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

## TL;DR

This paper proposes a flexible k-Inflated Negative Binomial mixture regression model for insurance claim frequency analysis, demonstrating its effectiveness in designing fairer rate-making systems compared to traditional models.

## Contribution

It introduces a novel k-Inflated Negative Binomial mixture regression model and applies it to insurance data, showing improved premium fairness over existing models.

## Key findings

- The new model provides more equitable premiums for policyholders.
- It outperforms traditional models in modeling claim frequency.
- The model is effective when claim counts are uniformly distributed in past data.

## 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 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.