The material here and in Chapter 12 has traditionally been presented under the heading of “survival models” with the accompanying notion that the techniques are useful only when studying lifetime distributions. Standard texts on the subject such as Klein and Moeschberger  and Lawless  contain examples that are exclusively oriented in that direction. However, as is shown in Chapters 11 and 12, the same problems that occur when modeling lifetime occur when modeling payment amount. The examples we present are of both types. Only a handful of references are presented, most of the results being well developed in the survival models literature. Readers wanting more detail and proofs should consult a text dedicated to the subject, such as the ones just mentioned.
In this chapter, it is assumed that the type of model is known but not the full description of the model. In Chapter 4, models were divided into two types—data dependent and parametric. The definitions are repeated here.
Definition 11.1 A data-dependent distribution is at least as complex as the data or knowledge that produced it, and the number of “parameters” increases as the number of data points or amount of knowledge increases.
Definition 11.2 A parametric distribution is a set of distribution functions, each member of which is determined by specifying one or more values called parameters. The number of parameters is fixed and finite.
Here, only two data-dependent distributions are considered. They ...