﻿ 共同效应结构下的分位数信度模型

# 共同效应结构下的分位数信度模型 Quantile Credibility Models under Common Effect Structure

Abstract: In non-life insurance application research, the classical Bühlmann credibility model cannot effectively reflect the tail information of distribution, and the policy between the risks is not completely independent. Therefore, considering the common effect between risks and combining the credibility theory from the perspective of quantile, the quantile credibility models under common effect structure are studied, which further expands the classical credibility model.

1. 引言

Bühlmann (1967) [1] 提出无分布信度模型，用经验数据的线性组合 ${\alpha }_{0}+\underset{i=1}{\overset{n}{\sum }}{\alpha }_{i}{X}_{i}$ 来逼近 ${\mu }_{n+1}\left(\theta \right)$ ，通过求解下面的期望损失函数：

2. 模型假设与准备

2.1. 准备知识

$\stackrel{^}{\xi }\left(p\right)=n\left(\frac{j}{n}-p\right){X}_{\left(j-1\right)}+n\left(p-\frac{j-1}{n}\right){X}_{\left(j\right)},其中\frac{j-1}{n}\le p\le \frac{j}{n},j=1,\cdots ,n.$

$X={\left({X}_{1}{}^{\prime },{X}_{2}{}^{\prime },\cdots ,{X}_{k}{}^{\prime }\right)}^{\prime }$ 为所有索赔数据组成的列向量， ${\stackrel{^}{\xi }}_{ij}\left(p\right)$ 为样本分位数的估计。我们的目标是通过对样本选取不同分位点，预测每份保单在各分位点第 $n+1$ 年的索赔。由于各保单之间存在某一特定效应，我们用随机变量 $\Lambda$ 来刻画此种共同效应，下面给出模型的基本假设。

2.2. 模型假设

$E\left({\stackrel{^}{\xi }}_{ij}\left(p\right)|{\Theta }_{i},\Lambda \right)={\Xi }_{p}\left({\Theta }_{i},\Lambda \right),Var\left({\stackrel{^}{\xi }}_{ij}\left(p\right)|{\Theta }_{i},\Lambda \right)={V}_{p}\left({\Theta }_{i},\Lambda \right).$

$\begin{array}{l}E\left[{\Xi }_{P}\left({\Theta }_{i},\Lambda \right)|\Lambda \right]={\Xi }_{1}\left(p,\Lambda \right),Var\left[{\Xi }_{P}\left({\Theta }_{i},\Lambda \right)|\Lambda \right]={S}_{2}^{2}\left(p,\Lambda \right),\\ E\left[{\Xi }_{1}\left(p,\Lambda \right)\right]={\Xi }_{1}\left(p\right),E\left[{S}_{2}^{2}\left(p,\Lambda \right)\right]={S}_{2}^{2}\left(p\right),Var\left[{\Xi }_{1}\left(p,\Lambda \right)\right]={\psi }_{1}\left(p\right),\\ E\left[{V}_{p}\left({\Theta }_{i},\Lambda \right)|\Lambda \right]={S}_{1}^{2}\left(p,\Lambda \right),E\left[{S}_{1}^{2}\left(p,\Lambda \right)\right]={S}_{1}^{2}\left(p\right).\end{array}$

2.3. 模型求解

$\underset{A,B}{\mathrm{min}}E\left[\left(\Xi \left(p\right)-A-B\stackrel{^}{\xi }\left(p\right)\right){\left(\Xi \left(p\right)-A-B\stackrel{^}{\xi }\left(p\right)\right)}^{\prime }\right]$ (1)

$\begin{array}{l}\Xi \left(p\right)={\left[{\stackrel{^}{\xi }}_{1,n+1}\left(p\right),\cdots ,{\stackrel{^}{\xi }}_{K,n+1}\left(p\right)\right]}^{\prime }\\ \stackrel{^}{\xi }\left(p\right)={\left[{\stackrel{^}{\xi }}_{1}{\left(p\right)}^{\prime },\cdots ,{\stackrel{^}{\xi }}_{K}{\left(p\right)}^{\prime }\right]}^{\prime }\\ {\stackrel{^}{\xi }}_{i}\left(p\right)={\left[{\stackrel{^}{\xi }}_{i1}\left(p\right),\cdots ,{\stackrel{^}{\xi }}_{in}\left(p\right)\right]}^{\prime }\end{array}$

