurance mathematics. Two widely used techniques are Panjer recursion and transform methods. Many authors have pointed out that aliasing errors imply the need to consider the whole distribution if transform methods are used, a potential drawback especially for heavy-tailed distributions. We investigate the magnitude of aliasing errors and show that this problem can be solved by a suitable change of measure.

Keywords

Total claim size distributionruin probabilitiesrandom sumsFourier transformationaliasingchange of measure
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 29 , Issue 2 , November 1999 , pp. 197 - 214
Copyright
Copyright © International Actuarial Association 1999

References

Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.Google Scholar
Asmussen, S. and Binswanger, K. (1997). Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27, 27–297.CrossRefGoogle Scholar
Beard, R.E., Pentikäinen, T. and Personen, E. (1984). Risk Theory: The Stochastic Basis of Insurance. (3rd ed.) Chapman and Hall, London.CrossRefGoogle Scholar
Buchwalder, M., Chevalier, E. and Kluppelberg, C. (1993). Approximation methods for the total claimsize distribution-an algorithmic and graphical presentation. Mitteilungen der Schweiz. Vereinigung der Versicherungsmathematiker, Heft 2/1993.Google Scholar
Bühlmann, H. (1984). Numerical evaluation of the compound Poisson distribution: recursion or fast Fourier transform? Scand. Actuarial J. 2, 2–116.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973). Functions of probability measures. J. Analyse Math. 26, 26–255.CrossRefGoogle Scholar
Embrechts, P., Goldie, C.M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete 49, 49–335.CrossRefGoogle Scholar
Embrechts, P., Grübel, R. and Pitts, S.M. (1993). Some applications of the fast Fourier transform algorithm in insurance mathematics. Statistica Neerlandica 47, 47–59.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, Th. (1997). Modelling Extremal Events. Springer, Berlin.CrossRefGoogle Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates of the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics 1, 1–55.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.CrossRefGoogle Scholar
Grübel, R. (1991). Algorithm AS 265: G/G/1 via Fast Fourier Transform. Applied Statistics (J. R. Stat. Soc. C) 40, 40–355.Google Scholar
Grübel, R. and Hermesmeier, R. (1999). Computation of compound distributions II: discretization errors and Richardson extrapolation. In preparation.Google Scholar
Hermesmeier, R. (1997). Transformationsmethoden und Diskretisierungsefjekte bei stochastischen Modellen in der Versicherungsmathematik. Doctoral thesis, University of Hannover.Google Scholar
Hipp, C. and Michel, R. (1990). Risikotheorie: Stochastische Modelle und Statistische Methoden, Verlag Versicherungswirtschaft e.V., Karlsruhe.Google Scholar
Johnson, N.L., Kotz, S. and Kemp, A.W. (1992). Univariate Discrete Distributions, 2nd edition. Wiley, New York.Google Scholar
Panjer, H.H. (1981). Resursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 12–22.CrossRefGoogle Scholar
Pitts, S.M. (1994). Nonparametric estimation of compound distributions with applications in insurance. Ann. Inst. Stat. Math. 46, 46–537.CrossRefGoogle Scholar
Schröter, K.J. (1995). Verfahren zur Approximation der Gesamtschadenverteilung: System-atisierung, Techniken und Vergleiche. Verlag Versicherungswirtschaft e.V., Karlsruhe.Google Scholar