A comparative analysis of several multivariate zero-inflated and zero-modified models with applications in insurance

Abstract.

Given that insurance companies often operate across multiple lines of insurance business, where claim frequencies on different lines are often correlated, it often becomes advantageous to employ multivariate count modeling where dependence between lines of business can be included in the modeling. Due to the operation of bonus-malus systems where there is a reward to the insurance policyholder for not claiming, claims data in automobile insurance often exhibits an excess of common zeros, a characteristic known as multivariate zero-inflation. In this article, we propose two approaches to address this feature. The first approach involves utilizing a multivariate zero-inflated model, where we artificially enhance the probability of common zeros based on standard multivariate count distributions. The second approach applies a multivariate zero-modified model, which separately handles the common zeros and the number of claims incurred in each line given that at least one claim occurs. We present several models under these frameworks, along with detailed inference procedures. In the applications section, we conduct a comprehensive comparative analysis of these models using data from an automobile insurance portfolio.

Keywords:

• Multivariate zero-inflated model
• multivariate zero-modified model
• expectation-maximization algorithm
• minorization-maximization algorithm
• automobile insurance
• ratemaking

Acknowledgments

The authors express their gratitude to the editor and two reviewers for their valuable comments and suggestions, which have significantly contributed to the improvement of this paper.

Disclosure statement

The authors report that there are no competing interests to declare.

Additional information

Funding

The research of Pengcheng Zhang was supported by the National Natural Science Foundation of China (No. 12301607), the Shandong Provincial Natural Science Foundation(No. ZR2023QA091) and the Taishan Scholars Program of Shandong Province (No. tsqnz20230620).