$\text{pro}\left(Y|L\left(X,1\right)\right)=E\left(Y\right)+Cov\left(Y,X\right)Co{v}^{-1}\left(X,X\right)\left(X-E\left(X\right)\right)$

${\left(A+BCD\right)}^{-1}={A}^{-1}-{A}^{-1}B{\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}$

$\begin{array}{l}\left(A+BCD\right)\left[{A}^{-1}-{A}^{-1}B{\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}\right]\\ \text{ }\text{\hspace{0.17em}}=I-B{\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}+BCD{A}^{-1}-BCD{A}^{-1}B{\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}\\ \text{ }\text{\hspace{0.17em}}=I+BCD{A}^{-1}-B\left(I+CD{A}^{-1}B\right){\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}{C}^{-1}CD{A}^{-1}\\ \text{ }\text{\hspace{0.17em}}=I+BCD{A}^{-1}-B\left(I+CD{A}^{-1}B\right){\left(I+CD{A}^{-1}B\right)}^{-1}CD{A}^{-1}\\ \text{ }\text{\hspace{0.17em}}=I+BCD{A}^{-1}-BCD{A}^{-1}\\ \text{ }\text{\hspace{0.17em}}=I\end{array}$

${\left(A+BCD\right)}^{-1}={A}^{-1}-{A}^{-1}B{\left({C}^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}$

1) ${\stackrel{^}{\xi }}_{i}\left(p\right)={\left[{\stackrel{^}{\xi }}_{i1}\left(p\right),\cdots ,{\stackrel{^}{\xi }}_{in}\left(p\right)\right]}^{\prime }$ 的期望为：

$E\left[{\stackrel{^}{\xi }}_{i}\left(p\right)\right]={\Xi }_{1}\left(p\right){1}_{n}.$

2) 向量 $\stackrel{^}{\xi }\left(p\right)={\left[{\stackrel{^}{\xi }}_{1}{\left(p\right)}^{\prime },\cdots ,{\stackrel{^}{\xi }}_{K}{\left(p\right)}^{\prime }\right]}^{\prime }$ 的协方差矩阵为：

$Cov\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]={I}_{k}\otimes \left({S}_{1}^{2}\left(p\right){I}_{n}+{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)+{\psi }_{1}\left(p\right){1}_{nk}{{1}^{\prime }}_{nk}.$

3) 矩阵 $Cov\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]$ 的逆矩阵为：

$\begin{array}{c}Co{v}^{-1}\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]=\frac{1}{{S}_{1}^{2}\left(p\right)}{I}_{k}\otimes \left({I}_{n}-\frac{{S}_{2}^{2}\left(p\right)}{{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)}{1}_{n}{{1}^{\prime }}_{n}\right)\\ -\frac{{\psi }_{1}\left(p\right)}{\left[{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)\right]\left[{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)+nk{\psi }_{1}\left(p\right)\right]}{1}_{nk}{{1}^{\prime }}_{nk}.\end{array}$

4) $\Xi \left(p\right)={\left[{\stackrel{^}{\xi }}_{1,n+1}\left(p\right),\cdots ,{\stackrel{^}{\xi }}_{K,n+1}\left(p\right)\right]}^{\prime }$$\stackrel{^}{\xi }\left(p\right)={\left[{\stackrel{^}{\xi }}_{1}{\left(p\right)}^{\prime },\cdots ,{\stackrel{^}{\xi }}_{K}{\left(p\right)}^{\prime }\right]}^{\prime }$ 的协方差矩阵为

$Cov\left[\Xi \left(p\right),\stackrel{^}{\xi }\left(p\right)\right]=\left({S}_{2}^{2}\left(p\right){{e}^{\prime }}_{i}+{\psi }_{1}\left(p\right){{1}^{\prime }}_{k}\right)\otimes {{1}^{\prime }}_{n}.$

1)

