3.1. Credibility estimator for the joint distribution function

Let us define the class of nonhomogeneous linear functions of the conditional joint distribution as:

(3.3) \begin{equation}\mathscr{L}=\left\{ \alpha _{0}\left( \textbf{y}\right)+\sum_{s=1}^{n}\alpha _{s}H_{s}\!\left( \textbf{y}\right)\right\}, \end{equation}

where $\alpha _{0}\!\left( \textbf{y}\right) $ is an unspecified multivariate function (in general representing the population information), and

\begin{equation*}H_{s}\!\left( \textbf{y}\right) =I_{\left\{\textbf{Y}\leq \textbf{y}\right\}}=\left\{\begin{array}{c@{\quad}c}1, & Y_{s}^{(1)}\leq y^{(1)},\cdots, Y_{s}^{(p)}\leq y^{(p)} \\[7pt]0, & \text{otherwise}\end{array}\right., \end{equation*}

for $\textbf{y}=\left( y^{(1)},\cdots, y^{(p)}\right) ^{\prime }$ and $s=1,\ldots, n$ , where $I_A$ represents the indicative function of event A, which is equal to 1 when A occurs, otherwise equal to 0. We aim to solve the following minimization problem:

(3.4) \begin{equation}\min_{\widehat{F}\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \in \mathscr{L}}\int_{\mathbb{R}^{p}}\mathbb{E}\!\left[ \left( F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) -\widehat{F}\!\left(\left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \right) ^{2}\right] d\textbf{y}.\end{equation}

The solution to problem (3.4) is summarized in the following theorem.

Theorem 3.1. In the Bayesian model settings for multivariate risks, as described by Assumptions 2.1 and 2.2, the optimal linear Bayesian estimator of the conditional joint distribution function $F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta}\right) $ solving problem (3.4) is given by:

(3.5) \begin{equation}\widehat{F}\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right)=Z_{N,n}F_{n}\!\left( \textbf{y}\right) +(1-Z_{N,n})F_{0}\!\left( \textbf{y}\right), \end{equation}

where

(3.6) \begin{equation}F_{n}\!\left( \textbf{y}\right) =\frac{1}{n}\sum_{i=1}^{n}H_{i}\!\left( \textbf{y}\right) \text{ }\end{equation}

and

\begin{equation*}F_{0}\!\left( \textbf{y}\right) =\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \right] =\int_{\mathbb{R}^{p}} F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\theta} \right) \pi \!\left( \boldsymbol{\theta}\right) d\boldsymbol{\theta}\end{equation*}

represent the empirical and aggregated joint distribution functions, respectively. The credibility factor $Z_{N,n}$ Footnote 7 is defined as:

(3.7) \begin{equation}Z_{N,n}=\frac{n\tau _{0}^{2}}{n\tau _{0}^{2}+\sigma _{0}^{2}}, \end{equation}

where

(3.8) \begin{equation}\tau _{0}^{2}=\int_{\mathbb{R}^{p}}\mathbb{V}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \right] d\textbf{y}\end{equation}

and

(3.9) \begin{equation}\sigma _{0}^{2}=\int_{\mathbb{R}^{p}}\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \left( 1-F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) \right) \right] d\textbf{y}. \end{equation}

We hereby designate $\widehat{F}\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta}\right) $ as the credibility estimate of the conditional distribution function $F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) $ and $Z_{N,n}$ as the credibility factor.

Remark 3.1. Noted that in Equation (3.3), none constraints were imposed on class $\mathscr{L}$ for estimating distribution functions. It is easy to observe that solving problem (3.4) would be quite challenging if we introduce some constraints. Given $\boldsymbol{\Theta}$ , due to the exchangeability of $\textbf{Y}_1,\textbf{Y}_2,\cdots, \textbf{Y}_n$ , it must hold that $\alpha_1=\alpha_2=\cdots=\alpha_n$ for the solution of (3.4). Therefore, the optimization problem (3.4) is actually equivalent to solving:

(3.10) \begin{equation}\min_{b\in \mathbb{R}}\int_{\mathbb{R}^{p}} \mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) -\alpha_{0}\left( \textbf{y}\right)-bF_{n}\!\left( \textbf{y}\right) \right] ^{2}d\textbf{y}. \end{equation}

Note that

\begin{eqnarray*}&&\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right)-\alpha _{0}\!\left( \textbf{y}\right) -bF_{n}\!\left( \textbf{y}\right) \right]^{2} \\[3pt]&=&\mathbb{V}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right)-\alpha _{0}\left( \textbf{y}\right) -bF_{n}\!\left( \textbf{y}\right) \right]+\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right)-\alpha _{0}\!\left( \textbf{y}\right) -bF_{n}\!\left( \textbf{y}\right) \right]\\[3pt]&\geq &\mathbb{V}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right)-\alpha _{0}\!\left( \textbf{y}\right) -bF_{n}\!\left( \textbf{y}\right) \right],\end{eqnarray*}

where the equality holds if and only if

\begin{equation*}\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) -\alpha_{0}\!\left( \textbf{y}\right) -bF_{n}\!\left( \textbf{y}\right) \right] =0.\end{equation*}

Therefore, $\alpha_{0}\!\left( \textbf{y}\right)$ should be chosen such that

\begin{equation*}\alpha _{0}\!\left( \textbf{y}\right) =\mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) -bF_{n}\!\left( \textbf{y}\right) \right] =aF_{0}\!\left( \textbf{y}\right), \end{equation*}

where $a=1-b.$ Thus, the optimization problem (3.4) is equivalent to finding the optimal estimate of the conditional distribution function $F\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)$ within the class:

\[\mathscr{L}^{*}=\left\{ aF_{0}(\textbf{y})+bF_{n}(\textbf{y})\right\} .\]

To ensure the candidates in $\mathscr{L}^{*}$ to be reasonable estimates, we need to impose the condition:

\[a\geq 0,\,b\geq 0,\,a+b=1,\]

under which $aF_{0}(\textbf{y})+bF_{n}(\textbf{y})$ remains a multivariate distribution function. To this regard, we should solve the following constrained optimization problem:

(3.11) \begin{equation}\left\{ \begin{aligned}&\min_{a,b\in \mathbb{R}}\int_{\mathbb{R}^{p}} \mathbb{E}\!\left[ F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) -aF_{0}(\textbf{y})-bF_{n}(\textbf{y})\right]^{2}d\textbf{y} \\[6pt]&\mbox{s.t.}\; a\geq 0,\,b\geq 0,\,a+b=1\end{aligned} \right. .\end{equation}

In accordance with Lemma 1 in Section 2 of the supplementary file, one can demonstrate that solving the constrained optimization problem (3.11) is equivalent to dealing with the unconstrained optimization problem (3.4).

Remark 3.2. For the case of one-dimensional claims, that is, $p=1$ , problem (3.4) can be written as:

(3.12) \begin{equation} \min_{\alpha_{0}({\cdot}),\alpha_{1},\ldots,\alpha_{n}\in \mathbb{R}}\int_{\mathbb{R}} \mathbb{E}\!\left[F\!\left(\left.y\right\vert{\Theta}\right)-\alpha_{0}(y)-\sum_{s=1}^{n}\alpha_{s}I(Y_{s}\leq y)\right]^{2}dy \end{equation}

yielding the credibility estimation of the univariate conditional distribution function as:

\[\widehat{F}\!\left(\left.y\right\vert{{\Theta}}\right)=Z_{N,n}F_{n}(y)+(1-Z_{N,n})F_{0}(y),\]

which is akin to the credibility estimation of the survival function stated in Equation (2.7) of Cai et al. (Reference Cai, Wen, Wu and Zhou2015).

