Loss Models: From Data to Decisions, 4th Edition

14.5 Compound models

For the method of moments, the first few moments can be matched with the sample moments. The system of equations can be solved to obtain the moment-based estimators. Note that the number of parameters in the compound model is the sum of the number of parameters in the primary and secondary distributions. The first two theoretical moments for compound distributions are

These results are developed in Chapter 9. The first three moments for the compound Poisson distribution are given in (7.10).

Maximum likelihood estimation is also carried out as before. The loglikelihood to be maximized is

When g_k is the probability of a compound distribution, the loglikelihood can be maximized numerically. The first and second derivatives of the loglikelihood can be obtained by using approximate differentiation methods as applied directly to the loglikelihood function at the maximum value.

EXAMPLE 14.8

Determine various properties of the Poisson–zero-truncated geometric distribution. This distribution is also called the Polya–Aeppli distribution.

For the zero-truncated geometric distribution, the pgf is

and therefore the pgf of the Polya–Aeppli distribution is

The mean is