3.1 Known mean case

In the variance-related premium principle, the risk premium is primarily composed of two components: the hypothetical mean $\mu(\Theta)$ and the process variance $\gamma(\Theta)$ . While the credibility estimate for the hypothetical mean has been extensively studied (c.f. (2.2)), obtaining a reliable estimate for the process variance component is equally crucial for accurate risk premium calculation.

Let $X_{1},\ldots,X_{n}$ denote independent and identically distributed (i.i.d.) samples conditional on the parameter $\Theta$ , and let $\gamma(\Theta) = \mathrm{Var}(X_1 |\Theta)$ represent the conditional variance of the population. Under the squared-error loss criterion, the optimal estimator of $\gamma(\Theta)$ is the posterior mean in the Bayesian framework:

(3.1) \begin{equation} \widehat{\gamma_B(\Theta)} = \mathbb{E}[\gamma(\Theta) \mid X_1,\ldots,X_n],\end{equation}

which serves as the Bayesian estimator of the process variance. In specific models, this Bayesian estimator can be expressed in a form equivalent to the classical credibility formula, demonstrating the close connection between Bayesian estimation and credibility theory.

The following example illustrates the Bayesian estimation of the process variance in the normal distribution case, showing how credibility theory can be applied in practical scenarios.

Example 3.1. Consider the case where, conditional on $\Theta = \theta$ , the observations $X_1, \ldots, X_n$ are independent and identically distributed as $N(\mu_0, \theta)$ , and the process variance parameter $\Theta$ itself follows an inverse gamma distribution $\mathrm{IG}(\alpha, \beta)$ . The conditional density of $X_i$ given $\Theta = \theta$ is

\begin{equation*}f_{\theta}(x) = \frac{1}{\sqrt{2\pi \theta}} \exp\Big\{-\frac{(x-\mu_0)^2}{2\theta}\Big\},\end{equation*}

while the prior density of $\Theta$ is

\begin{equation*}\pi_{\Theta}(\theta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \theta^{-(\alpha+1)} e^{-\beta/\theta}.\end{equation*}

By Bayes’ theorem, the posterior distribution of $\Theta$ given the observed sample is

\begin{equation*}\left[\Theta \mid X_1, \ldots, X_n \right] \sim \mathrm{IG}\Big(\alpha + \frac{n}{2}, \, \beta + \frac{1}{2} \sum_{i=1}^{n} (X_i - \mu_0)^2 \Big),\end{equation*}

which belongs to the inverse gamma conjugate family (see, e.g., Gelman et al. Reference Gelman, Carlin, Stern, Dunson, Vehtari and Rubin2013). The corresponding Bayesian estimator of the process variance is

\begin{equation*}\widehat{\gamma_B(\Theta)} = \frac{\beta + \frac{1}{2}\sum_{i=1}^{n} (X_i - \mu_0)^2}{\alpha + \frac{n}{2} - 1}.\end{equation*}

This Bayesian estimator can equivalently be expressed in the standard credibility form:

\begin{equation*}\widehat{\gamma_B(\Theta)} = w_n S_n^2 + (1 - w_n) \gamma_0,\end{equation*}

where

\begin{equation*}S_n^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu_0)^2, \quad\gamma_0 = \frac{\beta}{\alpha - 1}, \quad w_n = \frac{n}{n + 2(\alpha - 1)}.\end{equation*}

The posterior variance of $\gamma(\Theta)$ is given by

\begin{equation*}\mathrm{Var}[\gamma(\Theta) \mid X_1, \ldots, X_n] = \frac{\left(\beta + \frac{n}{2} S_n^2\right)^2}{\left(\alpha + \frac{n}{2} - 1\right)^2 \left(\alpha + \frac{n}{2} - 2\right)}.\end{equation*}

Remark 3.1. Note that, for $\Theta \sim \mathrm{IG}(\alpha, \beta)$ , the mean and variance are given by

\begin{equation*}\mathbb{E}[\Theta] = \frac{\beta}{\alpha - 1}, \qquad\mathrm{Var}(\Theta) = \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)}, \quad \alpha \gt 2.\end{equation*}

Hence, the posterior variance of the process variance $\gamma(\Theta)$ is well-defined only when $\alpha \gt 2$ .

Furthermore, the marginal distribution of each $X_i$ can be derived by integrating out $\Theta$ :

\begin{align*}f_{X_{i}}(x)& =\int_{0}^{\infty }\frac{1}{\sqrt{2\pi \theta }}\exp \Big\{-\frac{(x-\mu _{0})^{2}}{2\theta }\Big\}\frac{\beta ^{\alpha }}{\Gamma(\alpha )}\theta ^{-(\alpha +1)}e^{-\beta /\theta }\,d\theta \\[5pt] & =\frac{\Gamma \!\left( \alpha +\frac{1}{2}\right) }{\Gamma (\alpha )\sqrt{2\pi \beta }}\left[ 1+\frac{(x-\mu _{0})^{2}}{2\beta }\right] ^{-(\alpha+1/2)}.\end{align*}

This is precisely a Student’s t-distribution with $2\alpha$ degrees of freedom, location parameter $\mu_0$ , and scale parameter $\sqrt{\beta/\alpha}$ , that is,

\begin{equation*}X_i \sim t_{2\alpha} \big(\mu_0, \sqrt{\beta/\alpha} \big).\end{equation*}

This derivation explicitly shows how the heavy-tailed Inverse-Gamma prior for $\Theta$ induces a Student’s t-marginal distribution for $X_i$ , providing robustness in modeling and inference.

In Example 3.1, the weight $w_n$ plays a central role in determining the posterior mean estimator $\widehat{\Theta}_B$ of the process variance parameter $\Theta$ . It quantifies the relative influence of the sample variance $S_n^2$ on $\widehat{\Theta}_B$ compared to the prior mean $\mathbb{E}[\Theta] = \beta / (\alpha - 1)$ . It is straightforward to verify that $w_n \to 1$ as $n \to \infty$ and $w_n \to 0$ as $n \to 0$ . Consequently, for large sample sizes, the Bayesian estimator $\widehat{\Theta}_B$ is primarily driven by the observed data through $S_n^2$ , making it largely data-driven. In contrast, for small n, $w_n$ is smaller, and the prior mean exerts greater influence, reflecting the more significant role of prior information when observations are limited.

The weight $w_n$ therefore balances the contributions of the observed data and the prior information, allowing $\widehat{\Theta}_B$ to smoothly transition from reliance on the prior under scarce data conditions to reliance on the sample information as the number of observations increases. This property ensures a robust estimation procedure that accounts for both data variability and prior knowledge (Bühlmann, Reference Bühlmann1967; Bühlmann and Gisler, Reference Bühlmann and Gisler2005; Zhang, Reference Zhang2023).

In practical insurance applications, the conditional distribution of the population and the distribution of the risk parameter are often unknown, which generally prevents exact Bayesian computations. While the example in Example 3.1 is specific and admits an explicit solution only under conjugate assumptions, it provides useful insight: Bayesian estimation of conditional parameters can be interpreted as a weighted combination of the sample variance and the prior mean of the process variance, closely aligning with the principles of traditional credibility theory.