Remark 3.3. If the claims have discrete distributions, we only need to modify Equation (3.4) to

\begin{equation*}\min_{\alpha_{0}({\cdot}),\alpha_{1},\ldots,\alpha_{n}\in \mathbb{R}}\sum_{y}\mathbb{E}\!\left[p\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)-\alpha_{0}(\textbf{y})-\sum_{s=1}^{n}\alpha_{s}I(\textbf{Y}_{s}=\textbf{y})\right]^{2},\end{equation*}

where $p\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)=\mathbb{P}\!\left(\left.\textbf{Y}=\textbf{y}\right\vert\boldsymbol{\Theta}=\boldsymbol{\theta}\right)$ . This yields the corresponding credibility estimation as:

\begin{equation*}\widehat{p}\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)=Z_{N,n}p_{n}\left(\textbf{y}\right)+\left(1-Z_{N,n}\right)p_{0}\left(\textbf{y}\right),\end{equation*}

where $Z_{N,n}$ shares a resemblance with the results of Theorem 3.1, with the distinction of replacing the multivariate integration by multivariate summation.

It is noteworthy that the new credibility factor $Z_{N,n}$ is a deterministic function of sample size n, but independent of the samples. Additionally, both $F_{n}\!\left( {{\textbf{y}}}\right) $ and $F_{0}\!\left( {{\textbf{y}}}\right) $ conform to the fundamental properties of joint distribution functions. Consequently, it is evident that $\widehat{F}\!\left(\left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ possesses attributes such as nonnegativity, monotonicity, right-continuity for each component, and finite additivity. Moreover, it satisfies the following boundary conditions:

\begin{equation*}\lim_{{{y^{(1)}}}\rightarrow +\infty, \ldots, {{y^{(p)}}}\rightarrow +\infty}\left[ \widehat{F}\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) \right] =1\quad\end{equation*}

and

\begin{equation*}\text{ }\lim_{{{y^{(i)}\rightarrow }}-\infty \text{ }}\widehat{F}\!\left(\left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) =0\ \text{for some }i\in\left\{ {1,2,\cdots, p}\right\} .\quad\end{equation*}

Hence, the estimator $\widehat{F}\!\left( \left. {\textbf{y}}\right\vert\boldsymbol{\Theta} \right) $ also qualifies as a joint distribution function. As per Theorem 3.1, the estimation $\widehat{F}\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ of the conditional joint distribution function can be elegantly represented as a weighted combination of the empirical joint distribution function $F_{n}\!\left( {{\textbf{y}}}\right) $ and the integrated joint distribution function $F_{0}\!\left( {{\textbf{y}}}\right) $ .

The credibility factor $Z_{N,n}$ exhibits an increment with respect to the sample size n, and it fulfills the following asymptotic properties:

(3.13) \begin{equation}\lim_{n\rightarrow \infty }Z_{N,n}=1\quad \text{and }\quad\lim_{n\rightarrow 0}Z_{N,n}=0, \end{equation}

which enjoys a similar expression with the univariate standard Bühlmann model. Consequently, when the sample size is substantial, greater emphasis is conferred upon the empirical distribution function $F_{n}\!\left( {{\textbf{y}}}\right) $ , and vice versa.

3.2. Credibility estimator for the hypothetical mean

Utilizing the credibility estimator $\widehat{F}\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ , one can readily derive the credibility estimation for the hypothetical mean vector $\boldsymbol{\mu }(\boldsymbol{\Theta} )$ by applying the plug-in technique. As a result, one has the following proposition.

Proposition 3.1. By substituting the optimal linear estimate $\widehat{F}\!\left(\left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ of the distribution function into Equation (3.1), we obtain the credibility estimate of the conditional mean vector $\boldsymbol{\mu }(\boldsymbol{\Theta} )$ as follows:

\begin{equation*}\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}=Z_{N,n}\overline{\textbf{Y}}+(1-Z_{N,n}){\boldsymbol{\mu }_{0}}, \end{equation*}

where the credibility factor $Z_{N,n}$ is given by Equation (3.7).

It is evident that the estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ can be represented as a linear weighted average of the sample mean $\overline{\textbf{Y}}$ and the population mean $\boldsymbol{\mu }_{0}$ , where the weights satisfy the property (3.13). This form is very similar with the univariate case for classical Bühlmann model developed in Bühlmann (Reference Bühlmann1967). However, compared with the traditional multivariate credibility estimator given in Theorem 2.1, the credibility factor $Z_{N,n}$ is a scalar, which reduces the computational complexity of traditional multivariate credibility model.

Given the expression of the credibility estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ , we can calculate the conditional expectation of $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ as:

\begin{equation*}\mathbb{E}\!\left[ \left. \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}\right\vert\boldsymbol{\Theta} \right] =Z_{N,n}\mathbb{E}\!\left( \left. \overline{\textbf{Y}}\right\vert \boldsymbol{\Theta} \right) +(1-Z_{N,n})\boldsymbol{\mu }{_{0}=Z_{N,n}}\boldsymbol{\mu }(\boldsymbol{\Theta} )+(1-Z_{N,n})\boldsymbol{\mu }{_{0}.}\end{equation*}

By using the formula for double expectation, we can further find

\begin{equation*}\mathbb{E}\!\left[ \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}\right] =\mathbb{E}\!\left[ \mathbb{E}\!\left( \left. \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}\right\vert \boldsymbol{\Theta} \right) \right] =Z_{N,n}\mathbb{E}\!\left[ \boldsymbol{\mu }(\boldsymbol{\Theta} )\right] +(1-Z_{N,n})\boldsymbol{\mu }{_{0}=\boldsymbol{\mu }_{0}.}\end{equation*}

In the average sense, $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ is an unconditional unbiased estimation of $\mathbb{E}[\boldsymbol{\mu }(\boldsymbol{\Theta} )]$ , which is described as the following proposition.

Proposition 3.2. The credibility estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ is an unbiased estimator of the hypothetical mean vector $\boldsymbol{\mu }(\boldsymbol{\Theta} )$ , that is,

\begin{equation*}\mathbb{E}\!\left[ \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}\right] =\mathbb{E}\!\left[ \boldsymbol{\mu }(\boldsymbol{\Theta} )\right] =\boldsymbol{\mu }_{0}.\end{equation*}

As the estimation of $\boldsymbol{\mu }(\boldsymbol{\Theta} )$ , the mean quadratic error matrix of the credibility estimator $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ is given below.

Proposition 3.3. The mean quadratic error matrix of the credibility estimator $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ is given by:

\begin{equation*}\mathbb{E}\!\left[ \left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu }}(\boldsymbol{\Theta} )\right) \left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu}}(\boldsymbol{\Theta} )\right) ^{\prime }\right] =\frac{Z_{N,n}^{2}}{n}\Sigma_{0}+(1-Z_{N,n})^{2}T.\end{equation*}

Moreover, as $n\rightarrow \infty $ , the mean quadratic error matrix tends to a $p\times p$ zero matrix:

\begin{equation*}\mathbb{E}\!\left[ \left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu }}(\boldsymbol{\Theta} )\right) \left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu}}(\boldsymbol{\Theta} )\right) ^{\prime }\right] \rightarrow 0_{p\times p},\end{equation*}

and hence

\begin{equation*}\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu }}(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi} }^{2}\right] =\mathbb{E}\!\left[ \boldsymbol{\xi} ^{\prime }\left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu }}(\boldsymbol{\Theta} )\right) \left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-{\boldsymbol{\mu }}(\boldsymbol{\Theta} )\right) ^{\prime }\boldsymbol{\xi} \right] \rightarrow 0.\end{equation*}

where $||\cdot ||_{\boldsymbol{\xi} }^{2}$ represents the weighted Frobenius norm,Footnote 8 $\boldsymbol{\xi} =\left(\xi _{1},\xi _{2},\cdots, \xi _{p}\right) ^{\prime }$ represents the weight vector, and $\xi_{i}\geq 0$ , for $i=1,2,\cdots,p$ such that $\sum_{i=1}^{p}\xi_{i}=1$ .

