A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems

Abstract

Abstract 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 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: (1) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance k 1 = 1 $k_1=1$ and k 2 = 1 $k_2=1$ instead of k 1 = 0 $k_1=0$ and k 2 = 2 $k_2=2$ ). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.