$E\left[{\stackrel{^}{\xi }}_{i}\left(p\right)\right]=E\left[\left[\left(E{\stackrel{^}{\xi }}_{i}\left(p\right){|\Theta }_{i},\Lambda \right)\right]=E\left(\begin{array}{c}E\left({\Xi }_{p}\left({\Theta }_{i},\Lambda \right)|\Lambda \right)\\ ⋮\\ E\left({\Xi }_{p}\left({\Theta }_{i},\Lambda \right)|\Lambda \right)\end{array}\right)=E\left(\begin{array}{c}{\Xi }_{1}\left(p,\Lambda \right)\\ ⋮\\ {\Xi }_{1}\left(p,\Lambda \right)\end{array}\right)={\Xi }_{1}\left(p\right){1}_{n}.$

2) 记 $\Theta ={\left({\Theta }_{1},\cdots ,{\Theta }_{k}\right)}^{\prime }$ ，则

$\begin{array}{l}E\left[\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right]\text{=[}{\Xi }_{p}\left({\Theta }_{1},\Lambda \right){\text{1 ′}}_{n}\text{,}\cdots \text{,}{\Xi }_{p}\left({\Theta }_{k},\Lambda \right){\text{1 ′}}_{n}\text{] ′}={\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right)\text{,}\cdots \text{,}{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)\right]}^{\prime }\otimes {\text{1}}_{n}\\ Cov\left[\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right]=\text{diag}\left[{V}_{p}\left({\Theta }_{1},\Lambda \right){I}_{n}\text{,}\cdots \text{,}{V}_{p}\left({\Theta }_{k},\Lambda \right){I}_{n}\right]\\ Cov\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]=Cov\left[E\left(\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]+E\left[Cov\left(\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]\triangleq {Ι}_{1}+{Ι}_{2}\end{array}$

$\begin{array}{c}{Ι}_{1}=Cov\left[E\left(\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]\\ =E\left(Cov\left\{{\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right),\cdots ,{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)\right]}^{\prime }\otimes {1}_{n}|\Lambda \right\}\right)+Cov\left(E\left\{{\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right),\cdots ,{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)\right]}^{\prime }\otimes {1}_{n}|\Lambda \right\}\right)\\ =E\left[{S}_{2}^{2}\left(p,\Lambda \right)\right]{I}_{k}\otimes {1}_{n}{{1}^{\prime }}_{n}+Cov\left[{\Xi }_{1}\left(p,\Lambda \right){1}_{nk}\right]\\ ={S}_{2}^{2}\left(p\right){I}_{k}\otimes {1}_{n}{{1}^{\prime }}_{n}+{\psi }_{1}\left(p\right){1}_{nk}{{1}^{\prime }}_{nk}\\ {Ι}_{2}=E\left[Cov\left(\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]={S}_{1}^{2}\left(p\right){I}_{nk}\end{array}$

$Cov\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]\triangleq {Ι}_{1}+{Ι}_{2}={I}_{k}\otimes \left({S}_{1}^{2}\left(p\right){I}_{n}+{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)+{\psi }_{1}\left(p\right){1}_{nk}{{1}^{\prime }}_{nk}$

3) 根据引理2得

$\begin{array}{c}{\left({S}_{1}^{2}\left(p\right){I}_{n}\text{+}{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)}^{-1}=\frac{1}{{S}_{1}^{2}\left(p\right)}{I}_{n}-\frac{\left(p\right){1}_{n}{{1}^{\prime }}_{n}}{\frac{1}{{S}_{2}^{2}\left(p\right)}+\frac{1}{{S}_{1}^{2}\left(p\right)}{1}_{n}{{1}^{\prime }}_{n}}\\ =\frac{1}{{S}_{1}^{2}\left(p\right)}\left({I}_{n}-\frac{{S}_{2}^{2}\left(p\right)}{{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)}{1}_{n}{{1}^{\prime }}_{n}\right)\end{array}$

