Loss Models: From Data to Decisions, 4th Edition

14.2 Negative binomial

The moment equations are

(14.1)

and

(14.2)

with solutions −1 and . Note that this variance estimate is obtained by dividing by n, not n − 1. This is a common, though not required, approach when using the method of moments. Also note that, if s² < , the estimate of β will be negative, an inadmissible value.

EXAMPLE 14.3

(Example 14.1 continued) Estimate the negative binomial parameters by the method of moments.

The sample mean and the sample variance are 2.5 and 3.05 (verify this), respectively, and the estimates of the parameters are = 11.364 and = 0.22.

When compared to the Poisson distribution with the same mean, it can be seen that β is a measure of “extra-Poisson” ...