September 2012
Beginner to intermediate
536 pages
14h 40m
English
book
Loss Models: From Data to Decisions, 4th Edition
by Stuart A. Klugman, Harry H. Panjer, Gordon E. Willmot
Content preview from Loss Models: From Data to Decisions, 4th Edition
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9.4 Analytic results
For most choices of distributions of N and the Xjs, the compound distributional values can only be obtained numerically. Subsequent sections in this chapter are devoted to such numerical procedures.
However, for certain combinations of choices, simple analytic results are available, thus reducing the computational problems considerably.
EXAMPLE 9.7
(Compound negative binomial–exponential) Determine the distribution of S when the frequency distribution is negative binomial with an integer value for the parameter r and the severity distribution is exponential.
The mgf of S is
With a bit of algebra, this can be rewritten as
which is of the form
where
the pgf of the binomial distribution with parameters r and β/(1 + β), and M*X(z) is the mgf of the exponential distribution with mean θ(1 + β).
This transformation reduces the computation of the distribution function to the finite sum, that is,
When r = 1, S has a compound geometric distribution, and in this case the ...
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