$\begin{array}{c}Co{v}^{-1}\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]={I}_{k}\otimes {\left({S}_{1}^{2}\left(p\right){I}_{n}\text{+}{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)}^{-1}\\ -\frac{\left[{I}_{k}\otimes {\left({S}_{1}^{2}\left(p\right){I}_{n}\text{+}{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)}^{-1}\right]{1}_{nk}{{1}^{\prime }}_{nk}\left[{I}_{k}\otimes {\left({S}_{1}^{2}\left(p\right){I}_{n}\text{+}{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)}^{-1}\right]}{\frac{1}{{\psi }_{1}\left(p\right)}+{{1}^{\prime }}_{nk}\left[{I}_{k}\otimes {\left({S}_{1}^{2}\left(p\right){I}_{n}\text{+}{S}_{2}^{2}\left(p\right){1}_{n}{{1}^{\prime }}_{n}\right)}^{-1}\right]{1}_{nk}}\\ =\frac{1}{{S}_{1}^{2}\left(p\right)}{I}_{k}\otimes \left({I}_{n}-\frac{{S}_{2}^{2}\left(p\right)}{{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)}{1}_{n}{{1}^{\prime }}_{n}\right)\\ -\frac{{\psi }_{1}\left(p\right)}{\left[{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)\right]\left[{S}_{1}^{2}\left(p\right)+n{S}_{2}^{2}\left(p\right)+nk{\psi }_{1}\left(p\right)\right]}{1}_{nk}{{1}^{\prime }}_{nk}.\end{array}$

4) 因为 $Cov\left(\Xi \left(p\right),\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)=0$ ，则

$\begin{array}{c}Cov\left[\Xi \left(p\right),\stackrel{^}{\xi }\left(p\right)\right]=Cov\left[E\left(\Xi \left(p\right)|\Theta ,\Lambda \right),E\left(\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]+E\left[Cov\left(\Xi \left(p\right),\stackrel{^}{\xi }\left(p\right)|\Theta ,\Lambda \right)\right]\\ =Cov\left\{{\Xi }_{p}\left({\Theta }_{i},\Lambda \right),{\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right),\cdots ,{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)\right]}^{\prime }\otimes {1}_{n}\right\}\\ =E\left(Cov\left\{{\Xi }_{p}\left({\Theta }_{i},\Lambda \right),{\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right),\cdots ,{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)\right]}^{\prime }\otimes {1}_{n}|\Lambda \right\}\right)\\ +Cov\left(E\left[{\Xi }_{p}\left({\Theta }_{i},\Lambda \right)|\Lambda \right],{\left\{E\left[{\Xi }_{p}\left({\Theta }_{1},\Lambda \right)|\Lambda \right],\cdots ,E\left[{\Xi }_{p}\left({\Theta }_{k},\Lambda \right)|\Lambda \right]\right\}}^{\prime }\otimes {1}_{n}\right)\\ ={S}_{2}^{2}\left(p\right){{e}^{\prime }}_{i}\otimes {{1}^{\prime }}_{n}+Cov\left[{\Xi }_{1}\left(p,\Lambda \right),{\Xi }_{1}\left(p,\Lambda \right){1}_{nk}\right]={S}_{2}^{2}\left(p\right){{e}^{\prime }}_{i}\otimes {{1}^{\prime }}_{n}+{\psi }_{1}\left(p\right){{1}^{\prime }}_{nk}\\ =\left({S}_{2}^{2}\left(p\right){{e}^{\prime }}_{i}+{\psi }_{1}\left(p\right){{1}^{\prime }}_{k}\right)\otimes {{1}^{\prime }}_{n}\end{array}$

3. 具有共同效应结构的分位数信度模型

$\stackrel{^}{\Xi }\left(p\right)={Z}_{1}\stackrel{¯}{\stackrel{^}{\xi }}\left(p\right)+{Z}_{2}\stackrel{¯}{\stackrel{¯}{\stackrel{^}{\xi }}}\left(p\right)+{Z}_{3}{\Xi }_{1}\left(p\right)$

$\begin{array}{l}{Z}_{1}=\frac{n{S}_{2}^{2}\left(p\right)}{n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)},\\ {Z}_{2}=\frac{nk{\psi }_{1}\left(p\right){S}_{1}^{2}\left(p\right)}{\left[n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)\right]\left[nk{\psi }_{1}\left(p\right)+n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)\right]},\\ {Z}_{3}=1-{Z}_{1}-{Z}_{2}=\frac{{S}_{1}^{2}\left(p\right)}{nk{\psi }_{1}\left(p\right)+n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)}.\end{array}$

