## Abstract

Poisson random effect models with a shared random effect have been widely used in actuarial science for analyzing the number of claims. In particular, the random effect is a key factor in a posteriori risk classification. However, the necessity of the random effect may not be properly assessed due to the dual role of the random effect; it affects both the marginal distribution of the number of claims and the dependence among the numbers of claims obtained from an individual over time. We first show that the score test for the nullity of the variance of the shared random effect can falsely indicate significant dependence among the numbers of claims even though they are independent. To mitigate this problem, we propose to separate the dual role of the random effect by introducing additional random effects to capture the overdispersion part, which are called saturated random effects. In order to circumvent heavy computational issues by the saturated random effects, we choose a gamma distribution for the saturated random effects because it gives the closed form of marginal distribution. In fact, this choice leads to the negative binomial random effect model that has been widely used for the analysis of frequency data. We show that safer conclusions about the a posteriori risk classification can be made based on the negative binomial mixed model under various situations. We also derive the score test as a sufficient condition for the existence of the a posteriori risk classification based on the proposed model.

Original language | English |
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Pages (from-to) | 26-36 |

Number of pages | 11 |

Journal | Insurance: Mathematics and Economics |

Volume | 91 |

DOIs | |

State | Published - Mar 2020 |

### Bibliographical note

Funding Information:Woojoo Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( NRF-2016R1D1A1B03936100 ) and the Next-Generation BioGreen 21 program (Project No. PJ01337701 ), Rural Development Administration, Republic of Korea. Jae Youn Ahn was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government ( NRF-2017R1D1A1B03032318 ).

Publisher Copyright:

© 2019 Elsevier B.V.

## Keywords

- Claim frequency
- Dependence
- Experience ratemaking
- Negative binomial distribution
- Poisson random-effect model