To better utilize the principles of credibility theory for obtaining credibility estimates of process variance $\gamma(\Theta)$ , we assume that the hypothetical mean $\mu(\Theta)$ is known, that is, $\mu(\Theta) = \mu_{0}$ , and solve the following minimization problem:

(3.2) \begin{equation}\min_{a_{0}, a_{1}, \ldots, a_{n} \in\mathbb{R}} \mathbb{E}\left[ \left(\gamma(\Theta) - a_{0} - \sum_{i=1}^{n} a_{i} \left( X_{i} - \mu_{0} \right)^{2} \right) ^{2} \right] . \end{equation}

The objective of the optimization problem (3.2) is to construct a linear estimator for the process variance $\gamma(\Theta)$ by minimizing the expected squared deviation between $\gamma(\Theta)$ and a linear combination of the squared deviations $(X_i - \mu_0)^2$ . This formulation is equivalent to the classical credibility problem applied to the transformed variable $Y_i = (X_i - \mu_0)^2$ , where the estimation focus is shifted from the original observations $X_i$ to their squared deviations.

To characterize the statistical structure of this problem, let $Y_i = (X_i - \mu_0)^2$ for $i = 1, \ldots, n$ . Then, $Y_1, \ldots, Y_n$ are conditionally independent and identically distributed given $\Theta$ , with

\[\mathbb{E}(Y_i|\Theta) = \gamma(\Theta), \quad\mathrm{Var}(Y_i|\Theta) = k_4(\Theta) - \gamma^2(\Theta),\]

where $k_4(\Theta) = \mathbb{E}[(X_1 - \mu_0)^4|\Theta]$ . Taking expectations with respect to $\Theta$ yields

\[\mathbb{E}[\mathrm{Var}(Y_i|\Theta)] = k_4 - \lambda - \gamma_0^2 = v,\]

where $k_4=\mathbb{E}[k_4(\Theta)]$ , $\lambda = \mathrm{Var}[\gamma(\Theta)]$ and $\gamma_0 = \mathbb{E}[\gamma(\Theta)]$ .

According to the standard methods in credibility theory (Bühlmann, Reference Bühlmann1967; Bühlmann and Gisler, Reference Bühlmann and Gisler2005), the optimal linear credibility estimator in this setting is a weighted combination of the sample variance and the collective variance, with the weight determined by the relative magnitudes of the within-group and between-group variances. Consequently, we obtain the following corollary.

Corollary 3.1. Under Assumptions A1 and A2 of the Bayesian model, if $\mu(\Theta) = \mu_0$ , the optimal estimator of the process variance $\gamma(\Theta)$ obtained by solving (3.2) is given by

(3.3) \begin{equation}\widehat{\gamma_C(\Theta)}= w_n^{\ast} S_n^2 + \left(1 - w_n^{\ast}\right) \gamma_0,\end{equation}

where $\gamma_0 = \mathbb{E}[\gamma(\Theta)]$ denotes the collective (a priori) variance, $S_n^2 = \frac{1}{n} \sum_{i=1}^{n} (X_i - \mu_0)^2$ is the sample variance, and

(3.4) \begin{equation}w_n^{\ast} = \frac{n\lambda}{n\lambda + v}\end{equation}

is the credibility factor.

This corollary establishes a Bayesian–credibility framework for estimating the process variance $\gamma(\Theta)$ . The result shows that the optimal estimator is a weighted average of the sample variance $S_n^2$ and the collective variance $\gamma_0$ , with the credibility factor $w_n^{\ast}$ reflecting the relative importance of the observed data and the prior information. Specifically, $w_n^{\ast}$ depends on the ratio between the between-group variance $\lambda$ and the within-group variance v and increases with the sample size n. Hence, as more data become available, the estimator increasingly relies on the empirical sample variance, while for smaller samples, the estimation remains more influenced by the prior collective information.

This balance between prior belief and observed data lies at the core of credibility theory, providing a natural Bayesian interpretation of the credibility premium principle. Moreover, the formulation ensures that the estimator $\widehat{\gamma_C(\Theta)}$ is unbiased and efficient under the assumed conditional independence and distributional conditions, highlighting its practical relevance in actuarial science and risk management.

Remark 3.2. In Example 3.1, under the normal model with known mean and an inverse-gamma prior, the Bayesian estimator of the process variance admits the linear representation

\[\widehat{\gamma_B(\Theta)}=w_n S_n^2 + (1-w_n)\gamma_0,\]

which coincides exactly with the credibility estimator in Corollary 3.1. This agreement is structural rather than accidental. In the normal model with known mean, the variance parameter $\Theta$ is itself the scale parameter, and the likelihood depends on $\Theta$ only through the quadratic sufficient statistic $\sum_{i=1}^n (X_i-\mu_0)^2$ . The inverse-gamma prior is conjugate with respect to this quadratic structure, so posterior updating simply modifies the hyperparameters via functions of $\sum_{i=1}^n (X_i-\mu_0)^2$ . Consequently, the posterior mean $\mathbb{E}[\Theta\mid X_1,\ldots,X_n]$ is an affine function of $S_n^2$ , yielding an exact credibility-type formula.

In contrast, for a general one-parameter exponential family with conjugate prior, the process variance $\gamma(\Theta)=\mathrm{Var}(X_1\mid\Theta)$ is typically a nonlinear function of the structural parameter $\Theta$ . Although conjugacy ensures analytical tractability of the posterior distribution of $\Theta$ , it does not guarantee that the posterior mean $\mathbb{E}[\gamma(\Theta)\mid X_1,\ldots,X_n]$ can be expressed as a linear function of quadratic statistics such as $S_n^2$ . Unless the variance parameter coincides directly with the canonical scale parameter and the sufficient statistic has a purely quadratic form, such an exact credibility representation generally fails.

Nevertheless, this lack of exact equivalence does not diminish the relevance of the credibility estimator. The estimator derived from (3.2) is the optimal predictor within the class of linear estimators under squared-error loss. Hence, even when the Bayesian estimator is not exactly linear, the credibility estimator can be interpreted as its best linear approximation, depending only on low-order moment quantities $(\gamma_0,\lambda,v)$ . This moment-based structure provides robustness and computational simplicity, making the credibility approach a practically attractive alternative when full distributional specification is uncertain or when conjugacy does not preserve linearity.

A systematic characterization of the distribution–prior combinations under which the Bayesian estimator of the process variance admits an exact credibility-type linear representation remains an interesting open problem beyond the scope of the present paper.