3.3. Credibility estimator for the process variance

Note that

\begin{equation*}\Sigma(\boldsymbol{\Theta})=\int_{\mathbb{R}^{p}}\left[\left(\textbf{y}-\int_{\mathbb{R}^{p}}\textbf{y}dF\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)\right)\left(\textbf{y}-\int_{\mathbb{R}^{p}}\textbf{y}dF\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)\right)^{\prime}\right]dF\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right).\end{equation*}

We can first achieve the credibility estimator of the conditional variance based on $\widehat{F}\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta}\right) $ and then obtain the credibility estimator of $\Sigma (\boldsymbol{\Theta} )$ . For this purpose, let

\begin{equation*}U_{n}=\frac{1}{n}\sum_{i=1}^{n}\textbf{Y}_{i}\textbf{Y}_{i}^{\prime }\quad \mbox{and}\quad U_{0}=\int_{\mathbb{R}^{p}} \textbf{yy}^{\prime }dF_{0}\!\left( {\textbf{y}}\right).\end{equation*}

The following proposition can be reached.

Proposition 3.4. Under the Bayesian model settings stated in Assumptions 2.1 and 2.2, replacing $F\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ with $\widehat{F}\!\left( \left. {\textbf{y}}\right\vert \boldsymbol{\Theta} \right) $ in Equation (3.2), the credibility estimation of the conditional covariance matrix $\Sigma (\boldsymbol{\Theta} )$ is given by:

\begin{equation*}\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}=\omega _{1,n}\Sigma _{n}+\omega_{2,n}\Sigma _{0}+(1-\omega _{1,n}-\omega _{2,n})M_{0},\end{equation*}

where $\Sigma _{0}=\mathbb{E}\!\left[ \Sigma (\boldsymbol{\Theta} )\right] $ , $\Sigma _{n}=\frac{1}{n}\sum_{i=1}^{n}\left( \textbf{Y}_{i}-\overline{\textbf{Y}}\right) \left( \textbf{Y}_{i}-\overline{\textbf{Y}}\right) ^{\prime }$ is the sample covariance matrix, and

\begin{equation*}\omega _{1,n}=Z_{N,n}^{2},\quad \omega _{2,n}=(1-Z_{N,n})^{2},\quad M_{0}=\frac{1}{2}\left( U_{n}-\overline{\textbf{Y}}\boldsymbol{\mu }_{0}^{\prime }-\boldsymbol{\mu }_{0}\overline{\textbf{Y}}^{\prime }+U_{0}\right) .\end{equation*}

Notably, the credibility estimation of the conditional covariance matrix is consisting of three parts. According to Proposition 3.4 and property (3.13), it holds that $\omega_{1,n}\rightarrow1$ and $\omega_{2,n}\rightarrow0$ as $n\rightarrow\infty$ . Therefore, when n is large, the credibility estimate $\widehat{\Sigma_{N,n}(\boldsymbol{\Theta})}$ assigns more weight to sample covariance matrix $\Sigma_{n}$ . Conversely, more weight is assigned to $\Sigma_{0}$ and $M_{0}$ when the sample size is smaller.

3.4. Statistical properties of the estimators

The consistency and asymptotic normality of $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}$ are given in the following two theorems.

Theorem 3.2. Given $\boldsymbol{\Theta}$ , the estimators $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}$ are strongly consistent with respect to the conditional mean vector $\boldsymbol{\mu }(\boldsymbol{\Theta})$ and conditional covariance matrix $\Sigma (\boldsymbol{\Theta} )$ , respectively. That is, when $n\rightarrow \infty $ , we have

\begin{equation*}\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}\rightarrow \boldsymbol{\mu }(\boldsymbol{\Theta}),\,\,a.s.\quad \mbox{and}\quad \widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}\rightarrow\Sigma (\boldsymbol{\Theta} ),\,\,a.s.\end{equation*}

Theorem 3.3. Given $\boldsymbol{\Theta}$ , the estimator $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ is asymptotically normal when $n\rightarrow \infty $ , that is,

\begin{equation*}\sqrt{n}\left( \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta})\right) \overset{L}{\rightarrow }N(0,\Sigma (\boldsymbol{\Theta} )),\end{equation*}

where “ $\overset{L}{\rightarrow }$ ” means convergence in law/distribution.

Theorems 3.2 and 3.3 demonstrate that $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\Sigma _{N,n}(\boldsymbol{\Theta})}$ serve as good estimators of the conditional mean vector $\boldsymbol{\mu }(\boldsymbol{\Theta})$ and conditional covariance matrix $\Sigma (\boldsymbol{\Theta})$ . Both of them can be plugged in any continuous statistical functionals of these two quantities.

Next, we proceed with a theoretical comparison on the mean of the weighted $F$ -norms between our new credibility estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and the traditional credibility estimation $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ . To simplify the analysis, we first assume that $p=2$ and introduce the following notations:

\begin{equation*}\boldsymbol{\mu }(\boldsymbol{\Theta} )=\mathbb{E}[\textbf{Y}_{1}|\boldsymbol{\Theta} ]=\left(\begin{array}{c}{\mu }_{1}(\Theta ) \\[4pt]{\mu }_{2}(\Theta )\end{array}\right), \quad \Sigma {(\boldsymbol{\Theta} )}=\mathbb{V}[\textbf{Y}_{1}|\boldsymbol{\Theta} ]=\left(\begin{array}{c@{\quad}c}\sigma _{1}^{2}(\Theta ) & \nu _{2}(\Theta ) \\[4pt]\nu _{2}(\Theta ) & \sigma _{2}^{2}(\Theta )\end{array}\right)\end{equation*}

and

\begin{equation*}\boldsymbol{\mu }_{0}=\mathbb{E}(\boldsymbol{\mu }(\boldsymbol{\Theta} ))=\left(\begin{array}{c}{\mu }_{1} \\{\mu }_{2}\end{array}\right), \quad T=\mathbb{V}(\boldsymbol{\mu }(\boldsymbol{\Theta} ))=\left(\begin{array}{c@{\quad}c}\tau _{1}^{2} & \nu _{1} \\[4pt]\nu _{1} & \tau _{2}^{2}\end{array}\right), \quad \Sigma _{0}=\mathbb{E}(\Sigma (\boldsymbol{\Theta} ))=\left(\begin{array}{c@{\quad}c}\sigma _{1}^{2} & \nu _{2} \\[4pt]\nu _{2} & \sigma _{2}^{2}\end{array}\right) .\end{equation*}

where $\nu _{1}=\rho _{1}\tau _{1}\tau _{2}$ , $\nu _{2}=\rho _{2}\sigma_{1}\sigma _{2}$ , and

(3.14) \begin{equation}\rho _{1}=\frac{\mathbb{C}\unicode{x1D560}\unicode{x1D567}\!\left( {\mu }_{1}\left( \Theta \right), {\mu }_{2}\left( \Theta \right) \right) }{\sqrt{\mathbb{V}\!\left( {\mu }_{1}\left( \Theta \right) \right)\mathbb{V}\!\left( {\mu }_{2}\left( \Theta \right) \right) }},\quad\rho _{2}=\frac{\mathbb{E}\!\left[ \mathbb{C}\unicode{x1D560}\unicode{x1D567}\!\left(\left.Y_{1}^{(1)},Y_{1}^{(2)}\right\vert\boldsymbol{\Theta} \right) \right] }{\sqrt{\mathbb{E}\!\left[\mathbb{V}\!\left(\left. Y_{1}^{(1)}\right\vert\Theta_{1} \right) \right] \mathbb{E}\!\left[ \mathbb{V}\!\left(\left. Y_{1}^{(2)}\right\vert\Theta_{2} \right) \right] }}. \end{equation}

Here, $\rho _{1}$ and $\rho _{2}$ represent the two correlation coefficients denoting the inter-dependence among risks and the intra-dependence within each risk, respectively.

