Loss Models, 5th Edition

12Frequentist Estimation for Discrete Distributions

There are special considerations that apply when estimating parameters for discrete distributions. They are discussed in this chapter.

12.1 The Poisson Distribution

The principles of estimation discussed in Chapters 10 and 11 can be applied to frequency distributions. We now illustrate the methods of estimation by fitting a Poisson model.

Example 12.1

A hospital liability policy has experienced the number of claims over a 10-year period given in Table 12.1. Estimate the Poisson parameter using the method of moments and the method of maximum likelihood.

Table 12.1 The 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 12.2.

Table 12.2 Hospital liability claims by frequency.

Frequency (k)	Number of observations
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 12.2. Let denote the number ...