Remark 3.3. The credibility estimator $\widehat{\gamma_C(\Theta)}$ is unconditionally unbiased for $\gamma_0$ in the sense that

\[\mathbb{E}\big[\widehat{\gamma_C(\Theta)}\big]= w_n^\ast \, \mathbb{E}\big[S_n^2\big] + (1-w_n^\ast)\gamma_0= w_n^\ast \gamma_0 + (1-w_n^\ast)\gamma_0= \gamma_0 = \mathbb{E}\big[\gamma(\Theta)\big].\]

Its unconditional mean squared error can be written as

\begin{align*}\mathrm{MSE}\big(\widehat{\gamma_C(\Theta)}\big)&= (w_n^\ast)^2 \, \mathbb{E}\big[(S_n^2 - \gamma(\Theta))^2\big]+ (1-w_n^\ast)^2 \, \mathbb{E}\big[(\gamma_0 - \gamma(\Theta))^2\big]\\[5pt] &\quad + 2 w_n^\ast (1-w_n^\ast) \, \mathbb{E}\big[\big(S_n^2 - \gamma(\Theta)\big)(\gamma_0 - \gamma(\Theta))\big].\end{align*}

Since $\mathbb{E}[S_n^2 \mid \Theta] = \gamma(\Theta)$ , the cross term vanishes: $\mathbb{E}[(S_n^2 - \gamma(\Theta))(\gamma_0 - \gamma(\Theta))] = 0.$ Hence, the unconditional MSE simplifies to

\[\mathrm{MSE}\big(\widehat{\gamma_C(\Theta)}\big)= (w_n^\ast)^2 \, \mathbb{E}\big[\mathrm{Var}\big(S_n^2 \mid \Theta\big)\big]+ (1-w_n^\ast)^2 \lambda= \lambda (1-w_n^\ast)= w_n^\ast \frac{v}{n},\]

where v denotes the variance of $S_n^2$ and $\lambda$ the prior variance of $\gamma(\Theta)$ . It follows that $\mathrm{MSE}\big(\widehat{\gamma_C(\Theta)}\big) \to 0$ as $n \to \infty$ .

Furthermore, we can prove the consistency and asymptotic normality of the credibility estimation.

The proof is provided in Appendix A.3. This theorem establishes the large-sample properties of $\widehat{\gamma_C(\Theta)}$ under a fixed risk parameter $\Theta$ . The almost sure consistency ensures convergence to the true process variance, while asymptotic normality provides a basis for statistical inference in large samples. Together, these results justify the use of the proposed credibility estimator in practical applications where the sample size is moderate to large.

As discussed in Remark 3.2, we consider a hierarchical model in which, conditional on $\Theta = \theta$ , the observations $X_1, \ldots, X_n$ are i.i.d. $N(\mu_0, \theta)$ .

To explore the performance of the credibility estimator beyond conjugacy, we now replace the inverse-gamma prior with a non-conjugate specification. The results show that, even without a conjugate prior, the credibility estimator remains well-defined and closely approximates the Bayesian estimator, providing a practical and accurate alternative in such settings.

Example 3.2. Consider a hierarchical model in which, conditional on $\Theta = \theta$ , the observations $X_1, \ldots, X_n$ are i.i.d. $N(\mu_0, \theta)$ , and the prior distribution of $\Theta$ is $\mathrm{Gamma}(\alpha, \beta)$ with $\alpha, \beta \gt 0$ .

Under this model, we have $\gamma(\Theta) = \Theta$ . Moreover,

\[\mathbb{E}\big[(X_i - \mu_0)^2 \mid \Theta\big] = \Theta,\qquad\mathrm{Var}\big((X_i - \mu_0)^2 \mid \Theta\big) = 2\Theta^2.\]

It follows that the structural parameters in the credibility framework are

\[\gamma_0 = \mathbb{E}[\Theta] = \frac{\alpha}{\beta}, \quad\lambda = \mathrm{Var}(\Theta) = \frac{\alpha}{\beta^2}, \quad v = \mathbb{E}[2\Theta^2] = \frac{2\alpha(1+\alpha)}{\beta^2}.\]

The resulting credibility estimator is

\[\widehat{\gamma_C(\Theta)} = w_n^{\ast} S_n^2 + (1 - w_n^{\ast}) \gamma_0,\quad w_n^{\ast} = \frac{n}{n + 2(1+\alpha)}.\]

Since the Gamma prior is not conjugate to the normal likelihood with known mean, the posterior distribution of $\Theta$ is not available in closed form. Consequently, the Bayesian estimator $\widehat{\gamma_B(\Theta)} =\mathbb{E}[\Theta \mid X_1, \ldots, X_n]$ is no longer an affine function of $S_n^2$ and must be evaluated numerically.

In contrast, the credibility estimator remains explicit and depends only on the low-order moments $(\gamma_0, \lambda, v)$ . To quantify the discrepancy between the Bayesian and credibility approaches, we consider the unconditional mean squared error (MSE),

(3.5) \begin{equation}\mathrm{MSE}\big(\widehat{\gamma_v(\Theta)}\big)= \mathbb{E}\Big[\big(\widehat{\gamma_v(\Theta)} - \gamma(\Theta)\big)^2\Big],\quad v \in \{B, C\},\end{equation}

which is evaluated via Monte Carlo simulation.

Figure 1 reports the simulated MSEs under different sample sizes. For all values of n, the Bayesian estimator exhibits a slightly lower MSE than the credibility estimator. However, the difference remains modest. As the sample size increases, the gap further diminishes and becomes negligible for moderate to large samples (e.g., $n \geq 100$ ), indicating that the credibility estimator provides an accurate approximation to the Bayesian estimator in such settings.

Figure 1.

Unconditional mean squared error of the Bayesian and credibility estimators for different sample sizes.

To further investigate the distributional behavior of the Bayesian and credibility estimators at a fixed value of $\Theta = 1.5$ , we examine QQ-plots for two representative sample sizes, $n=30$ and $n=100$ . These plots allow us to visually assess how closely the empirical distributions of the two estimators align and to evaluate their approximate normality.

Figure 2 shows that the points are approximately aligned along the diagonal for both sample sizes, indicating that the Bayesian and credibility estimators are nearly normally distributed. Mild deviations are observed for $n=30$ , reflecting finite-sample differences, while for $n=100$ the points lie closer to the diagonal, demonstrating increasingly similar distributions and supporting their asymptotic normality.

Figure 2.

QQ-plots of the Bayesian and credibility estimators at $\Theta = 1.5$ for $n=30$ (left) and $n=100$ (right).

Example 3.2 highlights several important features arising in the non-conjugate setting. First, while the credibility estimator of the process variance retains an explicit closed-form expression, the corresponding Bayesian estimator is no longer analytically tractable and must be evaluated numerically, for instance via simulation-based methods.

Second, despite the loss of conjugacy, the credibility estimator continues to provide a close approximation to the Bayesian estimator. This is supported by the simulation results, which show that the two estimators have very similar mean squared errors, with only minor differences even for moderate sample sizes.

Finally, from a distributional perspective, both estimators exhibit asymptotic normality. As the sample size n increases, their empirical distributions become increasingly similar, further confirming that the credibility estimator serves as an accurate and computationally efficient proxy for the Bayesian estimator in large samples.

3.2 Unknown mean case

In the previous subsections, we considered the estimation of the process variance $\gamma(\Theta)$ under the assumption that the hypothetical mean $\mu(\Theta)$ is known. However, in many practical situations, this assumption does not hold, and the hypothetical mean must be estimated jointly with the process variance. This joint estimation increases the complexity of the risk premium evaluation $R(\Theta)$ , since both the mean and the variance must be inferred from the available data.

