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Finite Sum Evaluation of the Negative Binomial-Exponential Model*

Published online by Cambridge University Press:  29 August 2014

Harry H. Panjer  and
Gordon E. Willmot
Harry H. Panjer
Affiliation:
University of Waterloo, Ontario, Canada
Gordon E. Willmot
Affiliation:
University of Waterloo, Ontario, Canada
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The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.

The result is a generalization of Lundberg (1940).

Consider the distribution of

where X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution function

and N is an integer valued random variable with probability function

Then the distribution function of S is given by

If MX(t), MN(t) and MS(t) are the associated moment generating functions, then

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 12 , Issue 2 , December 1981 , pp. 133 - 137
Copyright
Copyright © International Actuarial Association 1981

Footnotes

*

This research was supported by the Natural Sciences and Engineering Research Council of Canada.

References

REFERENCES

Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions, National Bureau of Standards, Washington.Google Scholar
Johnson, N. L. and Kotz, S. (1969). Distributions in Statistics: Discrete Distributions, John Wiley and Sons, New York.Google Scholar
Lundberg, O. (1940). On Random Processes and their Application to Sickness and Accident Statistics, Almqvist and Wiksells, Uppsala.Google Scholar