## 5 Conclusions, Limitations, and Future Research  
In this article, we considered a family of mixed Poisson claim count regression models with varying dispersion and dependence parameters arising from non-conjugate mixing distributions for approximating overdispersed claim frequencies in non-life insurance. The flexibility in the choice of the mixing distribution combined with the proposed approach, which assumes that the mean and dispersion parameters of the model can be modelled in terms of risk factors, can provide an advantage relative to the majority of previous approaches in the literature, which have concentrated on mixing densities conjugate to the Poisson mixing and assumed that only the mean parameter can vary through covariates. From a practical business standpoint, the proposed modelling framework is beneficial for the insurance company, as it will result in an improved risk evaluation of policyholders who are more likely to have accidents, since the tail behaviour of mixed Poisson models is \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Specification Criteria Values**} \\ \hline  & DEV & AIC & SBC \\
**NBI** & 15,885.1 & 15,897.1 & 15,940.1 \\
**ZIP** & 16,052.2 & 16,062.2 & 16,098 \\
**PLN** & 15,859.4 & 15,871.4 & 15,914.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: ZIP regression model and NBI and PLN regression models comparison. similar to that of the mixing density and the majority of heavy-tailed mixing distributions are not conjugate to the Poisson. The PLN regression model with regression specifications on its mean and dispersion parameters was considered for expository purposes. Furthermore, we developed an efficient EM algorithm for maximum likelihood estimation of the parameters of the model. The implementation of the algorithm was illustrated by fitting the model to a real MTPL insurance data set. An interesting line for further research would be to extend the model to the multivariate case to permit inferences about the dependence structure between different types of overdispersed claim counts from the same and/or different types of coverage. However, it should be noted that the PLN model becomes more complicated in the two-dimensional setting due to algebraic intractability, which is a problem that is inherited from the univariate case. Moreover, modelling all the parameters of the PLN model in terms of covariates can further increase the computational burden in the high-dimensional setting. Finally, another fruitful future research direction is to include time series components to take into account both cross-dependence between different types of claims and time dependence. Conceptualization, G.T. ; methodology, G.T. ; software, G.T. ; formal analysis, G.T., N.H. and R.H.; investigation, G.T., N.H. and R.H.; data curation, G.T., N.H. and R.H.; writing--original draft preparation, G.T., N.H. and R.H.; writing--review and editing, G.T., N.H. and R.H.; supervision, G.T. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. **Data Availability Statement:** Please note that we cannot make the data supporting reported results publicly available due to data use agreement with the company which provided the data set. However, a toy data set along with the code for estimating the parameters of the PLN regression model with varying dispersion can be made available upon request. We would like to thank the handling editor and the two anonymous referees for their very helpful comments and suggestions that have significantly improved this article. Furthermore, we would like to thank the participants of the 13 th International Conference on Computational and Methodological Statistics. Finally, we would like to thank the undergraduate student [PERSON] for his interest in the research that was undertaken in this article and for his help in citing previous works. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: