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Pseudo Compound Poisson Distributions in Risk Theory

Published online by Cambridge University Press:  29 August 2014

W. Hürlimann
W. Hürlimann*
Affiliation:
Winterthur, Switzerland
*
Allgemeine Mathematik, Winterthur-Leben, CH-8400 Winterthur, Switzerland.
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Abstract

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Using Laplace transforms and the notion of a pseudo compound Poisson distribution, some risk theoretical results are revisited. A well-known theorem by Feller (1968) and Van Harn (1978) on infinitely divisible distributions is generalized. The result may be used for the efficient evaluation of convolutions for some distributions. In the particular arithmetic case, alternate formulae to those recently proposed by De Pril (1985) are derived and shown more adequate in some cases. The individual model of risk theory is shown to be pseudo compound Poisson. It is thus computable using numerical tools from the theory of integral equations in the continuous case, a formula of Panjer type or the Fast Fourier transform in the arithmetic case. In particular our results contain some of De Pril's (1986/89) recursive formulae for the individual life model with one and multiple causes of decrement. As practical illustration of the continuous case we construct a new two-parametric family of claim size density functions whose corresponding compound Poisson distributions are analytical finite sum expressions. Analytical expressions for the finite and infinite time ruin probabilities are also derived.

Keywords

Pseudo compound Poissonintegral equationinfinite divisibilitymultiple decrement modelruin probability
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 20 , Issue 1 , April 1990 , pp. 57 - 79
Copyright
Copyright © International Actuarial Association 1990

References

REFERENCES

Baker, T. H. (1977) The numerical treatment of integral equations. Oxford, Clarendon Press.Google Scholar
Bellman, R. and Roth, R. S. (1984) The Laplace Treansform. World Scientific.CrossRefGoogle Scholar
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itasca, IL.Google Scholar
Bühlmann, H. (1984) Numerical evaluation of the compound Poisson distribution: recursion or fast Fourier transform? Scand. Actuarial J., 116–126.CrossRefGoogle Scholar
De Pril, N. (1985) Recursions for convolutions of arithmetic distributions. ASTIN Bulletin 15, 135–139.CrossRefGoogle Scholar
De Pril, N. (1986) On the exact computation of the aggregate claims distribution in the individual life model. ASTIN Bulletin 16, 109–112.CrossRefGoogle Scholar
De Pril, N. and Vandenbroek, M. (1987) Recursions for the distribution of a life portfolio: a numerical comparison. Bull. of the Royal Association of Belgian Actuaries.Google Scholar
De Pril, N. (1988) Improved approximations for the aggregate claims distribution of a life insurance portfolio. Scand. Actuarial J.CrossRefGoogle Scholar
De Pril, N. (1989) The aggregate claims distribution in the individual model with arbitrary positive claims. ASTIN Bulletin 19, 9–24.CrossRefGoogle Scholar
Doetsch, G. (1976) Einführung in Theorie und Anwendung der Laplace Transformation, 3te Auflage. Birkhäuser Verlag Basel and Stuttgart. (English translation (1974) by Springer Verlag).CrossRefGoogle Scholar
Feller, W. (1968) An introduction to probability theory and its applications, vol. 1 and 2. New York: John Wiley.Google Scholar
Gerber, H. U. (1979) An introduction to mathematical Risk Theory. Huebner Foundation for Insurance Education, University of Pennsylvania.Google Scholar
Gerber, H.U. and Valderrama Ospina, A. (1987) A simple proof of Feller's characterization of the compound Poisson distributions. Insurance: Mathematics and Economics 6, 63–64.Google Scholar
Giffin, W.C. (1975) Transform techniques for probability modeling. Academic Press.Google Scholar
Hirschman, I.I. and Widder, D.V. (1965) La transformation de convolution. Paris: Gauthier-Villars.Google Scholar
Hürlimann, W. (1986) Error bounds for stop-loss premiums calculated with the fast Fourier transform. Scand. Actuarial J., 107–113.CrossRefGoogle Scholar
Hürlimann, W. (1989a) On maximum likelihood estimation for count data models. Appears in Insurance: Mathematics and Economics.Google Scholar
Hürlimann, W. (1989b) On linear combination of random variables and risk theory. 14. Symposium on Operations Research, Ulm, September 1989.Google Scholar
Jerri, A. J. (1985) Introduction to integral equations with applications. Monographs and textbooks in Pure and Applied Mathematics, vol. 93. New York and Basel: Marcel Dekker.Google Scholar
Kaas, R., van Heerwaarden, A.E. and Goovaerts, M.J. (1988) Between individual and collective model for the total claims. ASTIN Bulletin 18.CrossRefGoogle Scholar
Katti, S.K. (1967) Infinite divisibility of integer-valued random variables. Annals of Mathematical Statistics 38, 1306–1308.CrossRefGoogle Scholar
Kestemont, R.-M. and Paris, J. (1985) Sur l'ajustement du nombre de sinistres. Bulletin of the Association of Swiss Actuaries 85, 157–164.Google Scholar
Kestemont, R.-M. and Paris, J. (1987) On compound Poisson laws. Paper presented at the meeting on Risk Theory in Oberwolfach, West Germany.Google Scholar
Panjer, H.H. (1981) Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
Panjer, H. H. and Willmot, G.E. (1984) Models for the distribution of aggregate claims in risk theory. Transactions of the Society of Actuaries 36, 399–446.Google Scholar
Reimers, L. (1988) Letter to the Editor. ASTIN Bulletin 18.CrossRefGoogle Scholar
Schwartz, L. (1966) Théorie des distributions. Nouvelle édition, Hermann, Paris.Google Scholar
Steutel, F. (1970) Preservation of infinite divisibility under mixing and related results. Math. Centre Tracts 33. Amsterdam: Mathematical Centre.Google Scholar
Steutel, F. (1979) Infinite divisibility in theory and practice. Scandinavian Journal of Statistics 6, 57–64.Google Scholar
Ströter, B. (1985) The numerical evaluation of the aggregate claim density function via integral equations. Blätter der Deutschen Gesellschaft für Versicherungsmathematik, 1–13.Google Scholar
Sundt, B. and Jewell, W. (1981) Further results on recursive evaluation of compound distributions. ASTIN Bulletin 12, 27–39.CrossRefGoogle Scholar
Thyrion, P. (1969) Extension of the collective risk theory. Skandinavisk Aktuarietidskrift 52 (Suppl.), 84–98.Google Scholar
Van Harn, K. (1978) Classifying infinitely divisible distributions by functional equations. Math. Centre Tracts 103. Amsterdam: Mathematical Centre.Google Scholar
Widder, D.V. (1971) An introduction to transform theory. New York and London: Academic Press.Google Scholar
Willmot, G.E. (1988) Sundt and Jewell's family of discrete distributions. ASTIN Bulletin 18, 17–29.CrossRefGoogle Scholar