Loss Models: From Data to Decisions, 4th Edition

7.4 Effect of exposure on frequency

Assume that the current portfolio consists of n entities, each of which could produce claims. Let N_j be the number of claims produced by the jth entity. Then N = N₁ + ··· + N_n. If we assume that the N_js are independent and identically distributed, then

Now suppose the portfolio is expected to expand to n* entities with frequency N*. Then

Thus, if N is infinitely divisible, the distribution of N* will have the same form as that of N, but with modified parameters.

EXAMPLE 7.18

It has been determined from past studies that the number of workers compensation claims for a group of 300 employees in a certain occupation class has the negative binomial distribution with β = 0.3 and r = 10. Determine the frequency distribution for a group of 500 such individuals.

The pgf of N* is

which is negative binomial with β = 0.3 and r = 16.67.

For the (a, b, 0) class, all members except the binomial have this property. For the (a, b, 1) class, none of the members do. For compound distributions, it is the primary distribution that must be infinitely divisible. In ...