  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.
