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

*Payandeh Najafabadi Amir T.* () and
*MohammadPour Saeed*

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Payandeh Najafabadi Amir T.: Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, 1983963113, Tehran, Iran

MohammadPour Saeed: Headquarters of Iran Insurance Company, Tehran, Iran

*Asia-Pacific Journal of Risk and Insurance*, 2018, vol. 12, issue 2, 31

**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 k1=1$k_1=1$ and k2=1$k_2=1$ instead of k1=0$k_1=0$ and k2=2$k_2=2$). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.

**Keywords:** Negative Binomial regression; Poisson regression; mixture model; overdispersed behavior; heavy–tail behavior; inflated model; EM algorithm; rate–making system (search for similar items in EconPapers)

**Date:** 2018

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