Loss Models: From Data to Decisions, 4th Edition

5.3 Selected distributions and their relationships

5.3.1 Introduction

There are many ways to organize distributions into groups. Families such as Pearson (12 types), Burr (12 types), Stoppa (5 types), and Dagum (11 types) are discussed in Chapter 2 of [58]. The same distribution can appear in more than one system, indicating that there are many relations among the distributions beyond those presented here. The systems presented in Section 5.3.2 are particularly useful for actuarial modeling because all the members have support on the positive real line and all tend to be skewed to the right. For a comprehensive set of continuous distributions, the two volumes by Johnson, Kotz, and Balakrishnan [52] and [53] are a valuable reference. In addition, there are entire books devoted to single distributions (such as Arnold [5] for the Pareto distribution). Leemis and McQueston [66] present 76 distributions on one page with arrows showing all the various relationships.

5.3.2 Two parametric families

As noted when defining parametric families, many of the distributions presented in this section and in Appendix A are special cases of others. For example, a Weibull distribution with τ = 1 and θ arbitrary is an exponential distribution. Through this process, many of our distributions can be organized into groupings, as illustrated in Figures 5.2 and 5.3. The transformed beta family includes two special cases of a different nature. The paralogistic and inverse paralogistic distributions are created ...