7.3 Mixed frequency distributions
7.3.1 General mixed frequency distribution
Many compound distributions can arise in a way that is very different from compounding. In this section, we examine mixture distributions by treating one or more parameters as being “random” in some sense. This section expands on the ideas discussed in Section 6.3 in connection with the gamma mixture of the Poisson distribution being negative binomial.
We assume that the parameter is distributed over the population under consideration and that the sampling scheme that generates our data has two stages. First, a value of the parameter is selected. Then, given that parameter value, an observation is generated using that parameter value.
In automobile insurance, for example, classification schemes attempt to put individuals into (relatively) homogeneous groups for the purpose of pricing. Variables used to develop the classification scheme might include age, experience, a history of violations, accident history, and other variables. Because there will always be some residual variation in accident risk within each class, mixed distributions provide a framework for modeling this heterogeneity.
Let P(z|θ) denote the pgf of the number of events (e.g., claims) if the risk parameter is known to be θ. The parameter, θ, might be the Poisson mean, for example, in which case the measurement of risk is the expected number of events in a fixed time period.
Let U(θ) = Pr(Θ ≤ θ) be the cdf of Θ, where Θ is the risk parameter, ...