Proposition 3.5. Consider $p=2$ and $\boldsymbol{\xi} =(\xi _{1},\xi _{2})^{\prime }$ . Based on Theorem 2.1 and Proposition 3.1, the mean of the weighted F-norms for the error of estimates $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ can be expressed as:

(3.15) \begin{equation}\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi} }^{2}\right] =\frac{n\tau _{0}^{4}\left( \xi _{1}\sigma _{1}^{2}+\xi _{2}\sigma _{2}^{2}\right)}{\left( n\tau _{0}^{2}+\sigma _{0}^{2}\right) ^{2}}+\frac{\sigma_{0}^{4}\left( \xi _{1}\tau _{1}^{2}+\xi _{2}\tau _{2}^{2}\right) }{\left(n\tau _{0}^{2}+\sigma _{0}^{2}\right) ^{2}} \end{equation}

and

(3.16) \begin{equation}\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi} }^{2}\right] =\Delta_{1}+\Delta _{2}+\Delta _{3}+\Delta _{4}, \end{equation}

respectively, where

\begin{align*} \Delta _{1} & =\frac{n\delta _{1}\left( \xi _{1}\left(\tau _{1}^{2}\delta _{2}-\nu_{1}\delta _{3}\right)^{2}+\xi _{2}\left(\nu _{1}\sigma _{1}^{2}-\nu _{2}\tau_{1}^{2}\right)^{2}\right) }{\left( \delta _{1}\delta _{2}-\delta _{3}^{2}\right)^{2}}, \\[5pt] \Delta _{2} & =\frac{n\delta _{2}\left( \xi _{1}\left(\tau _{2}^{2}\delta _{1}-\nu_{1}\delta _{3}\right)^{2}+\xi _{2}\left(\nu _{1}\sigma _{2}^{2}-\nu _{2}\tau_{2}^{2}\right)^{2}\right) }{\left( \delta _{1}\delta _{2}-\delta _{3}^{2}\right)^{2}}, \\[5pt]\Delta _{3} & =\frac{1}{({-}2\nu _{1})^{-1}\left( \delta _{1}\delta _{2}-\delta_{3}^{2}\right) ^{2}}\times \left[ n\nu _{1}\sigma _{1}^{2}\sigma_{2}^{2}(\xi _{2}\delta _{1}+\xi _{1}\delta _{2})-n\nu _{2}\left( \xi_{2}\tau _{1}^{2}\sigma _{2}^{2}\delta _{1}\right. \right. \\[5pt]& \quad \left. \left. +\xi _{1}\tau _{2}^{2}\sigma _{1}^{2}\delta_{2}-\delta _{3}\left( \nu _{2}(\xi _{2}\tau _{1}^{2}+\xi _{1}\tau_{2}^{2})-\nu _{1}\left(\xi _{2}\sigma _{1}^{2}+\xi _{1}\sigma _{2}^{2}\right)\right)\right) \right], \\[5pt]\Delta _{4} & =\frac{\xi _{2}\left(\sigma _{2}^{2}-n\tau _{2}^{2}\right)\delta _{1}\tau_{2}^{2}+\xi _{1}\left(\sigma _{1}^{2}-n\tau _{1}^{2}\right)\delta _{2}\tau_{1}^{2}-\left(\nu _{2}-n\nu _{1}\right)\left(\xi _{1}\tau _{1}^{2}+\xi _{2}\tau_{2}^{2}\right)\delta _{3}}{\delta _{1}\delta _{2}-\delta _{3}^{2}}\end{align*}

and

\begin{equation*}\delta _{1}=n\tau _{1}^{2}+\sigma _{1}^{2},\quad \delta _{2}=n\tau_{2}^{2}+\sigma _{2}^{2},\quad \delta _{3}=n\nu _{1}+\nu _{2}.\end{equation*}

Compared to the traditional multivariate credibility estimation $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ , the mean of the weighted F-norms for the error of the new credibility estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta})}$ is more straightforward to compute. The main reason is that the credibility factor ${Z}_{C,n}$ in the traditional model is a matrix, while in our estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ , the corresponding credibility factor $Z_{N,n}$ is a scalar. This scalar form simplifies the computation process and enhances user-friendliness and interpretability.

Indeed, according to the result of Proposition 3.5, when $p>2$ , the mean of the weighted F-norms for the error of $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ can be calculated as:

\begin{align*}\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi} }^{2}\right] & =\mathbb{E}\!\left[ \xi _{1}\left( \widehat{{\mu }_{N,n}^{(1)}(\Theta )}-{\mu }_{1}(\Theta )\right) ^{2}\right] +\cdots +\mathbb{E}\!\left[ \xi_{p}\!\left( \widehat{{\mu }_{N,n}^{(p)}(\Theta )}-\boldsymbol{\mu }_{p}(\Theta )\right) ^{2}\right] \\& =\frac{n\tau _{0}^{4}\left( \xi _{1}\sigma _{1}^{2}+\cdots +\xi _{p}\sigma_{p}^{2}\right) }{\left( n\tau _{0}^{2}+\sigma _{0}^{2}\right) ^{2}}+\frac{\sigma _{0}^{4}\left( \xi _{1}\tau _{1}^{2}+\cdots +\xi _{p}\tau_{p}^{2}\right) }{\left( n\tau _{0}^{2}+\sigma _{0}^{2}\right) ^{2}}.\end{align*}

The traditional multivariate credibility estimation $\widehat{\boldsymbol{\mu}_{C,n}(\boldsymbol{\Theta})}$ minimizes problem (2.2), and thus, in the sense that the matrix $A-B \leq0$ is equivalent to $B-A$ being a positive definite matrix, it is evident that

\[\mathbb{E}\!\left[\left(\widehat{\boldsymbol{\mu}_{C,n}(\boldsymbol{\Theta})}-\boldsymbol{\mu}(\boldsymbol{\Theta})\right) \left(\widehat{\boldsymbol{\mu}_{C,n}(\boldsymbol{\Theta})}-\boldsymbol{\mu}(\boldsymbol{\Theta})\right)^{\prime}\right]\leq \mathbb{E}\!\left[\left(\widehat{\boldsymbol{\mu}_{N,n}(\boldsymbol{\Theta})}-\boldsymbol{\mu}(\boldsymbol{\Theta})\right) \left(\widehat{\boldsymbol{\mu}_{N,n}(\boldsymbol{\Theta})}-\boldsymbol{\mu}(\boldsymbol{\Theta})\right)^{\prime}\right].\]

In other words, $\widehat{\boldsymbol{\mu}_{N,n}(\boldsymbol{\Theta})}$ is dominated by $\widehat{\boldsymbol{\mu}_{C,n}(\boldsymbol{\Theta})}$ in terms of the mean squared loss, which will be illustrated in Section 4 by two numerical examples. However, it would be interesting to see that our method perform competitively with the traditional one when the sample size is moderate or large.

For the case of $p=2$ , a comparison of the mean of the weighted F-norms for the error for both credibility estimations leads to the following conclusion.

Proposition 3.6. In Equation (3.14), if $\rho _{1}=\rho _{2}=0$ (i.e., $\nu _{1}=\nu _{2}=0$ ), then we have

(3.17) \begin{equation}\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi} }^{2}\right] \leq\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi }}^{2}\right] .\end{equation}

In accordance with Proposition 3.6, it can be readily demonstrated that when $p>2$ and $\rho_{1}=\cdots=\rho_{p}=0$ , the p-dimensional random vector $\textbf{Y}$ demonstrates pairwise independence among its components. The mean of the weighted F-norm for the error in traditional multivariate credibility estimation is smaller than that for the error in the new estimator. However, when the correlation coefficients $\rho_i\neq0 $ for some $i=1,\cdots,p$ , the expression for the mean of the weighted F-norm for the error in traditional credibility estimation becomes significantly intricate, rendering a direct comparison of their magnitudes. In the next section dedicated to numerical simulations, we will employ Monte Carlo methods to perform the comparison.

