## 4 Empirical Study
### Results with LGPIF (H = 1679)
As outlined in Section 4.2, we account for scenarios where the covariates associated with the negative binomial outcome \(N_{h}\) are complete, while the continuous covariate \(\mathbf{X}^{S}\) for the claim amount outcome is subject to Non-Differential Berkson (NDB) measurement errors. Our focus will center on the log-normal outcome \(\tilde{Y}_{h}\) and its associated NDB covariate \(\mathbf{X}^{S*}\) within the hierarchical modeling framework. **Comparisons [Model (B) vs. Model (C) vs. Model (D)]:** As illustrated in Figure 4, this experiment evaluates which candidate models, Model (C) and -(D), yield results closest to the gold standard, Model (A). Specifically, we compare Model (C), utilizing the Gustafson correction, with Model (D), which employs the SIMEX correction. Model (A), -(B), and -(C) are developed within the Bayesian hierarchical GLM framework, while Model (D) is constructed in the conventional GLM framework, providing a frequentist perspective. Additionally, a sensitivity analysis for the claim amount model is conducted to determine the optimal scaling factor value \(\zeta\), leveraging the prior knowledge: \(\mathbf{\Sigma}_{j}^{2}=(1-\zeta)\hat{\lambda}_{j}^{2}\). Table 2 presents a comparison of the marginal posterior means of the scale parameter \(\sigma_{j}^{2}\) obtained from Model (A), -(B), and -(C). Each hierarchical GLM describes the log-normal outcome: \(\bar{Y}_{h}|\mathbf{X}^{S}\sim\ \mathbf{LogN}(\mu_{h}=ln\big{(}E[\bar{Y}_{h}| \mathbf{X}^{S}]\big{)}-\frac{1}{2}\sigma_{j}^{2},\ \sigma_{j}^{2})\). Additional comparisons of the marginal posterior means of the GLM coefficients \(\mathbf{\beta}_{j}\) are provided in Tables 3 and 4, which also include the results from the sensitivity analysis on the optimal scaling factor \(\zeta\). We first examine Model (B), which incorporates the NDB covariate \(\mathbf{x}^{S*}\), introducing model risk. Our investigation focuses on how the parameter estimates--the scale parameter \(\sigma^{2}\) and the GLM coefficients \(\mathbf{\beta}\)--vary with increasing error rates \(R_{\varepsilon_{x}}\), reflecting different degrees of model risk. As seen in Tables 2-4, estimates from Model (B) consistently show an upward trend in response to higher error rates (\(R_{\varepsilon_{x}}:0.01\to R_{\varepsilon_{x}}:0.40\)). Both the scale parameter \(\sigma^{2}\) and the GLM coefficients \(\beta_{0},\beta_{2}\) (intercept and binary covariate) inflate, while \(\beta_{1}\) (linked to \(\mathbf{x}^{S*}\)) decreases across all clusters \(j=1,\ldots,6\). This pattern aligns with the expectation that as the error rate \(R_{\varepsilon_{x}}\) increases, the added noise results in greater variability in the outcomes, causing a proportional rise in the scale parameter \(\sigma^{2}\). The consistent directional shift in the coefficient \(\beta_{1}\), tied to the NDB covariate \(\mathbf{x}^{S*}\), indicates that \(\mathbf{x}^{S*}\) follows a systematic trend characteristic of NDB error. Turning to Model (C), which applies the Gustafson correction technique, we observe that as the error rate \(R_{\varepsilon_{x}}\) increases, the parameter estimates from Model (C) progressively align with those of the gold standard, Model (A), with the values falling within the credible intervals established by Model (A). This alignment is a promising indicator of our correction method's effectiveness. Note that a key aspect of our error correction technique is the selection of an optimal scaling factor \(\zeta\). Tables 2-4 highlight that Model (C) performs best within the scaling factor range \(0.5\leq\zeta\leq 0.7\). This suggests that the relationship between  
Figure 5: Model (A) Result V: A histogram of the overall expected aggregate claim amount on a log scale, overlaid with the individual cluster-wise distributions \(lnS_{h}|\mathbf{X}^{F},\mathbf{X}^{S}\).