Loss Models: From Data to Decisions, 4th Edition

14.6 Effect of exposure on maximum likelihood estimation

In Section 7.4, the effect of exposure on discrete distributions is discussed. When aggregate data from a large group of insureds is obtained, maximum likelihood estimation is still possible. The following example illustrates this fact for the Poisson distribution.

EXAMPLE 14.9

Determine the maximum likelihood estimate of the Poisson parameter for the data in Table 14.8.

Table 14.8 Automobile claims by year.

Year	Exposure	Claims
1986	2,145	207
1987	2,452	227
1988	3,112	341
1989	3,458	335
1990	3,698	362
1991	3,872	359

Let Let λ be the Poisson parameter for a single exposure. If year κ has e_k exposures, then the number of claims has a Poisson distribution with parameter λe_k. If n_k is the number of claims in year κ, the likelihood function is

The maximum likelihood estimate is found by

In this example the answer is what we expected it to be: the average number of claims per exposure. This technique will work for any distribution in the (a, b, 0)⁵ and compound classes. But care must be taken in the interpretation of the model. For example, if we use a negative binomial distribution, we are assuming that each exposure ...