3.5. Application in various premium principles

In nonlife insurance, premium calculation principles are real functionals mapping risk variables to the set of nonnegative real numbers. For a p-dimensional risk $\textbf{Y}$ with conditional distribution function $F\!\left( \left. \textbf{y}\right\vert \boldsymbol{\Theta} \right) $ , consider the aggregate risk in for such insurance portfolio:

\begin{equation*}Z=a_{1}Y^{(1)}+\cdots +a_{p}Y^{(p)}=\textbf{a}^{\prime }\textbf{Y},\end{equation*}

where $\textbf{Y}=\left( Y^{(1)},Y^{(2)},\cdots,Y^{(p)}\right) ^{\prime }$ and $\textbf{a}=\left( a_{1},\cdots,a_{p}\right) ^{\prime }$ are the weights for these business lines such that $a_{i}\geq 0$ , $i=1,2,\cdots, p$ . Guerra and Centeno (Reference Guerra and Centeno2010) introduced variance-related premium principleFootnote 9 of risk Z with its expression given by:

(3.18) \begin{equation}H_{Z}\!\left( \boldsymbol{\Theta} \right) =\left( 1+\lambda_{1} \right) \mathbb{E}\!\left(Z|\boldsymbol{\Theta} \right) +g\!\left( \mathbb{V}\!\left( Z|\boldsymbol{\Theta} \right) \right) =\left(1+\lambda_{1} \right) \textbf{a}^{\prime }\boldsymbol{\mu }\left( \boldsymbol{\Theta} \right) +g\!\left(\textbf{a}^{\prime }\Sigma \!\left( \boldsymbol{\Theta} \right) \textbf{a}\right), \end{equation}

where $\lambda_{1} \geq 0$ signifies the nonnegative safety loading factor for the expectation and $g\,:\,\mathbb{R}_{+}\mapsto \mathbb{R}_{+}$ is an increasing function such that $g(0)=0$ .

By substituting $\boldsymbol{\mu }\!\left( \boldsymbol{\Theta} \right) $ and $\Sigma \!\left(\boldsymbol{\Theta} \right) $ with their credibility estimators $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}$ , respectively, a credibility estimation of $H_{Z}\!\left( \boldsymbol{\Theta} \right) $ is achieved as:

\begin{equation*}\widehat{H_{Z}\!\left( \boldsymbol{\Theta} \right) }=\left( 1+\lambda_{1} \right) \textbf{a}^{\prime }\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}+g\!\left( \textbf{a}^{\prime }\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}\textbf{a}\right) .\end{equation*}

It is easy to show that the estimation $\widehat{H_{Z}\!\left( \boldsymbol{\Theta} \right) }$ is consistent estimator for risk premium $H_{Z}\!\left( \boldsymbol{\Theta} \right) $ according to Theorem 3.2 and the continuity of function g. Based on the premium principle given in Equation (3.18), we obtain credibility estimates of risk premiums under some commonly used actuarial premium principles as follows:

• • If we set $g(x)=0$ and $\lambda_{1} \geq 0$ , then

  (3.19) \begin{equation}\widehat{H_{Z}^{(1)}\left( \boldsymbol{\Theta} \right) }=\left( 1+\lambda_{1} \right) \textbf{a}^{\prime }\left[ Z_{N,n}\overline{\textbf{Y}}+(1-Z_{N,n}){\boldsymbol{\mu }_{0}}\right] \end{equation}
  is the credibility estimation of risk premium under the expected value premium principle.

• • If we set $g(x)=\lambda_{2} \cdot x$ and $\lambda _{1} =0$ , where $\lambda_{2} \geq 0$ signifies the nonnegative safety loading factor for the variance term, then

  (3.20) \begin{eqnarray}\widehat{H_{Z}^{(2)}\left( \boldsymbol{\Theta} \right) } &=&\textbf{a}^{\prime }\left[ Z_{N,n}\overline{\textbf{Y}}+(1-Z_{N,n}){\boldsymbol{\mu }_{0}}\right] \notag \\[5pt]&&\quad +{{\lambda_{2} \cdot \textbf{a}^{\prime }\left[ \omega _{1,n}\Sigma _{n}+\omega_{2,n}\Sigma _{0}+(1-\omega _{1,n}-\omega _{2,n})M_{0}\right] \textbf{a}}}\end{eqnarray}
  is the credibility estimation of risk premium under the variance premium principle.

• • If we set $g(x)={{\lambda_{2}\cdot}}\sqrt{x}$ and $\lambda_{1} =0$ , then we derive the credibility estimation of risk premium under the standard deviation premium principle as:

  (3.21) \begin{eqnarray}\widehat{H_{Z}^{(3)}\left( \boldsymbol{\Theta} \right) } &=&\textbf{a}^{\prime }\left[Z_{N,n}\overline{\textbf{Y}}+(1-Z_{N,n}){\boldsymbol{\mu }_{0}}\right] \notag \\[5pt]&&\quad +{{\lambda_{2} \cdot \sqrt{\textbf{a}^{\prime }\left(\omega _{1,n}\Sigma _{n}+\omega_{2,n}\Sigma _{0}+(1-\omega _{1,n}-\omega _{2,n})M_{0}\right)\textbf{a}}}}.\end{eqnarray}

Of course, we can also obtain the credibility estimation of the exponential premium principle charged for Z as follows:

• • By plug in the new credibility estimation (3.5), the credibility estimation of risk premium under the exponential premium principle is given by:

  (3.22) \begin{eqnarray}\widehat{H_{Z}^{(4)}\left( \boldsymbol{\Theta} \right) }&=&\frac{1}{\beta}\log\!\left[\int_{\mathbb{R}^{p}} e^{\beta \textbf{a}^{\prime}\textbf{y}}d\widehat{F}\!\left(\left.\textbf{y}\right\vert\boldsymbol{\Theta}\right)\right]\notag\\[5pt]&=& \frac{1}{\beta}\log\!\left[Z_{N,n}\int_{\mathbb{R}^{p}} e^{\beta \textbf{a}^{\prime}\textbf{y}}dF_{n}(\textbf{y})+\left(1-Z_{N,n}\right)\int_{\mathbb{R}^{p}} e^{\beta \textbf{a}^{\prime}\textbf{y}}dF_{0}(\textbf{y})\right]\notag \\[5pt]&=&\frac{1}{\beta }\log \!\left[Z_{N,n}L_{n}+(1-Z_{N,n})L{_{0}}\right], \end{eqnarray}
  where $\beta >0$ , $L_{n}=\frac{1}{n}\sum_{i=1}^{n}\exp \!\left( \beta \textbf{a}^{\prime }\textbf{Y}_{i}\right) $ , and $L{_{0}=}\mathbb{E}\!\left[ \exp \!\left(\beta \textbf{a}^{\prime }\textbf{Y}_{1}\right) \right] .$

Clearly, compared to traditional multivariate credibility estimates, our estimations have broader applicability, as they can be naturally applied to most premium principles in nonlife insurance and provide corresponding credibility estimators based on the credibility estimate of conditional distribution function.

4.1. Bivariate exponential-gamma model

Assuming that, for a given $\boldsymbol{\Theta}=\boldsymbol{\theta} $ , where $\boldsymbol{\Theta}=\left(\Theta_{1},\Theta_{2}\right)^{\prime}$ and $\boldsymbol{\theta}=\left(\theta_{1},\theta_{2}\right)^{\prime}$ , $\textbf{Y}_{1},\textbf{Y}_{2},\cdots, \textbf{Y}_{n}$ are independently and identically distributed according to the bivariate exponential distribution, with the cumulative joint distribution function given by:

\begin{equation*}F\!\left( \left. {y^{(1)},y^{(2)}}\right\vert \boldsymbol{\Theta}=\boldsymbol{\theta} \right)=1-\exp \!\left\{ -\frac{y^{(1)}}{\theta _{1}}\right\} -\exp \!\left\{ -\frac{y^{(2)}}{\theta _{2}}\right\} +\exp \!\left\{ -\frac{y^{(1)}}{\theta _{1}}-\frac{y^{(2)}}{\theta _{2}}-\frac{\rho _{2}}{\theta _{1}\theta _{2}}y^{(1)}y^{(2)}\right\}, \end{equation*}

and the corresponding probability density function is

\begin{equation*}f\!\left( \left. {y^{(1)},y^{(2)}}\right\vert \boldsymbol{\Theta}=\boldsymbol{\theta} \right) =\left[ -\frac{\rho _{2}}{\theta _{1}\theta _{2}}+\left( \frac{1}{\theta _{1}}+\frac{\rho _{2}}{\theta _{1}\theta _{2}}y^{(2)}\right) \left( \frac{1}{\theta _{2}}+\frac{\rho _{2}}{\theta _{1}\theta _{2}}y^{(1)}\right) \right]e^{-\frac{y^{(1)}}{\theta _{1}}-\frac{y^{(2)}}{\theta _{2}}-\frac{\rho _{2}}{\theta _{1}\theta _{2}}y^{(1)}y^{(2)}},\end{equation*}

where $y^{(1)},y^{(2)}\geq 0$ and $\rho_{2}\in [-1,1]$ .

