## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
\begin{cases}x_{j(h)}^{S}|x_{j(h)}^{S}\sim \ \mathbf{N}(\mathbf{x}_{j(h)^{*}}^{S},\ \mathbf{\bar{x}}_{j}^{2})\\ \mathbf{\bar{x}}_{j}^{2}\sim\ \text{undetermined}\\ x_{j(h)}^{S}|z_{j(h)}^{S}\sim\ \mathbf{N}(\kappa_{\bar{\eta}0}+\kappa_{j1}z_{j(h) ^{*}}^{S},\ \lambda_{j}^{2S})\\ \kappa_{j}\sim\ \mathbf{MVN}(\bar{\kappa},\ \lambda_{j}^{2S}\mathbf{\bar{x}}_{ \kappa})\\ \lambda_{j}^{2S}\sim\ \mathbf{InvGa}(c_{0}^{S},d_{0}^{S})\\ z_{j(h)}^{S}\sim\ \mathbf{Bernoulli}(\pi_{j}^{S})\\ \pi_{j}^{S}\sim\ \mathbf{Beta}(g_{0}^{S},h_{0}^{S})\end{cases}  \tag{26e}\]  
Within the hierarchical framework established in Equation (26), both models for \(N_{h}|\mathbf{X}^{F}\) and \(\bar{Y}_{h}|\mathbf{X}^{S*}\) are elaborated to compute \(S_{h}|\mathbf{X}\). However, this section will focus solely on modeling the claim amount \(\bar{Y}_{h}|\mathbf{X}^{S*}\) for demonstration purposes. This is based on the assumption that the claim amount model \(\bar{Y}_{h}|\mathbf{X}^{S*}\) is solely influenced by the NDB covariate \(\mathbf{x}^{*}\). For simplicity, we will denote the covariate matrix \(\mathbf{X}^{S*}\) and the continuous covariate vector \(\mathbf{x}^{S*}\) as \(\mathbf{X}^{*}\) and \(\mathbf{x}^{*}\), respectively. We will start by constructing the complete joint model outlined in Equation (25) to derive the model for the claim amount \(\bar{Y}_{h}|\mathbf{X}^{*}\). This entails identifying previously defined 'linking component' (also known as measurement model, termed by [PERSON] [23]) that connects \(\mathbf{x}^{*}\) to \(\mathbf{x}\) within a hierarchical framework. We define the linking component under the assumption that \(\mathbf{x}^{*}\) is a normally distributed random variable:  
\[f_{N}(x_{h}^{*}|x_{h})=\frac{1}{\sqrt{2\pi\mathbf{\bar{x}}_{j}^{2}}}\exp\{ \frac{-(x_{h}^{*}-x_{h})^{2}}{2\mathbf{\bar{x}}_{j}^{2}}\} \tag{27}\]  
where \(x_{h}^{*}|x_{h}=x_{h}+\epsilon_{j}\sim\mathbf{N}(x_{h},\mathbf{\bar{x}}_{j}^ {2})\), \(\mathbf{\bar{x}}_{j}^{2}:V(\mathbf{x}^{*}|\mathbf{x})\), \(\epsilon_{j}\sim\mathbf{N}(0,\sigma_{j\epsilon}^{2})\), and \(\mathbf{\bar{x}}_{j}^{2}=\sigma_{x}^{2}+\sigma_{j\epsilon}^{2}\). This suggests that the prior knowledge of the dispersion \(\mathbf{\bar{x}}_{j}^{2}\) in Equation (27) captures the cluster-specific traits of the NDB covariate, derived from the relationship between the NDB covariate \(\mathbf{x}^{*}\) and the true covariate \(\mathbf{x}\). However, initially estimating \(\mathbf{\bar{x}}_{j}^{2}\) is challenging because both the true covariate variance \(\sigma_{x}^{2}\) and the cluster-wise error variance \(\sigma_{j\epsilon}^{2}\) are unknown. To complete the joint model in Equation (25), we define the _outcome_ and _covariate_ components (referred to as 'exposure model' in [PERSON]'s terminology [23]) as outlined below:  
\[f_{LogN}(\bar{Y}_{h}|x_{h},z_{h})= \frac{1}{\bar{Y}_{h}\sqrt{2\pi\sigma_{j}^{2}}}\exp\{-\frac{1}{2} \Big{[}\frac{ln\bar{Y}_{h}-(\beta_{j0}+\beta_{j1}x_{h}+\beta_{j2}z_{h})}{ \sigma_{j}}\Big{]}^{2}\} \tag{28a}\] \[f_{N}(x_{h}|z_{h})= \frac{1}{\sqrt{2\pi\lambda_{j}^{2}}}\exp\{\frac{-(x_{h}-\{\kappa_{ j0}+\kappa_{j1}z_{h}\})^{2}}{2\lambda_{j}^{2}}\} \tag{28b}\]  
where \(\bar{Y}_{h}|x_{h},z_{h}\sim\mathbf{LogN}(\mathbf{X}_{h}\beta_{j},\sigma_{j}^{ 2})\), \(\sigma_{j}^{2}:V(\bar{Y}|\mathbf{X})\), \(x_{h}|z_{h}\sim\mathbf{N}(\kappa_{j0}+\kappa_{j1}z_{h},\ \ \lambda_{j}^{2})\), and \(\lambda_{j}^{2}:V(\mathbf{x}|\mathbf{z})\) as shown in Equations (26c) and (26e). Note that the three terms in Equations (27) and (28) are essential building blocks, as they must be multiplied to form the complete joint model specified in Equation (25). However, the two terms in Equation (28) are largely theoretical since the true covariate \(\mathbf{x}\) remains unknown.