## 4 Empirical Study
### Results with LGPIF (H = 1679)
\begin{table}
\begin{tabular}{c|c|c} \hline \hline
**Model** & **Intercept** & **Parameter Estimates \(\beta_{\beta\beta}\)** \\ \hline  & \(\beta_{0}\)\({}_{j-1}\) & 7.18 with 95\% Credible Interval: \(\{4.98\leq\beta_{0}\}_{j-1}\leq 9.32\}\) \\ Model (A) & \(\beta_{0}\)\({}_{j-2}\) & 7.69 with 95\% Credible Interval: \(\{5.50\leq\beta_{0}\}_{j-2}\leq 9.71\}\) \\ Gold & \(\beta_{0}\)\({}_{j-3}\) & 6.37 with 95\% Credible Interval: \(\{4.11\leq\beta_{0}\}_{j-3}\leq 8.86\}\) \\ standard & \(\beta_{0}\)\({}_{j-4}\) & 7.36 with 95\% Credible Interval: \(\{3.83\leq\beta_{0}\}_{j-4}\leq 10.52\}\) \\  & \(\beta_{0}\)\({}_{j-5}\) & 7.14 with 95\% Credible Interval: \(\{3.83\leq\beta_{0}\}_{j-5}\leq 10.32\}\) \\  & \(\beta_{0}\)\({}_{j-6}\) & 7.36 with 95\% Credible Interval: \(\{4.87\leq\beta_{0}\}_{j-6}\leq 10.46\}\) \\ \hline  & Error rate \(R_{c_{i}}\) in \(\mathbf{x}^{*}\): 0.01 & Error rate \(R_{c_{i}}\) in \(\mathbf{x}^{*}\): 0.01 & Error rate \(R_{c_{i}}\) in \(\mathbf{x}^{*}\): 0.40 \\ \hline  & \(\beta_{0}\)\({}_{j-1}\) & 7.28 & 8.89 & 9.40 \\ Model (B) & \(\beta_{0}\)\({}_{j-2}\) & 7.72 & 9.14 & 9.51 \\ Before & \(\beta_{0}\)\({}_{j-3}\) & 6.44 & 8.82 & 9.28 \\ Correction & \(\beta_{0}\)\({}_{j-4}\) & 7.33 & 9.28 & 9.52 \\  & \(\beta_{0}\)\({}_{j-5}\) & 7.12 & 9.37 & 9.57 \\  & \(\beta_{0}\)\({}_{j-6}\) & 7.46 & 9.11 & 9.72 \\ \hline  & \(\zeta:0.5\) & \(\zeta:0.6\) & \(\zeta:0.7\) & \(\zeta:0.5\) & \(\zeta:0.6\) & \(\zeta:0.7\) \\ \hline  & \(\beta_{0}\)\({}_{j-1}\) & 6.0 The challenge, however, lies in identifying the optimal range of \(\zeta\) values for a specific error rate \(R_{ex}\) in the covariate \(\mathbf{x}^{*}\). Our findings from this LGPIF experiment show that when \(\zeta\) deviates from the \(0.5\leq\zeta\leq 0.7\) range, Model (C)'s corrections become ineffective, often performing worse than the erroneous Model (B). This worsens as the error rate \(R_{ex}\) increases, which is expected, because a higher error rate implies greater bias, making it much harder for the model to correct. Interestingly, our experimental results suggest that the optimal \(\zeta\) range can be effectively identified by evaluating the L PPD for each modeling result, as the L PPD reflects the degree of predictive performance. This reflects how closely the estimated parameter values align with the gold standard. Therefore, by searching for the modeling results with the maximum L PPD, we can systematically pinpoint the optimal \(\zeta\), leading to enhanced correction performance without requiring access to the gold standard data. Our LGPIF experimental results are further analyzed in Figure 6 and Table 5 for the error rate \(R_{ex}=0.01\), Figure 7 and Table 6 for the error rate \(R_{ex}=0.10\), and Figure 8 and Table 7 for the error rate \(R_{ex}=0.40\). For comparisons, Tables 5-7 present (i) LPDP, (ii) SSPE, (iii) SAPE, and (iv) Kullback-Leibler Divergence (\(D_{KL}\)) for the individual claim amount model \(f(\tilde{Y}_{h}|\mathbf{X}^{5})\) as well as CTEs for the aggregate claim amount model \(f(S_{h}|\mathbf{X}^{\tilde{F}},\mathbf{X}^{5})\) within each error rate scenario. The results from Model (D), traditional GLM with SIMEX correction, are also included. Figures 6-8 display histograms of the testing set across all clusters \(j=1,\ldots,6\), comparing the out-of-sample prediction curves from Model (C) (blue) against the gold standard Model (A) (red) and the erroneous Model (B) (dotted). The prediction curve for Model (D) is omitted due to its subpar performance. As expected, Tables 5-7 show that the gold standard model, Model (A), achieves the highest LPDP value of \(-16\),\(155.90\), while the naive model with model risk, Model (B), consistently exhibits the lowest LPDP values across datasets with varying error rates. Note that the LPDP for the GLM-based SIMEX method is unavailable, as L PPD calculations require posterior densities.