Further, we assume that $\theta _{1}$ and $\theta _{2}$ are independent of each other and follow gamma prior with density functions:

\begin{equation*}\pi_{\Theta_{1}} (\theta _{1})=\frac{\beta _{1}^{\alpha _{1}}}{\Gamma (\alpha _{1})}\theta _{1}^{\alpha _{1}-1}e^{-\beta _{1}\theta _{1}},\quad \theta_{1},\alpha _{1},\beta _{1}>0,\end{equation*}

and

\begin{equation*}\pi_{\Theta_{2}} (\theta _{2})=\frac{\beta _{2}^{\alpha _{2}}}{\Gamma (\alpha _{2})}\theta _{2}^{\alpha _{2}-1}e^{-\beta _{2}\theta _{2}},\quad \theta_{2},\alpha _{2},\beta _{2}>0\end{equation*}

respectively.

Set $\alpha _{1}=2.4$ , $\alpha _{2}=3.2$ , $\beta _{1}=5.4$ , $\beta _{2}=2.8$ , and $\rho _{2}=0.2$ . By utilizing Equations (2.1), (3.8), and (3.9) and using the Monte Carlo integration method,Footnote 10 the values for various structural parameters are approximately obtained as follows:

\begin{equation*}{\boldsymbol{\mu }_{0}}\approx \left(\begin{array}{c}0.0263 \\[5pt]0.0424\end{array}\right), \quad {T}\approx \left(\begin{array}{c@{\quad}c}0.0015 & 0 \\[5pt]0 & 0.0017\end{array}\right), \quad {\Sigma _{0}}\approx \left(\begin{array}{c@{\quad}c}0.0133 & 0.0043 \\[5pt]0.0043 & 0.0355\end{array}\right), \end{equation*}

and

\begin{equation*}{\tau _{0}^{2}}\approx 0.0026,\qquad {\sigma _{0}^{2}}\approx 0.0387.\end{equation*}

Furthermore, the corresponding conditional mean of the weighted F-norm for the error are obtained as:

(4.1) \begin{equation}\textrm{MSE}_{k,n}(\boldsymbol{\theta} )=\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{k,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert_{\boldsymbol{\xi} }^{2}\bigg|\boldsymbol{\Theta}=\boldsymbol{\theta} \right], \quad k=C\,\,or\,\,N. \end{equation}

Taking $\boldsymbol{\xi} =(0.4,0.6)^{\prime }$ and different values of $(\theta_{1},\theta _{2})$ and n, the simulation is repeated 10,000 times, and the simulation results are given in Table 1.

Table 1. Conditional mean of the weighted F-norm for the error for different values of $(\theta _{1},\theta _{2})$ and n.

Based on the simulation results presented in Table 1, some observations can be made as follows:

• • When the sample size n is given, as the $\theta_{1}$ and $\theta _{2}$ increase, the conditional mean of the weighted F-norms for the errors of $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ and $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ also increase.

• • For a given set of $(\theta_{1},\theta_{2})$ , both estimators exhibit a decreasing trend in their conditional mean of the weighted F-norms for the errors as the sample size n increases, indicating that they both satisfy the consistency property.

• • Note that the true means of $\Theta_1$ and $\Theta_2$ are 4/9 and 8/7, respectively. For various settings of $(\theta_{1},\theta_{2})$ taking values around $(4/9,8/7)$ , $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ has lower values of conditional mean of the weighted F-norms compared with $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ . As the sample size grows, the difference becomes negligible. This demonstrates great performances of our proposed estimators.

To verify the asymptotic normality of $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ , we take different values of $(\theta _{1},\theta _{2})$ and n and repeat the simulation 10,000 times, where we continue to set $\alpha _{1}=2.4$ , $\alpha _{2}=3.2$ , $\beta _{1}=5.4$ , $\beta _{2}=2.8$ , and $\rho _{2}=0.2$ . The corresponding Q-Q plots are shown in Figures 1–6.

Figure 1. Q-Q plot for $\theta_{1}=0.3,\theta_{2}=1.0$ .

Figure 2. Q-Q plot for $\theta_{1}=0.5,\theta_{2}=1.3$ .

Figure 3. Q-Q plot for $\theta_{1}=0.7,\theta_{2}=1.6$ .

Figure 4. Q-Q plot for $n=5$ .

Figure 5. Q-Q plot for $n=30$ .

Figure 6. Q-Q plot for $n=50$ .

Figures 1–3. display the Q-Q plots for different values of $(\theta _{1},\theta _{2})$ with a fixed sample size of $n=20$ . It is evident that the variations in $(\theta _{1},\theta _{2})$ have minimal impact on the asymptotic normality of $\widehat{{\mu }_{N,n}^{(1)}(\Theta )}$ and $\widehat{{\mu }_{N,n}^{(2)}(\Theta )}$ when the sample size remains constant, where $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}=\left( \widehat{{\mu }_{N,n}^{(1)}(\Theta )},\widehat{{\mu }_{N,n}^{(2)}({\Theta} )}\right) ^{\prime }$ .

On the other hand, Figures 4–6 show the Q-Q plots for $\theta _{1}=0.5$ and $\theta _{2}=1.3$ with sample sizes of 5, 30, and 50, respectively. It is apparent that the Q-Q plot of the estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ obtained in this paper gradually approaches a straight line as the sample size increases, indicating that $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ fulfills with asymptotic normality.

4.2. Multivariate normal-normal model

Given the joint distribution function of the sample and the prior distribution $\pi({\cdot})$ , we can in general obtain the values of $\tau_{0}^{2}$ and $\sigma_{0}^{2}$ according to Equations (3.8) and (3.9). However, since there is no explicit cumulative joint distribution function for multivariate normal distribution, we need to estimate the structure parameters $\tau_{0}^{2}$ and $\sigma_{0}^{2}$ with moment method.

First, we generate a sample of $\boldsymbol{\Theta} $ randomly with multivariate normal distribution $N_{2}(\boldsymbol{\mu }_{0},T)$ for the sample size m. Then, based on this sample, we randomly generate m groups of samples with a sample size of n with multivariate normal distribution $N_{3}(\boldsymbol{\Theta},{{\Sigma_{0} }})$ and then obtain the estimations of $\tau _{0}^{2}$ and $\sigma_{0}^{2}$ as:

(4.2) \begin{equation}\widehat{\tau _{0}^{2}}=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty}\int_{-\infty }^{\infty }\frac{1}{m-1}\sum_{i=1}^{m}\left(F_{n}^{(i)}\left( y^{(1)},y^{(2)},y^{(3)}\right) -\overline{F_{n}\!\left(y^{(1)},y^{(2)},y^{(3)}\right) }\right) ^{2}dy^{(1)}dy^{(2)}dy^{(3)}\end{equation}

and

(4.3) \begin{equation}\widehat{\sigma _{0}^{2}}=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty}\int_{-\infty }^{\infty }\frac{1}{m}\sum_{i=1}^{m}\left( F_{n}^{(i)}\left(y^{(1)},y^{(2)},y^{(3)}\right) \left( 1-F_{n}^{(i)}\left(y^{(1)},y^{(2)},y^{(3)}\right) \right) \right) dy^{(1)}dy^{(2)}dy^{(3)},\end{equation}

where $F_{n}^{(i)}\left( y^{(1)},y^{(2)},y^{(3)}\right) $ represents the empirical joint distribution function of the i-th sample, $i=1,\ldots, m$ , and

\begin{equation*}\overline{F_{n}\!\left( y^{(1)},y^{(2)},y^{(3)}\right) }=\frac{1}{m}\sum_{i=1}^{m}F_{n}^{(i)}\left( y^{(1)},y^{(2)},y^{(3)}\right) .\end{equation*}