Nevertheless, the Bühlmann credibility estimator $\widehat{\mu_{C}(\Theta)}$ in (2.2) already provides an optimal estimator of the hypothetical mean when the prior distribution is unknown. Therefore, in the context of variance-related premium principles, we will not further address the estimation of $\mu(\Theta)$ . Instead, we directly substitute its Bühlmann credibility estimate when required. Our primary focus remains on developing a credibility estimator for the process variance $\gamma(\Theta)$ .

Different from the previous subsection, when the hypothetical mean is unknown, the key idea is to replace the term $(X_i - \mu_0)^2$ with its centered counterpart $(X_i - \overline{X}_n)^2$ , where $\overline{X}_n$ is the sample mean. To formalize this approach, we introduce the following quantities:

\[u_{4}(\Theta)= \mathbb{E}\!\left[\left.(X_1 - \mu(\Theta))^4\right|\Theta\right], \qquad\delta_{n}(\Theta)= \mathrm{Var}\big[\tilde{S}_n^2\big|\Theta\big], \qquad\delta_{n}= \mathbb{E}\!\left[\delta_{n}(\Theta)\right],\]

where $\tilde{S}_n^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \overline{X}_n)^2$ denotes the unbiased sample variance. Here, the quantity $\lambda$ appearing later in Theorem 3.2 refers to the between-group variability of the process variance $\gamma(\Theta)$ , as defined previously. The above notation summarizes the conditional fourth central moment and the conditional and unconditional variability of $\tilde{S}_n^{2}$ , which play a key role in the derivation of the credibility estimator for the unknown mean case.

This leads to the following theorem, which generalizes the credibility estimation of the process variance to the case of an unknown hypothetical mean.

Theorem 3.2. Under the Bayesian model with unknown mean $\mu(\Theta)$ , the optimal linear estimator of the process variance $\gamma(\Theta)$ is obtained by solving

(3.6) \begin{equation}\min_{a_{0},a_{1},\ldots,a_{n}\in\mathbb{R}}\mathbb{E}\!\left[\left(\gamma(\Theta)- a_{0}- \sum_{i=1}^{n} a_{i} (X_i - \overline{X}_n)^2\right)^{2}\right].\end{equation}

The solution is given by the credibility-weighted estimator

(3.7) \begin{equation}\widetilde{\gamma_C(\Theta)}= t_{n}^{\ast}\, \tilde{S}_n^{2}+ (1 - t_{n}^{\ast})\, \gamma_{0},\end{equation}

where $\tilde{S}_n^2$ is the unbiased sample variance of the observations, $\gamma_{0} = \mathbb{E}[\gamma(\Theta)]$ denotes the collective (a priori) variance, $\lambda = \mathrm{Var}[\gamma(\Theta)]$ is the between-group variance of the process variance, and $t_{n}^{\ast} = \frac{\lambda}{\lambda + \delta_{n}}$ is the credibility factor balancing the individual sample variance and the collective variance.

The proof of Theorem 3.2 is provided in Appendix A.4. This result extends the classical credibility framework to the case where the hypothetical mean is unknown. By using the centered observations $(X_i - \overline{X}_n)^2$ , the estimator retains its interpretability and computational simplicity, while effectively adjusting for the uncertainty introduced by estimating the mean. As in the known-mean case, the credibility factor $t_{n}^{\ast}$ reflects the relative weight assigned to the observed sample variance and the collective (prior) variance, ensuring a balanced integration of empirical data and prior structure.

Remark 3.4. By evaluating the conditional expectation and fourth-order moment of $\tilde{S}_n^2$ given $\Theta$ under the hierarchical framework (Benhamou, Reference Benhamou2018), we have

\[\delta_{n}(\Theta) = \mathrm{Var}\big(\tilde{S}_n^2 \mid \Theta\big)= \frac{u_4(\Theta)}{n} - \frac{n-3}{n(n-1)} \,\gamma^2(\Theta),\]

and its unconditional expectation

\[\delta_n = \mathbb{E}[\delta_n(\Theta)]= \frac{u_4}{n} - \frac{(n-3)(\lambda+\gamma_0^2)}{n(n-1)}= \frac{1}{n} (u_4 - \lambda - \gamma_0^2) + \frac{2(\lambda+\gamma_0^2)}{n(n-1)}.\]

Consequently, the credibility factor can be expressed as

\[t_n^* = \frac{\lambda}{\lambda + \delta_n}= \frac{n}{\,n + \kappa_n\,},\]

where $\kappa_n = (u_4 - \lambda - \gamma_0^2)/\lambda + 2\,(\lambda + \gamma_0^2)/[\lambda (n-1)]$ is a constant of order O(1) with respect to n.

This form explicitly shows the dependence of $t_n^*$ on the sample size n: $t_n^*$ increases with n and approaches 1 as $n\to\infty$ . Intuitively, as the sample size grows, the credibility estimator gives more weight to the sample component and less to the prior, whereas for small n, the prior receives more weight, consistent with standard credibility theory.

Remark 3.5. The credibility estimation $\widetilde{\gamma_C(\Theta)}$ is an unconditionally unbiased estimator of $\gamma_{0}$ in the sense of $\mathbb{E}\left[ \widetilde{\gamma_C(\Theta)}\right] =\mathbb{E}\left[\gamma(\Theta)\right] =\gamma_{0}.$ In addition, the unconditional mean square error of $\widetilde{\gamma_C(\Theta)}$ is given by:

\[\mathrm{MSE}\left( \widetilde{\gamma_C(\Theta)}\right) =\mathbb{E}\left[\left( \widetilde{\gamma_C(\Theta)}-\gamma(\Theta)\right) ^{2}\right]=\frac{\lambda\delta_{n}}{\lambda+\delta_{n}}=\lambda\left( 1-t_{n}^{\ast}\right) =t_{n}^{\ast}\delta_{n},\]

and we have $\mathrm{MSE}\left( \widetilde{\gamma_C(\Theta)}\right)=O\left( \frac{1}{n}\right) \rightarrow0$ as $n\rightarrow\infty$ .

The remark addresses two fundamental properties of the credibility estimator $\widetilde{\gamma_C(\Theta)}$ . This property ensures that, on average, the estimator neither overestimates nor underestimates the true parameter value $\gamma_{0}$ . The MSE formula shows that as the sample size n increases, the MSE decreases at a rate proportional to $1/n$ , asymptotically approaching zero as $n\rightarrow\infty$ . This indicates that the estimator’s accuracy improves with larger sample sizes, which is a desirable characteristic.

Example 3.3. Suppose that, conditional on the parameter $\Theta = \theta$ , the samples $X_{1}, \ldots, X_{n}$ are independent and identically distributed, each following a normal distribution $N(\mu(\theta), \gamma(\theta))$ with conditional density

\begin{equation*}f_{\theta}(x)= \frac{1}{\sqrt{2\pi \gamma(\theta)}}\exp\!\left[-\frac{(x - \mu(\theta))^{2}}{2\gamma(\theta)}\right],\quad x \in \mathbb{R}.\end{equation*}

The parameter $\Theta$ is assumed to be a random variable following an unknown prior density $\pi_{\Theta}(\theta)$ .

