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Cumulants of Convolution—Mixed Distributions

Published online by Cambridge University Press:  29 August 2014

Alan Brown
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Consider a risk process which is characterised by three stochastic variables

(1) the number of accidents, N,

(2) the number of claims per accident, C, and

(3) the amount of a claim, X.

Let Y be a random variable denoting the total loss in a given period.

Suppose that

and

If Pr represents the probability that exactly r claims occur in the period, then Kupper [4] has shown on certain simplifying assumptions that

where , the probability of exactly r claims in n accidents, is given by

and for γ < n

Further

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 9 , Issue 1-2 , January 1977 , pp. 59 - 63
Copyright
Copyright © International Actuarial Association 1977

References

[1]Beard, R. E., Pentikainen, T. & Pesonen, E., 1969. Risk Theory.Google Scholar
[2]Cox, D. R. & Miller, H. D., 1965. The Theory of Stochastic Processes.Google Scholar
[3]Kendall, M. G. & Stuart, A., 1963. The Advanced Theory of Statistics.Google Scholar
[4]Kupper, J., 1963. Some Aspects of the Cumulative Risk, Astin Bulletin, III, 1.Google Scholar
[5]Thyrion, P., 1960. Note sur les distributions ‘par grappes’, Bulletin de l'Association Royale des Actuaries Beiges.Google Scholar