## 4 Numerical Illustration  
This study was based on a subset of claim frequency data from a pool of MTPL insurance policies observed for 3.5 years from a major Greek insurance company. A total of 14,143 observations with complete records (i.e., with availability of all the explanatory variables) were taken for our analysis. The response variable is the number of claims at fault registered for each insured vehicle. In addition, a subset of explanatory variables with the highest predictive power for the response variable was chosen based on exploratory analysis. In particular, we considered the following covariates: the age of the driver (AD), the horsepower (HP) of their car, and the age of their car (AC). Additionally, we grouped the levels of each a priori rating variable with respect to risk profiles with similar claim frequency in order to balance homogeneity as well as sufficiency of the volume of data in each cell. The summary of the explanatory variables and their corresponding groupings with the number of observations in each category along with the descriptive statistics for claim counts are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Statistic**} & \multirow{2}{*}{**Value**} & \multicolumn{2}{c}{**Age of the**} & \multicolumn{2}{c}{**Horsepower of**} & \multicolumn{2}{c}{**Age of the**} \\  & & \multicolumn{2}{c}{**Driver (AD)**} & \multicolumn{2}{c}{**the Car (HP)**} & \multicolumn{2}{c}{**Car (AC)**} \\ \hline \# Obser- & 14,143 & C1: & 3238 & C1: & 5042 & C1: & 4318 \\ Minimum & 0 & C2: & 10,905 & C2: & 9101 & C2: & 9825 \\ Median & 0 & & - & - & - & - \\ Mean & 0.4827 & & - & - & - & - \\ Variance & 0.6988 & & - & - & - & - \\ Maximum & 12 & & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of claim counts and the size of the different categories of the explanatory variables. In the following subsection, we fit the Poisson-lognormal (PLN) regression model on the number of claims. Moreover, we will compare its fit with those of the classic negative binomial type I (NBI) distribution, which has been used in an abundance of actuarial settings for approximating claim counts, for the case when regression components are introduced on its mean and dispersion parameters. Finally, the high presence of zeros in the MTPL data set motivates the use of zero-inflated models, which can provide a parsimonious yet powerful way to handle data sets that contain a large number of zeros. In this study, the zero-inflated Poisson (ZIP) regression model will be used as a benchmark for comparison. * The NBI regression model with varying dispersion is derived as follows. Consider policyholder \(i\), \(i=1,\ldots,n\), whose number of claims, denoted as \(k_{i}\), with \(k_{i}=0,1,2,3,\ldots\), are independent. In addition, assume that \(k_{i}|,z_{i}\) follows a Poisson distribution with pmf given by Equation (1), and \(z_{i}\) follows a Gamma distribution with pdf given by \[g(z_{i};\phi_{i})=\frac{z_{i}^{\frac{1}{\theta_{i}}-1}}{\frac{1}{\phi_{i}}} \frac{1}{\phi_{i}}\exp\Bigl{(}-\frac{z_{i}}{\phi_{i}}\Bigr{)},\] (29) where \(\phi_{i}>0\). Parameterization (29) ensures that \(E(z_{i})=1\), and hence the model is identifiable.