Heavy-Tailed Loss Frequencies from Mixtures of Negative Binomial and Poisson Counts

Heavy-tailed random variables have been used in insurance research to model both loss frequencies and loss severities, with substantially more emphasis on the latter. In the present work, we take a step toward addressing this imbalance by exploring the class of heavy-tailed frequency models formed by continuous mixtures of Negative Binomial and Poisson random variables. We begin by defining the concept of a calibrative family of mixing distributions (each member of which is identifiable from its associated Negative Binomial mixture), and show how to construct such families from only a single member. We then introduce a new heavy-tailed frequency model -- the two-parameter ZY distribution -- as a generalization of both the one-parameter Zeta and Yule distributions, and construct calibrative families for both the new distribution and the heavy-tailed two-parameter Waring distribution. Finally, we pursue natural extensions of both the ZY and Waring families to a unifying, four-parameter heavy-tailed model, providing the foundation for a novel loss-frequency modeling approach to complement conventional GLM analyses. This approach is illustrated by application to a classic set of Swedish commercial motor-vehicle insurance loss data.