According to Theorem 3.2, the credibility estimation (3.7) still holds under this scenario. In this case, the credibility estimator provides the best linear approximation to the Bayesian estimator of the process variance $\gamma(\Theta)$ , thereby offering a practical and robust alternative when the exact prior distribution is unavailable.

Due to the fact that the excess kurtosis of a normal distribution is zero, we have

\[u_{4}(\Theta) = 3\gamma^{2}(\Theta),\]

and thus

\[\delta_{n}(\Theta) = \frac{2\gamma^{2}(\Theta)}{n-1}.\]

Accordingly, the credibility factor $t_{n}^{\ast}$ can be simplified as

\begin{equation*}t_{n}^{\ast}= \frac{\lambda}{\lambda + 2(\lambda + \gamma_{0}^{2})/(n-1)}= \frac{(n-1)\lambda}{(n-1)\lambda + 2(\lambda + \gamma_{0}^{2})}.\end{equation*}

Therefore, the credibility estimation of $\gamma(\Theta)$ can be expressed as

(3.8) \begin{equation}\widetilde{\gamma_C(\Theta)}= \frac{(n-1)\lambda\,{\tilde{S}_n^{2}}}{(n-1)\lambda + 2(\lambda + \gamma_{0}^{2})}+ \frac{2(\lambda + \gamma_{0}^{2})\,\gamma_{0}}{(n-1)\lambda + 2(\lambda + \gamma_{0}^{2})}.\end{equation}

This formulation highlights an important property of credibility estimation: even without explicit assumptions about the prior distribution, the estimator $\widetilde{\gamma_C(\Theta)}$ maintains a functional form that is closely aligned with the Bayesian estimator. Consequently, it provides a robust and interpretable approximation that effectively balances empirical data information with collective-level variance characteristics.

This example demonstrates how credibility estimation can be effectively applied to normally distributed data when the prior distribution is unknown. By leveraging the properties of the normal distribution and simplifying the problem, the example derives a credibility factor and estimator that appropriately balance the influence of sample data and prior information. Notably, because the kurtosis under the normal distribution is zero, the credibility estimator is elegantly simplified, ensuring that the estimation process remains both efficient and accurate.

Remark 3.6. In general, when the conditional distribution of the risk variable X given $\Theta$ has zero excess kurtosis (i.e., a kurtosis equal to 3), it follows that

\[u_4(\Theta) = \mathbb{E}\!\left[\left.(X - \mu(\Theta))^4\right|\Theta\right] = 3 \gamma^2(\Theta),\]

where $\gamma(\Theta) = \mathrm{Var}[X \mid \Theta]$ . Under this condition, the credibility estimator can be simplified to the form given in (3.8).

While the normal distribution is the most well-known and canonical example exhibiting zero excess kurtosis, it is not the only one. Other symmetric distributions, including certain parameterizations of the generalized normal (exponential power) distribution, may also have zero excess kurtosis. This highlights that the simplification of the credibility estimator based on the kurtosis property is applicable beyond the normal case. For further discussion on distributions with zero excess kurtosis, see, for example, Bühlmann (Reference Bühlmann2007).

Overall, this remark emphasizes that the key condition for the simplified form of the credibility estimator is the kurtosis of the conditional distribution rather than normality per se, thereby extending the applicability of the result to a broader class of symmetric distributions.

Theoretically, regardless of whether the hypothetical mean is known, the Bayesian estimator of the process variance is optimal under the squared loss function. Specifically, the Bayesian estimator of $\gamma(\Theta)$ is given by

(3.9) \begin{equation}\widetilde{\gamma _{B}(\Theta )}=\mathbb{E}\!\left[ \gamma (\Theta )\,\middle|\,X_{1},X_{2},\ldots ,X_{n}\right] =\int_{\Omega }\gamma (\theta )\,\pi\!\left( \theta \,\middle|\,X_{1},X_{2},\ldots ,X_{n}\right) d\theta ,\end{equation}

where $\Omega$ denotes the support of $\Theta$ . However, the posterior density $\pi\!\left( \theta \,\middle|\, X_{1}, X_{2}, \ldots, X_{n} \right)$ is typically unknown because the conditional distribution $f_{\theta}(x)$ or the prior distribution $\pi_{\Theta}(\theta)$ is unspecified. Consequently, the Bayesian estimator $\widetilde{\gamma _{B}(\Theta )}$ cannot be obtained in closed form.

However, with a sufficiently large sample size, we can establish the asymptotic properties of both the Bayesian and credibility estimators. Before presenting these results, it is useful to recall a fundamental structural feature of credibility estimators: they are typically expressed as a convex combination of a constant (representing prior or collective information) and a moment-based estimator computed from the sample. As the sample size increases, the weight placed on the sample-based component tends to one, while the weight placed on the constant term diminishes. This property implies that credibility estimators automatically recover the classical moment estimators in large samples, which in turn guarantees their consistency and asymptotic normality.

More importantly, credibility estimators are designed to perform well in small samples. By optimally combining prior information with limited empirical data, they achieve a smaller mean squared error than purely sample-based estimators when n is small. Thus, credibility estimation not only satisfies the desirable large-sample properties of consistency and asymptotic Gaussianity but also offers strong small-sample efficiency, which is a central motivation for the development of credibility theory.

Details of the proof can be found in Appendix A.5. Theorem 3.3 confirms that both the credibility and Bayesian estimators possess desirable large-sample properties. Consistency ensures convergence to the true process variance, while asymptotic normality provides a foundation for statistical inference in settings with sufficiently large sample sizes.

To provide a concise summary of the similarities and differences between the Bayesian and credibility estimators introduced in Theorems 3.3, we include the following compact comparison table. This table highlights key features, including consistency, use of prior information, optimality, model dependence, closed-form availability, and typical MSE behavior.

Table 2.

Comparison of Bayesian and credibility estimators of the process variance $\gamma(\Theta)$ .

Table 2 Long description

A table comparing Bayesian and credibility estimators of the process variance across six features. The table has two columns labeled Bayesian and Credibility, and six rows labeled Consistency, Prior/structural info, Optimality, Model dependence, Closed-form, and MSE. Row 1: Consistency, Bayesian: Yes; converges to γ(Θ), Credibility: Yes; converges to γ(Θ). Row 2: Prior/structural info, Bayesian: Uses prior information, Credibility: Uses structural parameters. Row 3: Optimality, Bayesian: Minimises expected squared loss over all functions, Credibility: Minimises loss over sample-summary class. Row 4: Model dependence, Bayesian: Requires full likelihood and prior; may lack closed form, Credibility: Depends on low-order moments; always closed form. Row 5: Closed-form, Bayesian: Only in certain special conjugate cases, Credibility: Always available. Row 6: MSE, Bayesian: Typically smaller, Credibility: Typically larger.

