Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-18T22:30:29.585Z Has data issue: false hasContentIssue false
  • English
  • Français

Recursions for Compound Distributions*

Published online by Cambridge University Press:  29 August 2014

H. H. Panjer  and
G. E. Willmot
H. H. Panjer
Affiliation:
University of Waterloo, Ontario, Canada
G. E. Willmot
Affiliation:
University of Waterloo, Ontario, Canada
Rights & Permissions [Opens in a new window]

Extract

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.

Various methods for developing recursive formulae for compound distributions have been reported recently by Panjer (1980, including discussion), Panjer (1981), Sundt and Jewell (1981) and Gerber (1982) for a class of claim frequency distributions and arbitrary claim amount distributions. The recursions are particularly useful for computational purposes since the number of calculations required to obtain the distribution function of total claims and related values such as net stop-loss premiums may be greatly reduced when compared with the usual method based on convolutions.

In this paper a broader class of claims frequency distributions is considered and methods for developing recursions for the corresponding compound distributions are examined. The methods make use of the Laplace transform of the density of the compound distribution.

Consider the class of claim frequency distributions which has the property that successive probabilities may be written as the ratio of two polynomials. For convenience we write the polynomials in terms of descending factorial powers. For obvious reasons, only distributions on the non-negative integers are considered.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 13 , Issue 1 , June 1982 , pp. 1 - 12
Copyright
Copyright © International Actuarial Association 1982

Footnotes

*

This research was supported by the Natural Sciences and Engineering Research Council of Canada. The authors are indebted to an anonymous referee for a number of comments that greatly improved this paper.

References

Buhlmann, H. (1970). Mathematical Methods in Risk Theory. Springer-Verlag: New York.Google Scholar
Gerber, H. U. (1982). On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums. Insurance: Mathematics and Economics, 1, 13–18.Google Scholar
Goovaerts, M. J. and Van Wouwe, M. (1981). The generalized Waring distribution as a mixed Poisson with a generalized gamma mixing distribution, Bulletin K. V.B.A., 95–98.Google Scholar
Irwin, J. O. (1965). A unified derivation of some well-known frequency distributions. Journal of the Royal Statistical Society, Series A, 118, 394–404.Google Scholar
Irwin, J. O. (1968). The generalized Waring distribution applied to accident theory. Journal of the Royal Statistical Society, Series A, 130, 205–225.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1969). Distribution in Statistics: discrete distributions. John Wiley: New York.Google Scholar
Panjer, H. H. (1980). The aggregate claims distribution and stop-loss reinsurance. Transactions of the Society of Actuaries, 32, 523–535.Google Scholar
Panjer, H. H. (1981). Recursive evaluation of a family of compound distribution. ASTIN Bulletin 12, 22–26.CrossRefGoogle Scholar
Seal, H. (1978). Survival probabilities: The goal of risk theory. John Wiley: New York.Google Scholar
Skellam, J. G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as a variable between the sets of trials. Journal of the Royal Statistical Society, Series B, 10, 257–261.Google Scholar
Sundt, B. and Jewell, W. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin, 12, 27–39.CrossRefGoogle Scholar