## 4 Numerical Illustration
### Modelling Results  
The ML estimates of the parameters (all the parameters were statistically significant at a 5% threshold) for the NBI and PLN regression models with varying dispersion and the ZIP regression model are presented in Table 2. Note that variable selection can be performed for all the models by selecting the best predictor for parameter \(\mu_{i}\) using backward elimination. This can be done by including all available explanatory variables present in the data set and testing whether the exclusion of each variable will result in lower global deviance (DEV), Akaike information criterion (AIC), and Schwartz Bayesian criterion (SBC) values. Subsequently, in the case of the NBI and PLN models, we can take all the variables selected for the parameter \(\mu_{i}\) and continue variable selection for the parameter \(\phi_{i}\) by performing forward selection, where we can test which explanatory variable would lead to a further decrease of the DEV, AIC, and SBC values when added to parameter \(\phi_{i}\). Additionally, if different subsets of explanatory variables result in very similar values of DEV, AIC, and SBC, we should chose the simpler model with less predictors to avoid overfitting. Regarding our data, as we can see from Table 2, the explanatory variables AD, HP, and AC were chosen for \(\mu_{i}\), and only the variable AD was chosen for \(\phi_{i}\). From the results in Table 2, we observe that the values of the estimated regression coefficients of the variables AD, HP, and AC have a similar effect (positive and/or negative) on parameter \(\mu_{i}\) in the case of all the models, and the same observation can be made for parameter \(\phi_{i}\) in the case of the NBI and PLN models. Finally, we rely on normalized quantile residuals [13] as an exploratory graphical tool to help us evaluate the adequacy of the fit of the NBI, ZIP, and PLN models. For these discrete response distributions, the normalized (randomized) quantile residuals are defined as \(\hat{r}_{i}=\Phi^{-1}(u_{i})\), where \(\Phi^{-1}\) is the inverse cumulative distribution function of a standard normal distribution and where \(u_{i}\) is defined as a random value from the uniform distribution on the interval \(\left[F_{i}(k_{i}-1|\mathbf{\theta}^{(r+1)}),F_{i}(k_{i}|\mathbf{\theta}^{(r+1)})\right]\), where \(F_{i}\) is the cumulative distribution function estimated for the \(i\)th policyholder, and where \(\mathbf{\theta}^{(r+1)}\) contains the estimated model parameters after the EM algorithm has reached the global maximum, and \(k_{i}\) is the corresponding observation. The fit of the claim count model can be evaluated \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **NBI** & **ZIP** & **PLN** \\ \hline \multicolumn{4}{c}{Coeff. \(\mathbf{\beta}_{1}\)} & \multicolumn{2}{c}{Coeff. \(\mathbf{\beta}_{1}\)} & \multicolumn{2}{c}{Coeff. \(\mathbf{\beta}_{1}\)} \\ \hline Intercept & \(-0.4729\) & Intercept & \(-0.1277\) & Intercept & \(-0.4709\) \\ AD & & CS & & CS & \\ C2 & \(-1.2390\) & C2 & \(-1.2454\) & C2 & \(-1.2360\) \\ HP & & HP & & HP & \\ C2 & \(1.0378\) & C2 & \(0.9892\) & C2 & \(1.0469\) \\ AC & & AC & & AC & \\ C2 & \(-0.6481\) & C2 & \(-0.6398\) & C2 & \(-0.6586\) \\ \hline \multicolumn{4}{c}{Coeff. \(\mathbf{\beta}_{2}\)} & \multicolumn{2}{c}{Coeff. \(\mathbf{\beta}_{2}\)} \\ \hline Intercept & \(-2.4935\) & Prob. \(\pi\) & \(0.3032\) & Intercept & \(-1.0969\) \\ AD & & & CS & \\ C2 & \(0.8878\) & & & C2 & \(0.3481\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter estimates of the ZIP regression model and the NBI and PLN regression models with varying dispersion. by means of the usual quantile-quantile plots. Specifically, if the data indeed follow the assumed distribution, then the residual on the quantile-quantile plot will fall approximately on a straight line. Figure 1 shows the normalized (random) quantiles for the ZIP regression model and the NBI and PLN claim frequency regression models with varying dispersion.