7.1 Compound frequency distributions

A larger class of distributions can be created by the processes of compounding any two discrete distributions. The term compounding reflects the idea that the pgf of the new distribution PS(z) is written as

(7.1) equation

where PN(z) and PM(Z) are called the primary and secondary distributions, respectively.

The compound distributions arise naturally as follows. Let N be a counting random variable with pgf PN(z). Let M1, M2, … be i.i.d. counting random variables with pgf PM(Z). Assuming that the Mjs do not depend on N, the pgf of the random sum S = M1 + M2 + ··· + MN (where N = 0 implies that S = 0) is PS(z) = PN[PM(z)]. This is shown as


In insurance contexts, this distribution can arise naturally. If N represents the number of accidents arising in a portfolio of risks and {Mk : k = 1, 2, …, N} represents the number of claims (injuries, number of cars, etc.) from the accidents, then S represents the total number of claims from the portfolio. This kind of interpretation is not necessary to justify the use of a compound distribution. If a compound distribution fits data well, that may be enough justification itself. Also, there are other motivations for these distributions, as presented in Section 7.18.


Demonstrate that any zero-modified distribution ...

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