Based on the properties of multivariate normal distribution and Example 2.2, the matrices are defined as follows:

\begin{equation*}T=\left(\begin{array}{c@{\quad}c@{\quad}c}\tau _{1}^{2} & \rho _{1}\tau _{1}\tau _{2} & \rho _{2}\tau _{1}\tau _{3} \\[4pt]\rho _{1}\tau _{1}\tau _{2} & \tau _{2}^{2} & \rho _{3}\tau _{2}\tau _{3} \\[4pt]\rho _{2}\tau _{1}\tau _{3} & \rho _{3}\tau _{2}\tau _{3} & \tau _{3}^{2}\end{array}\right), \quad \Sigma _{0}=\left(\begin{array}{c@{\quad}c@{\quad}c}\sigma _{1}^{2} & \rho _{4}\sigma _{1}\sigma _{2} & \rho _{5}\sigma_{1}\sigma _{3} \\[4pt]\rho _{4}\sigma _{1}\sigma _{2} & \sigma _{2}^{2} & \rho _{6}\sigma_{2}\sigma _{3} \\[4pt]\rho _{5}\sigma _{1}\sigma _{3} & \rho _{6}\sigma _{2}\sigma _{3} & \sigma_{3}^{2}\end{array}\right), \end{equation*}

and let $\boldsymbol{\mu }_{0}=(1.4,0.8,1.2)^{\prime }$ . By varying the elements of matrices T and $\Sigma _{0}$ with different values, the unconditional mean of the weighted F-norms for the errors of $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ and $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ are computed, where the corresponding formulas are given by:

\begin{equation*}\textrm{MSE}_{k,n}=\mathbb{E}\!\left[ \left\vert \left\vert \widehat{\boldsymbol{\mu }_{k,n}(\boldsymbol{\Theta} )}-\boldsymbol{\mu }(\boldsymbol{\Theta} )\right\vert \right\vert _{\boldsymbol{\xi}}^{2}\right], \quad k=C\,\,or\,\,N,\end{equation*}

where we set $\boldsymbol{\xi} =(0.3,0.5,0.2)^{\prime }$ . Upon conducting 1000 repetitions of the procedure, the corresponding means have been computed and reported in Table 2.

Table 2. Comparison of the unconditional means for different cases.

Table 2 reports the mean value of $\textrm{MSE}_{k,n}$ after undergoing 1000 repetitions. Additionally, in Case 1, the matrices T and $\Sigma_{0}$ are assumed to hold the following values:

(4.4) \begin{equation} T=\left(\begin{array}{c@{\quad}c@{\quad}c}0.2^{2} & 0.006 & 0.008 \\[4pt]0.006 & 0.3^{2} & 0.012 \\[4pt]0.008 & 0.012 & 0.4^2\end{array}\right),\quad\Sigma_{0}=\left(\begin{array}{c@{\quad}c@{\quad}c}0.4^{2} & 0.06 & 0.024 \\[4pt]0.06 & 1.5^{2} & 0.09 \\[4pt]0.024 & 0.09 & 0.6^{2}\end{array}\right),\end{equation}

where $\rho_{i}=0.1$ , for $i=1,2,\ldots,6$ . In Case 2 and Case 3, we set $\rho_{i}=0.5$ and 0.9 for $i=1,\cdots,6$ , respectively, while keeping other values unchanged. In Case 4 and Case 5, we take $(\tau_{1}, \tau_{2}, \tau_{3})= (0.4,0.6,0.8)$ and $(0.8,2.0,1.2)$ , respectively, while other values remain unchanged. In Case 6 and Case 7, we consider $(\sigma_{1},\sigma_{2},\sigma_{3})=(0.6,1.7,0.8)$ and $(0.8,1.9,1.0)$ , respectively, while other values remain unchanged. The following observations can be made based on the simulation results presented in Table 2:

• • According to the simulation outcomes in Cases 1–3, $\textrm{MSE}_{C,n}$ diminishes with escalating correlation coefficients. In contrast, the influence on $\textrm{MSE}_{N,n}$ remains minimal.

• • The results obtained from simulations in Cases 1, 4, and 5 reveal a noteworthy trend: with all other factors unchanged, the value of $\textrm{MSE}_{C,n}$ increases with the values of $(\tau_{1}, \tau_{2}, \tau_{3})$ . On the contrary, the impact on $\textrm{MSE}_{N,n}$ is particularly pronounced in the context of smaller sample sizes; however, as the sample size grows, this effect diminishes. Moreover, as the parameters $(\tau_{1}, \tau_{2},\tau_{3})$ increase, the gap between the values of $\textrm{MSE}_{C,n}$ and $\textrm{MSE}_{N,n}$ gradually narrows. Furthermore, with larger values of n, situations may arise where $\textrm{MSE}_{N,n}$ becomes smaller than $\textrm{MSE}_{C,n}$ , which is in favor of the potential application of our proposed estimator.

• • Based on the simulation outcomes from Case 1, Case 6, and Case 7, both $\textrm{MSE}_{C,n}$ and $\textrm{MSE}_{N,n}$ exhibit an increasingness in $(\sigma _{1},\sigma _{2},\sigma _{3})$ . Besides, $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ converges more quickly as the sample size increases, and its performance becomes comparable to that of the traditional credibility estimator $\widehat{\boldsymbol{\mu }_{C,n}(\boldsymbol{\Theta} )}$ .

According to Equations (4.1) and (4.4), Q-Q plots are implemented for different values of $(\theta _{1},\theta _{1},\theta _{3})$ and n, which are depicted in Figures 7–12. It is readily seen that the simulation results based on the multivariate normal-normal model exhibit similarities to those under the bivariate exponential-gamma model. In particular irrespective of the specific values of $(\theta _{1},\theta _{2},\theta _{3})$ , the asymptotic normality of $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ remains largely unaffected by changes in the sample size. However, when $(\theta _{1},\theta _{2},\theta _{3})$ is fixed, an interesting trend emerges as the sample size increases: the Q-Q plot of the estimation $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta})}$ becomes progressively closer to a straight line. This observation indicates that $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ satisfies asymptotic normality, bolstering its practicability as a new credibility estimator.

Figure 7. Q-Q plot for $\theta_{1}=2,\theta_{2}=0.2,\theta_{3}=1.4$ .

Figure 8. Q-Q plot for $\theta_{1}=2.2,\theta_{2}=0.4,\theta_{3}=1.6$ .

Figure 9. Q-Q plot for $\theta_{1}=2.4,\theta_{2}=0.6,\theta_{3}=1.8$ .

Figure 10. Q-Q plot for $n=5$ .

Figure 11. Q-Q plot for $n=20$ .

Table 3. Annual loss data related to fire insurance.

Figure 12. Q-Q plot for $n=50$ .

4.3. Case analysis

In this part, we shall adopt two explicit insurance scenarios to show the performances of our proposed estimator. At the meantime, the predicted values based on the traditional credibility method will be also reported for the first application under some specified premium principles.

4.3.1. Application I

To demonstrate the specific application of the new estimators presented in this paper, the data on page 117 of Bühlmann and Gisler (Reference Bühlmann and Gisler2005) from some fire insurance business are used. Table 3 shows the related data for five portfolios of insurance policies in five policy years, where $Y_{ij}^{(1)}$ and $Y_{ij}^{(2)}$ represent the total loss and loss rateFootnote 11 of the i-th policy portfolio in the j-th year, respectively.

