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Heavy-tailed random variables have been used in insurance research to modelboth loss frequencies and loss severities, with substantially more emphasis onthe latter. In the present work, we take a step toward addressing thisimbalance by exploring the class of heavy-tailed frequency models formed bycontinuous mixtures of Negative Binomial and Poisson random variables. We beginby defining the concept of a calibrative family of mixing distributions (eachmember of which is identifiable from its associated Negative Binomial mixture),and show how to construct such families from only a single member. We thenintroduce a new heavy-tailed frequency model -- the two-parameter ZYdistribution -- as a generalization of both the one-parameter Zeta and Yuledistributions, and construct calibrative families for both the new distributionand the heavy-tailed two-parameter Waring distribution. Finally, we pursuenatural extensions of both the ZY and Waring families to a unifying,four-parameter heavy-tailed model, providing the foundation for a novelloss-frequency modeling approach to complement conventional GLM analyses. Thisapproach is illustrated by application to a classic set of Swedish commercialmotor-vehicle insurance loss data.

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