This table allows the reader to clearly visualize the relationship between the two estimators. It demonstrates that while both $\widetilde{\gamma_C(\Theta)}$ and $\widetilde{\gamma_B(\Theta)}$ are consistent and asymptotically valid, the Bayesian estimator generally achieves smaller mean squared errors due to its optimality among all measurable functions and its model-dependent flexibility. Overall, the table provides a concise and accessible illustration of the theoretical properties discussed above, thereby enhancing both the readability and practical understanding of the estimators.

Although Bayesian estimation typically has a smaller mean squared error than credibility estimation, credibility estimation is more practical. Even when the population and sample distributions are unknown, credibility estimation still provides explicit expressions. In contrast, Bayesian estimation heavily relies on the specific forms of the population and sample distributions. Moreover, even if these distributions are known, Bayesian estimation may not yield a closed-form solution and often requires numerical integration methods, such as Markov Chain Monte Carlo (MCMC), for computation. Therefore, in practical applications, credibility estimation is generally preferred over Bayesian estimation.

4.1 Numerical comparisons

The variance-related premium principle has significant applications in insurance and risk management. By considering both the expected value and the variance of the risk, this principle provides a more comprehensive evaluation of risk compared to traditional pure premium principles. This section discusses practical applications of the variance-related premium principle, highlighting how credibility estimation can be effectively used to estimate risk premiums.

Definition 4.1. Under the Bayesian models, the credibility estimation of the risk premium $R(\Theta)$ by the “plug-in” principle is defined by:

(4.1) \begin{equation}\widetilde{R_C(\Theta)}=\widehat{\mu_{C}(\Theta)}+ g\left(\widehat{\mu_{C}(\Theta)},\widetilde{\gamma_C(\Theta)}\right) \end{equation}

where $\widehat{\mu_{C}(\Theta)}$ and $\widetilde{\gamma_C(\Theta)}$ are the credibility estimators for the hypothetical mean $\mu(\Theta)$ and the process variance $\gamma(\Theta)$ given in (2.2) and (3.7), respectively.

The plug-in credibility estimator defined above provides a unified framework for constructing risk premium estimates under various premium principles by substituting the credibility estimators of the hypothetical mean and process variance into the functional form of the premium principle. This formulation not only preserves the interpretability of the classical credibility approach but also enables its extension to a broader class of variance-related premium principles. To highlight the generality and practical advantages of this framework, it is instructive to compare it with the method proposed by Yong et al. (Reference Yong, Zeng and Zhang2025), which focuses specifically on the variance premium principle.

Remark 4.1. Yong et al. (Reference Yong, Zeng and Zhang2025) investigated credibility estimation under the classical variance premium principle, where the risk premium is expressed as

\[R(\Theta)=\mu(\Theta)+\rho\gamma(\Theta).\]

Specifically, they considered a class of transformed quadratic functions of the sample,

\[\Phi = \Biggl\{ a_{0} + \sum_{i=1}^{n} a_{i} X_{i} + \sum_{i=1}^{n} b_{i} X_{i}^{2} + \sum_{i=1}^{n} \sum_{j=1}^{n} c_{ij} X_{i} X_{j} \;\Big|\; a_{i}, b_{i}, c_{ij} \in \mathbb{R} \Biggr\},\]

and obtained the credibility estimator by minimizing the expected squared error:

\[\min_{\widehat{R(\Theta)} \in \Phi} \mathbb{E}\!\left[ \bigl(\mu(\Theta)+\rho\gamma(\Theta)-\widehat{R(\Theta)}\bigr)^2 \right].\]

This provides a consistent estimate for the variance-based risk premium and represents a significant extension of Bühlmann’s classical credibility theory. Notably, as mentioned on page 6 of Yong et al. (2025), their credibility estimation involves at least 14 structural parameters, which are generally nontrivial to estimate. Furthermore, because the two components $\mu(\Theta)$ and $\gamma(\Theta)$ are generally dependent, an exact computation of the variance of $R(\Theta)$ is challenging.

Building on this foundation, the framework proposed in this paper introduces several key features. First, we separate the estimation of the hypothetical mean $\mu(\Theta)$ and the process variance $\gamma(\Theta)$ . Since the mean and variance have different dimensions, estimating them separately is both necessary and statistically reasonable. The process variance is constructed as a linear combination of transformed sample data, minimizing squared error in a manner analogous to Bühlmann’s approach for the mean. This separation enhances analytical tractability and stability.

Second, our method requires only a small number of easily interpretable hyperparameters. As shown in Table 6 of Subsection 4.2, the framework involves only five structural parameters, all of which can be readily estimated using an empirical Bayes approach. This parsimonious structure facilitates practical implementation and reliable estimation even with limited data, while maintaining flexibility to accommodate different types of premium functions.

Finally, by combining the separately estimated $\mu(\Theta)$ and $\gamma(\Theta)$ through a plug-in variance-related premium function

\[R(\Theta) = \mu(\Theta) + g(\mu(\Theta),\gamma(\Theta)),\]

our framework naturally extends to a broader class of premium principles, including the variance, standard deviation, and modified variance principles. This approach preserves desirable statistical properties such as conditional unbiasedness, mean squared error convergence, consistency, and asymptotic normality, which can be established analytically.

Both the Bayesian estimator in Definition 2.1 and the credibility estimator in (4.1) share a common analytical structure. Each consists of a location term, representing the estimated expected loss, and a risk loading term, determined by the function g(x, y), which captures the adjustment for higher-order uncertainty.

The Bayesian estimator computes these components using posterior expectations of the model parameters, providing an exact posterior risk premium under the assumed prior-likelihood framework. In contrast, the credibility estimator replaces these posterior quantities with their credibility estimates of the hypothetical mean and process variance given in (2.2) and (3.7). This yields a practical, data-driven approximation that preserves the Bayesian interpretation while avoiding explicit prior specification and posterior computation.

Thus, both estimators belong to the same family, distinguished primarily by how expectations are evaluated: the Bayesian estimator serves as the theoretical benchmark, whereas the credibility estimator offers an operational alternative suited for actuarial applications with limited data or a preference for computational simplicity.

Table 3 summarizes the explicit forms of both estimators for several common premium principles, highlighting their structural coherence and the credibility estimator’s role as a natural extension of the Bayesian approach.

Table 3.

Bayesian and credibility estimators of variance-related risk premiums.

Table 3 Long description

Panel A: The table compares Bayesian and credibility estimators for different premium principles. Panel B: The premium principles include Net, Expected value, Variance, Standard deviation, and Modified variance. Panel C: For each premium principle, the table provides formulas for both Bayesian and credibility estimators. Panel D: The formulas involve variables such as rho, mu, gamma, and Theta, with specific operations like expectation and division.

The variance-related premium principle extends traditional actuarial pricing methodologies by incorporating not only the expected loss but also the associated uncertainty, typically represented by the variance. The estimators summarized in Table 3 offer a practical means of implementing this generalized principle. To demonstrate their application, we present a numerical example in which the latent risk variable follows a Gamma distribution, while the observed claim amounts are conditionally exponential given the risk parameter.

