  This article introduces a k-Inflated Negative Binomial mixture
distribution/regression model as a more flexible alternative to zero-inflated
Poisson distribution/regression model. An EM algorithm has been employed to
estimate the model's parameters. Then, such new model along with a Pareto
mixture model have been employed to design an optimal rate--making system.
Namely, this article employs number/size of reported claims of Iranian third
party insurance dataset. Then, it employs the k-Inflated Negative Binomial
mixture distribution/regression model as well as other well developed counting
models along with a Pareto mixture model to model frequency/severity of
reported claims in Iranian third party insurance dataset. Such numerical
illustration shows that: ({\bf 1}) the k-Inflated Negative Binomial mixture
models provide more fair rate/pure premiums for policyholders under a
rate--making system; and ({\bf 2}) in the situation that number of reported
claims uniformly distributed in past experience of a policyholder (for instance
$k_1=1$ and $k_2=1$ instead of $k_1=0$ and $k_2=2$). The rate/pure premium
under the k-Inflated Negative Binomial mixture models are more appealing and
acceptable.