${\stackrel{¯}{\stackrel{^}{\xi }}}_{i}\left(p\right)=\frac{1}{n}\underset{j=1}{\overset{n}{\sum }}{\stackrel{^}{\xi }}_{ij}\left(p\right),\stackrel{¯}{\stackrel{¯}{\stackrel{^}{\xi }}}\left(p\right)=\frac{1}{k}\underset{i=1}{\overset{k}{\sum }}{\stackrel{¯}{\stackrel{^}{\xi }}}_{i}\left(p\right).$

$\begin{array}{c}\stackrel{^}{\Xi }\left(p\right)=E\left[\Xi \left(p\right)\right]+Cov\left[\Xi \left(p\right),\stackrel{^}{\xi }\left(p\right)\right]Co{v}^{-1}\left[\stackrel{^}{\xi }\left(p\right),\stackrel{^}{\xi }\left(p\right)\right]\left[\stackrel{^}{\xi }\left(p\right)-E\left(\stackrel{^}{\xi }\left(p\right)\right)\right]\\ ={\Xi }_{1}\left(p\right)+\frac{1}{n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)}\left[{S}_{2}^{2}\left(p\right){{e}^{\prime }}_{i}\otimes {{1}^{\prime }}_{n}+\frac{{\psi }_{1}\left(p\right){S}_{1}^{2}\left(p\right)}{nk{\psi }_{1}\left(p\right)+n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)}{{1}^{\prime }}_{nk}\right]\left[\stackrel{^}{\xi }\left(p\right)-{\Xi }_{1}\left(p,\Lambda \right){1}_{nk}\right]\\ ={\Xi }_{1}\left(p\right)+\frac{1}{n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)}\left[n{S}_{2}^{2}\left(p\right)\left(\stackrel{¯}{{\stackrel{^}{\xi }}_{i}}\left(p\right)-{\Xi }_{1}\left(p\right)\right)+\frac{nk{\psi }_{1}\left(p\right){S}_{1}^{2}\left(p\right)}{nk{\psi }_{1}\left(p\right)+n{S}_{2}^{2}\left(p\right)+{S}_{1}^{2}\left(p\right)}\left(\stackrel{¯}{\stackrel{¯}{\stackrel{\wedge }{\xi }}}\left(p\right)-{\Xi }_{1}\left(p\right)\right)\right]\\ ={Z}_{1}\stackrel{¯}{{\stackrel{^}{\xi }}_{i}}\left(p\right)+{Z}_{2}\stackrel{¯}{\stackrel{¯}{\stackrel{^}{\xi }}}\left(p\right)+{Z}_{3}{\Xi }_{1}\left(p\right)\end{array}$

4. 总结

NOTES

*第一作者。

#通讯作者。

[1] Bühlmann, H. (1967) Experience Rating and Credibility. ASTIN Bulletin, 5, 199-207.
https://doi.org/10.1017/S0515036100008989

[2] Bühlmann, H. and Straub, E. (1970) Glaubwürdigkeit für schadensätze. Bulletin of the Swiss Association of Actuaries, 70, 111-133.

[3] Yeo, K.L. and Valdez, E.A. (2006) Claim Dependence with Common Effects in Credibility Models. Insurance Mathematics & Economics, 38, 609-629.
https://doi.org/10.1016/j.insmatheco.2005.12.006

[4] Wen, L., Wu, X. and Zhou, X. (2009) The Credibility Premiums for Models with Dependence Induced by Common Effects. Insurance Mathematics & Economics, 44, 19-25.
https://doi.org/10.1016/j.insmatheco.2008.09.005

[5] Wang, Z. and Wen, L. (2011) Regression Credibility Models with Random Common Effects. Chinese Journal of Applied Probability & Statistics, 27, 312-322.

[6] Zhang, Q. and Chen, P. (2018) Credibility Estimators with Dependence Structure over Risks and Time under Balanced Loss Function. Statistica Neerlandica, 72, 153-179.

[7] Pitselis, G. (2013) Quantile Credibility Models. Insurance Mathematics & Economics, 52, 477-489.
https://doi.org/10.1016/j.insmatheco.2013.02.011

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