Example 4.1. Let $\Theta=\theta$ denote the underlying risk parameter, and assume that $X_{1},\ldots,X_{n}$ are conditionally i.i.d. $\mathrm{Exp}(\theta)$ with density

\[f_{\theta}(x)=\theta e^{-\theta x}, \qquad x\gt0.\]

Hence,

\[\mu(\Theta)=\mathbb{E}(X\mid\Theta)=\Theta^{-1},\qquad\gamma(\Theta)=\mathrm{Var}(X\mid\Theta)=\Theta^{-2}.\]

The prior distribution is taken as $\Theta\sim\Gamma(\alpha,\beta)$ with density

\[\pi_{\Theta}(\theta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\theta^{\alpha-1}e^{-\beta\theta},\qquad \theta\gt0,\ \alpha\gt4,\]

where $\alpha\gt4$ ensures the existence of the required higher-order moments. Under the Exponential-Gamma hierarchy, the marginal distribution of $X_i$ is

\[f_{X_i}(x)=\alpha\beta^{\alpha}(x+\beta)^{-(\alpha+1)},\qquad x\gt0,\]

that is, a Pareto (Type II)/Lomax distribution (Lemonte and Cordeiro, Reference Lemonte and Cordeiro2013), exhibiting the heavy-tailed behavior induced by the mixing. Moreover, $\mu(\Theta)$ follows an inverse-gamma $(\alpha,\beta)$ distribution.

Given the sample $\mathbf{X}=(X_{1},\ldots,X_{n})$ with mean $\overline{X}_{n}$ , conjugacy implies

\[[\Theta\mid\mathbf{X}]\sim\Gamma(\alpha+n,\beta+n\overline{X}_{n}),\]

so that the Bayesian estimators of the hypothetical mean and process variance are

\[\mathbb{E}[\mu(\Theta)\mid\mathbf{X}]= \frac{\beta+n\overline{X}_{n}}{\alpha+n-1},\qquad\mathbb{E}[\gamma(\Theta)\mid\mathbf{X}]= \frac{(\beta+n\overline{X}_{n})^{2}} {(\alpha+n-1)(\alpha+n-2)}.\]

Consequently, all premium principles expressed in terms of $(\mu(\Theta),\gamma(\Theta))$ admit closed Bayesian forms. For example, the variance premium estimator is

\[\widetilde{R_2^B(\Theta)}= \mathbb{E}[\mu(\Theta)\mid\mathbf{X}]+\rho_{2}\,\mathbb{E}[\gamma(\Theta)\mid\mathbf{X}].\]

The prior moments are given by

\[\mu_0 = \frac{\beta}{\alpha-1}, \quad\tau^2 = \frac{\beta^2}{(\alpha-1)^2(\alpha-2)}, \quad\gamma_0 = \frac{\beta^2}{(\alpha-1)(\alpha-2)}, \quad\lambda = \frac{2\beta^4(2\alpha-5)}{(\alpha-1)^2(\alpha-2)^2(\alpha-3)(\alpha-4)}.\]

For exponential claims, the fourth conditional central moment leads to

\[u_4(\Theta) = \frac{9}{\Theta^4}, \quad\delta_n(\Theta) = \frac{2(4n-3)}{\Theta^4 n(n-1)}, \quad\delta_n = \frac{2(4n-3)\,\beta^4}{n(n-1)(\alpha-1)(\alpha-2)(\alpha-3)(\alpha-4)}.\]

The associated credibility factors for the hypothetical mean and process variance are

\begin{equation*}Z_{n}=\frac{n\tau ^{2}}{n\tau ^{2}+\gamma _{0}}=\frac{n}{n+\alpha -1},\text{}t_{n}^{\ast }=\frac{\lambda }{\lambda +\delta _{n}}=\frac{1}{1+\frac{(4n-3)}{n(n-1)}\cdot \frac{(\alpha -1)(\alpha -2)}{2\alpha -5}}.\end{equation*}

These expressions provide all quantities required to compute both the Bayesian and credibility estimators for the considered premium principles. Their accuracy can be assessed through the conditional mean squared error (CMSE),

\[\mathrm{CMSE}_{k,i}\!\left[\widetilde{R_{i}^{k}(\Theta)}\right]=\mathbb{E}\!\left[\left(\widetilde{R_{i}^{k}(\Theta)}-R_{i}(\Theta)\right)^{2}\Big|\Theta\right],\]

and the unconditional mean squared error (UMSE),

\[\mathrm{UMSE}_{k,i}\!\left[\widetilde{R_{i}^{k}(\Theta)}\right]=\mathbb{E}\!\left[\left(\widetilde{R_{i}^{k}(\Theta)}-R_{i}(\Theta)\right)^{2}\right],\]

where $k=B$ or C and $i=0,1,2,3,4$ correspond to the net, expected value, variance, standard deviation, and modified variance premium principles.

For the parameter choices $\alpha=6$ , $\beta=4$ , $\rho_{1}=\rho_{2}=0.2$ , and $\rho_{3}=\rho_{4}=0.3$ , Monte Carlo simulations in R are conducted to compute the conditional mean squared errors (conditional MSEs, CMSEs) for risk levels $\Theta=0.6$ and $\Theta=1.0$ across sample sizes $n=10,50,200$ . The resulting conditional MSEs are reported in Table 4.Footnote ¹ Here, we only display the results for $i=2,3,4$ because the premiums for $i=0$ and $i=1$ coincide under both Bayesian and credibility approaches and including them would be redundant.

Table 4.

Conditional MSEs of Bayesian and credibility estimators.

Table 4 Long description

A table comparing conditional mean squared errors (CMSEs) of Bayesian and credibility estimators for different sample sizes and risk levels. The table has 6 columns and 6 rows, including headers. The columns are labeled with risk levels (Theta = 0.6 and Theta = 1.0) and sample sizes (n = 10, n = 50, n = 200). The rows are labeled with different estimators (CMSE_B,2, CMSE_C,2, CMSE_B,3, CMSE_C,3, CMSE_B,4, CMSE_C,4). Each cell contains a numerical value representing the CMSE for the corresponding estimator, sample size, and risk level. The values decrease as the sample size increases, indicating that the errors decline substantially with larger sample sizes. The errors are consistently smaller for larger values of Theta, reflecting higher precision for less volatile risks. The Bayesian and credibility estimators exhibit almost identical performance for large sample sizes, but the Bayesian estimator generally achieves slightly lower mean squared errors for small sample sizes.

Furthermore, to evaluate the unconditional mean squared errors (unconditional MSEs, UMSEs), values of $\Theta$ are randomly generated from the Gamma prior. Two different parameter settings $(\alpha,\beta)$ are considered, and numerical simulations are conducted to compute the unconditional MSEs for each credibility estimator under the various premium principles. The resulting unconditional MSEs are reported in Table 5.

Table 5.

Unconditional MSEs of Bayesian and credibility estimators.

