Loss Models: From Data to Decisions, 4th Edition

14.1 Poisson

The principles of estimation discussed earlier in this chapter for continuous models can be applied equally to frequency distributions. We now illustrate the methods of estimation by fitting a Poisson model.

EXAMPLE 14.1

A hospital liability policy has experienced the number of claims over a 10-year period given in Table 14.1. Estimate the Poisson parameter using the method of moments and the method of maximum likelihood.

Table 14.1 Number of hospital liability claims by year.

Year	Number of claims
1985	6
1986	2
1987	3
1988	0
1989	2
1990	1
1991	2
1992	5
1993	1
1994	3

These data can be summarized in a different way. We can count the number of years in which exactly zero claims occurred, one claim occurred, and so on, as in Table 14.2.

Table 14.2 Hospital liability claims by frequency.

Frequency (k)	Number of observations (nk)
0	1
1	2
2	3
3	2
4	0
5	1
6	1
7+	0

The total number of claims for the period 1985–1994 is 25. Hence, the average number of claims per year is 2.5. The average can also be computed from Table 14.2. Let n_k denote the number of years in which a frequency of exactly k claims occurred. The expected frequency (sample mean) is

where n_k represents the number of observed values at frequency k. Hence the method-of-moments ...