In order to obtain the credibility estimator, the structure parameters $\boldsymbol{\mu }_{0}$ , T, $\Sigma _{0}$ , $\tau _{0}^{2}$ , and $\sigma _{0}^{2}$ need to be estimated based on the dataset. With moment estimation method, the corresponding estimations of $\boldsymbol{\mu }_{0}$ , T, and $\Sigma _{0}$ are given by:

\begin{equation*}\widehat{\boldsymbol{\mu }}_{0}=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}\textbf{Y}_{ij},\,\,\widehat{\Sigma }_{0}=\frac{1}{m(n-1)}\sum_{i=1}^{m}\sum_{j=1}^{n}\left(\textbf{Y}_{ij}-\overline{\textbf{Y}}_{i}\right) \left( \textbf{Y}_{ij}-\overline{\textbf{Y}}_{i}\right) ^{\prime },\end{equation*}

and

\begin{equation*}\widehat{T}=\frac{1}{n(m-1)}\left[ \sum_{i=1}^{m}n\!\left( \overline{\textbf{Y}}_{i}-\widehat{\boldsymbol{\mu }}_{0}\right) \left( \overline{\textbf{Y}}_{i}-\widehat{\boldsymbol{\mu }}_{0}\right) ^{\prime }-(m-1)\widehat{\Sigma }_{0}\right], \end{equation*}

where $m=n=5$ , and the estimations of $\tau _{0}^{2}$ and $\sigma _{0}^{2}$ are given by (4.2) and (4.3), respectively. Based on the dataset in Table 3, those estimators can be calculated as $\widehat{\tau _{0}^{2}}=0.5450$ , $\widehat{\sigma _{0}^{2}}=0.9591$ , and

\begin{equation*}\widehat{\boldsymbol{\mu }}_{0}=\left(\begin{array}{c}1.2276 \\[4pt]0.8068\end{array}\right), \quad \widehat{T}=\left(\begin{array}{c@{\quad}c}1.3669 & 0.0864 \\[4pt]0.0864 & 0.0607\end{array}\right), \quad \widehat{\Sigma _{0}}=\left(\begin{array}{c@{\quad}c}0.3795 & 0.2692 \\[4pt]0.2692 & 0.3547\end{array}\right) .\end{equation*}

Based on these results and Equations (3.19)–(3.22), the corresponding credibility estimations under the new estimator and the classical method based on various premium principles are shown in Tables 4–6, where we set $\lambda =0.2$ , $\rho=0.24$ , and $\beta =0.54$ in those four premium principles, and the estimates of $U_{0}$ and $L_{0}$ are given by:

(4.5) \begin{equation}\widehat{U_{0}}=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}\textbf{Y}_{ij}\textbf{Y}_{ij}^{\prime }\,\,\mbox{and}\,\,\widehat{L_{0}}=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}\exp \!\left( \beta \textbf{a}^{\prime }\textbf{Y}_{ij}\right). \end{equation}

The values in Tables 4 through 6 illustrate both of the two credibility estimators of risk premium derived for different weight matrix $\textbf{a}$ and different premium principles. Specifically, Tables 4 and 5 display the predictions using the novel credibility estimates $\widehat{\boldsymbol{\mu }_{N,n}(\boldsymbol{\Theta} )}$ and $\widehat{\Sigma _{N,n}(\boldsymbol{\Theta} )}$ , whereas in Table 6, owing to the challenge of estimating the conditional covariance matrix $\Sigma (\boldsymbol{\Theta} )$ within traditional multivariate credibility method, we can only report the performance of the expected value premium principle, where the expression for $\widehat{H_{Z}^{(1)}(\boldsymbol{\Theta} )}$ based on the traditional multivariate credibility method is

\begin{equation*}\widehat{H_{Z}^{(1)}\left( \boldsymbol{\Theta} \right) }=\left( 1+\lambda_{1} \right) \textbf{a}^{\prime }\left[ {Z}_{C,n}\overline{\textbf{Y}}+(I_{p}-{Z}_{C,n}){\boldsymbol{\mu }_{0}}\right].\end{equation*}

It can be noted that the outcomes predicted by the new estimator are slightly different with those calculated from the traditional method with respect to $\widehat{H_{Z}^{(1)}(\boldsymbol{\Theta})}$ . While for the other three types of premium principles, our method can be easily applied, but the traditional multidimensional credibility is hard to put to use.

Table 4. Our credibility estimations based on various premium principles.

Table 5. Our credibility estimations based on various premium principles.

Table 6. The traditional credibility estimation based on the expected value premium principle.

4.3.2. Application II

To illustrate the practical application of the newly proposed credibility estimates obtained in this paper in higher dimensions, we use the “ClaimsLong” datasetFootnote 12 from the “insuranceData” package in R software; see Wolny-Dominiak and Trzesiok (Reference Wolny-Dominiak and Trzesiok2022). In this dataset, we treat the drivers’ age as the risk profile and categorize it into six classes, that is, 1, 2, 3, 4, 5, 6, with 1 stands for the youngest and 6 stands for the oldest. We use $\textbf{Y}_{i}=\left( Y_{i}^{(1)},\cdots, Y_{i}^{(6)}\right) ^{\prime }$ to model the claim severities for these six classes in year i, and we assume a payout of 1000 yuan for each claim. Consequently, $\textbf{Y}_{i}$ represents the claim amount in year i, calculated as the number of claims in year i divided by the total number of policies and then multiplied by 1000. To obtain estimates of the structural parameters, we employ the Bootstrap method to resample the dataset. After obtaining m sets of data, we estimate the structural parameters using the same method as the previous case and the estimated values are obtained as $\widehat{\tau _{0}^{2}}=12,219.09$ , $\widehat{\sigma _{0}^{2}}=33,728.27$ ,

\begin{equation*}\widehat{\boldsymbol{\mu }}_{0}=\left(19.3750,38.6250,48.0500,46.7222,25.0972,15.8972\right) ^{\prime },\end{equation*}

And

\begin{equation*}\widehat{\Sigma }_{0}=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}422.0484 & 812.1451 & 996.7712 & 970.2049 & 528.0890 & 342.2654 \\[5pt]812.1451 & 1584.5024 & 1943.0102 & 1893.6903 & 1027.0936 & 660.1979 \\[5pt]996.7712 & 1943.0102 & 2409.5705 & 2339.4193 & 1266.2069 & 814.4658 \\[5pt]970.2049 & 1893.6903 & 2339.4193 & 2275.9546 & 1231.5569 & 792.0600 \\[5pt]528.0890 & 1027.0936 & 1266.2069 & 1231.5569 & 668.0973 & 430.0422 \\[5pt]342.2654 & 660.1979 & 814.4658 & 792.0600 & 430.0422 & 279.8111\end{array}\right) .\end{equation*}

With these values and Equations (3.19)–(3.22), the corresponding credibility estimations of the aggregate risk under various premium principles are reported in Table 7, where the weight vector is set as $\textbf{a}=(0.1,0.2,0.2,0.2,0.2,0.1)^{\prime}$ and the estimates of $U_{0}$ and $L_{0}$ are given by Equation (4.5). For the sake of comparisons, we define the expression for the safety loading coefficient as follows:

Table 7. Credibility estimations for the aggregate risk under various premium principles.

\begin{equation*}\textrm{Safety loading coefficient} = \frac{ \textrm{Various premium} - \textrm{Net premium}}{\textrm{Net premium}}.\end{equation*}

However, the traditional multidimensional credibility estimator developed by Jewell (Reference Jewell1973) is challenging to apply to this dataset due to the computational complexity involved in calculating the credibility factor matrix, which requires inverting a high-dimensional matrix with six rows and six columns. Although the obtained credibility factor in this study necessitates the calculation of integrals involving multivariate functions, in practical applications, two approaches can be employed to address the integration computation issue. First, if samples from multiple policy contracts are available, the estimation of credibility factors can be obtained based on samples resembling policy contracts outlined in Application II. The second approach involves utilizing Markov Chain Monte Carlo methods to numerically integrate the multivariate function. In comparison with traditional multivariate credibility methods, we believe that the outcomes of this study hold greater practical applicability.