Table 5 Long description

A table comparing unconditional mean squared errors of Bayesian and credibility estimators for different sample sizes and parameter settings. The table has six columns and six rows, including headers. The columns are labeled with different sample sizes (n = 10, n = 50, n = 200) for two parameter settings (alpha = 5, beta = 3 and alpha = 6, beta = 4). The rows are labeled with different estimators (UMSE_B_2, UMSE_C_2, UMSE_B_3, UMSE_C_3, UMSE_B_4, UMSE_C_4). Each cell contains a numerical value representing the unconditional mean squared error for the corresponding estimator and sample size. The values decrease as the sample size increases, indicating that the errors decline substantially with larger sample sizes. The table provides a detailed comparison of the performance of Bayesian and credibility estimators under different conditions.

From Tables 4 to 5, it is evident that both conditional MSEs and unconditional MSEs decline substantially as the sample size n increases. The errors are consistently smaller for larger values of $\Theta$ , reflecting the higher precision attainable for less volatile risks. Moreover, while the Bayesian and credibility estimators exhibit almost identical performance for large sample sizes, the Bayesian estimator generally achieves slightly lower mean squared errors when the sample size is small.

Finally, we examine the asymptotic behavior of the credibility estimator $\widetilde{R_{i}^{C}(\Theta)}$ for $\alpha=6$ , $\beta=4$ , $\rho_{1}=\rho_{2}=0.2$ , $\rho_{3}=\rho_{4}=0.3$ , and $\Theta=1.4$ . Figure 3 presents kernel density estimates of $\widetilde{R_{i}^{C}(\Theta)}$ for sample sizes $n=5$ and $n=50$ , with dashed lines indicating the theoretical premium values. As the sample size increases from $n=5$ to $n=50$ , the empirical distributions become more symmetric and more concentrated around the theoretical values, providing clear numerical evidence in favor of the estimator’s asymptotic normality.

Figure 3.

Kernel density estimates of credibility estimators for different sample sizes.

Under this hierarchical specification, the marginal distribution of $X_i$ follows a Pareto (Type II) or Lomax distribution, which naturally arises from the exponential–gamma conjugate structure. Furthermore, since $\mu(\Theta) = 1/\Theta$ , the transformed random variable $\mu(\Theta)$ follows an inverse Gamma distribution with parameters $(\alpha, \beta)$ (Klugman et al., Reference Klugman, Panjer and Willmot2019). This alternative representation provides valuable insight into the mean–variance relationship inherent in the exponential–gamma model.

4.2 Practical applications

To illustrate the implementation of the proposed credibility framework, we use the dataset fremotor2freq9907u Footnote ² available in the CASdatasets package in R (Rizzo, Reference Rizzo2019; Dutang and Charpentier, Reference Dutang and Charpentier2024). Each policy is treated as a separate credibility unit associated with a latent risk parameter $\Theta_i$ ( $i=1,\ldots,K$ ). Conditional on $\Theta_i$ , the observed annual claim counts $\{X_{ij}\}_{j=1}^{n_i}$ are assumed i.i.d. with conditional distribution $f_{\theta_i}(x)$ . Additional implementation details, including data processing, structural parameter estimation, and bootstrap approximation of $\delta_{n_i}$ and $\lambda$ , are provided in Appendix A.6.

Based on the sample, we estimate the five structural parameters required for the credibility model, namely $\mu_{0}$ , $\gamma_{0}$ , $\tau^{2}$ , $\delta_{n_{i}}$ , and $\lambda$ . Following Bühlmann and Gisler (Reference Bühlmann and Gisler2005, p. 94), $\mu_{0}$ is the portfolio-level mean, $\gamma_{0}$ is the within-policy variance, and $\tau^{2}$ is the between-policy variance.

The resulting unbiased estimates of the structural parameters are summarized in Table 6. The policy-specific dispersion parameters $\widehat{\delta}_{n_i}$ can be obtained from these estimates as described in Appendix A.6, but are not listed here for brevity.

Given the estimates in Table 6, we next compute credibility estimates of risk premiums using the sample. At this stage, the structural parameters are treated as known, and the resulting credibility estimators correspond to empirical Bayes estimators in the sense of Robbins (Reference Robbins1964), with actuarial developments discussed in Mashayekhi (Reference Mashayekhi2002).

Table 6.

Structural parameter estimates.

Table 6 Long description

A table with two rows and five columns. The first row contains the headers: Parameter and Estimate. The second row lists the parameters and their corresponding estimates: gamma sub 0, 0.1680; tau squared, 0.0585; delta, 0.2345; lambda, 0.1801; mu sub 0, 0.1635.

Before calculating premiums, we determine the safety loading coefficients $\rho_i$ ( $i = 1, \ldots, 4$ ), which link risk-based estimates to actual premium levels under different principles. Following Yong et al. (Reference Yong, Zeng and Zhang2025), these coefficients are calibrated via a value-at-risk (VaR) approach at confidence level $p = 0.9$ . Let Y denote the aggregate claim amount. The loadings are defined by

where $\mathbb{E}[Y]$ , $\mathrm{Var}(Y)$ , and $\mathrm{VaR}_{p}(Y)$ are estimated using, respectively, the sample mean $\overline{X}_{i}$ , the sample variance $\widetilde{S}_{n_i}^{2}$ , and the empirical p-quantile of policy i computed from the sample ( $i = 1,2,\ldots,K$ ). This VaR-based calibration ensures that, under each premium principle, the premiums cover aggregate claims at the specified confidence level.

Using the estimated structural parameters and calibrated safety loadings $\rho_i$ , we then compute the credibility estimator in (4.1) under the net, expected value, variance, standard deviation, and modified variance principles. The sample still contains a large number of policies, so it is not feasible to report results for all contracts. For illustration, we select seven policies from the sample and report their credibility-adjusted premium estimates in Table 7.

Table 7.

Estimated premiums for seven sampled policies under different premium principles.

Table 7 Long description

A table with seven rows and six columns comparing estimated premiums for seven sampled policies under different premium principles. The columns are labeled with Policy IDs: PN148469, PN191262, PN105635, PN62080, PN76584, PN38328, and PN6536. The rows are labeled with different premium principles: Net premium principle, Expected value, Variance principle, Standard deviation, and Modified variance. Each cell contains a numerical value representing the estimated premium under the respective principle. Row 1: Net premium principle, 0.7122, 0.4206, 0.2139, 0.1867, 0.1656, 0.2502, 0.1352. Row 2: Expected value, 2.1774, 1.2859, 0.6538, 0.5708, 0.5064, 0.7648, 0.4133. Row 3: Variance principle, 2.0351, 1.4360, 1.0377, 0.8798, 0.7639, 1.0642, 0.6048. Row 4: Standard deviation, 1.3793, 1.0051, 0.7404, 0.6696, 0.6143, 0.7735, 0.5327. Row 5: Modified variance, 1.0160, 0.8154, 0.8439, 0.7939, 0.7563, 0.7823, 0.7033.

Table 7 shows the resulting premiums under the different principles. As expected, the net premium corresponds to the pure risk expectation, while the other principles incorporate additional loadings through the parameters $\rho_i$ . The variance and modified variance principles produce higher premiums for policies with more volatile claim experience, reflecting their sensitivity to underlying risk heterogeneity and the ability of the credibility-adjusted framework to differentiate premiums accordingly.