## 1 Introduction
### Research Questions  
Suppose we are dealing with a group policy where each policy \(h\) can generate multiple claims [5]. For example, each policy \(h\) is taken out by a different company, such as an oil company, a car manufacturer, etc., resulting in \(h=1,\ldots,H\) unique policies in the insurer's portfolio. Each company holds a single group policy \(h\) to protect its multiple assets. Let the individual claim amount \(Y_{hi},\ i=1,2,\ldots,N_{h}(t)\) associated with \(N_{h}(t)\) different claims during a policy period \(t\) for a single policy \(h\) be log-normally distributed (i.e., \(lnY_{hi}\sim\mathbf{N}(\mu(t),\ \sigma(t)^{2})\)). Assuming the policy period \(t\) is fixed and short, as is typical in a non-life insurance context, we can ignore \(t\) for simplicity. Under this assumption, the claim count \(N_{h}\) for policy \(h\) remains limited and follows a negative binomial distribution [3]. The individual claim amount \(Y_{hi}\), as well as the mean claim amount \(\bar{Y}_{h}\) for a policy \(h\) follows a log-normal distribution, assuming the claim count is not large enough for the Central Limit Theorem to apply. Each policy, treated as an individual observation, has a unique \(N_{h}\) value because it is linked to a distinct group of insured assets requiring insurance protection. Hence, for each policy \(h\), the aggregate claim amounts \(S_{h}\) received by an insurer can be defined as a log-normal convolution: \(S_{h}=\Sigma_{i=1}^{N_{h}}Y_{hi}=Y_{h1}+Y_{h2}+\ldots+Y_{N_{h}N_{h}}=N_{h}\bar {Y}_{h}\). In order to determine the future value of the expected aggregate claim amounts \(E[S_{h}]\) for a policy \(h\), the traditional risk modeling principle applies the _frequency-severity_ approach [6], which involves modeling the frequency and severity of claims (the number of claims and the size of each claim) separately. Resting on the assumption that the summands \(Y_{hi}\) are mutually independent and identically distributed (i.i.d.) to maintain homogeneity, the expected aggregate claim amount \(E[S_{h}]=E[Y_{h1}]+E[Y_{h2}]+\ldots+E[Y_{hN_{h}}]\) for a policy \(h\) can be simplified as:  
\[E[S_{h}]=E[N_{h}]\times E[\bar{Y}_{h}]\ \ \text{for a policy $h$ by the Frequency-Severity approach.} \tag{1}\]  
Now, let \(\mathbf{X}=\{\mathbf{X}^{F},\mathbf{X}^{S}\}\) represent the covariate matrix that is statistically significant to understand \(N_{h}\), \(Y_{hi}\), \(\bar{Y}_{h}\), and \(S_{h}\). The inclusion of covariates \(\mathbf{X}\) introduces new, potentially unknown structures into the data space of \(N_{h}\), \(Y_{hi}\), \(\bar{Y}_{h}\), and \(S_{h}\), altering the underlying distributional properties of the individual summand \(\bar{Y}_{h}|\mathbf{X}_{h}^{S}\)[7]. As a result, their convolutions \(E[S_{h}|\mathbf{X}]\) may be re-organized into hierarchical structures, where observations at one level can be grouped into others. To be specific, based on the group policy assumption, we can assume that \(Y_{h1},Y_{h2},\ldots,Y_{hN_{h}}\) are influenced by the same covariate vector \(\mathbf{X}_{h}^{S}=\{x_{h}^{S},x_{h}^{S}\}\), and thus the inclusion of \(\mathbf{X}\) still leads to \(E[S_{h}|\mathbf{X}]=E[N_{h}|\mathbf{X}_{h}^{F}]\times E[\bar{Y}_{h}|\mathbf{X }_{h}^{S}]\), as shown in Equation (1). However, if each claim amount \(Y_{hi}\) is influenced by individually different covariate vector \(\mathbf{X}_{hi}^{S}=\{x_{hi}^{S},x_{hi}^{S}\}\), \(i=1,\ldots,N_{h}\), the summands--\(Y_{h1}|\mathbf{X}_{h1}^{S},Y_{h2}|\mathbf{X}_{h2}^{S},\ldots Y_{hN_{h}}| \mathbf{X}_{hN_{h}}^{S}\)--may remain independent, but the assumption of the identically distributed observations (homogeneity) is no longer valid (since the distribution of each summand may have different parameter values). This heterogeneity introduced by \(\mathbf{X}\) renders \(E[S_{h}|\mathbf{X}]\
eq E[N_{h}|\mathbf{X}_{h}^{F}]\times E[Y_{hi}|\mathbf{X }_{hi}^{S}]\), making these log-normal convolutions analytically intractable under the traditional risk modeling principle [6].