assignmentutor-lab™ 为您的留学生涯保驾护航 在代写风险理论Risk theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论Risk theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|风险理论代写Risk theory代考|The Simple Credibility Model

As the simplest example, assume as in Sect. $1.1$ that $X_1, X_2, \ldots$ are i.i.d. with distribution $F_\zeta$ given $Z=\zeta$, and write
$\mu(\zeta)=\mathbb{E}[X \mid Z=\zeta], \sigma^2(\zeta)=\mathbb{V} \operatorname{ar}[X \mid Z=\zeta], \sigma^2=\mathbb{E} \sigma^2(Z), \tau^2=\mathbb{V} \operatorname{ar} \mu(Z)$
The collective premium is $H_{\text {Coll }}=\mathbb{E} \mu(Z)$ and the Bayes premium is
$$H_{\text {Bayes }}=\mathbb{E}\left[\mu(Z) \mid X_1, \ldots, X_n\right]$$
The Bayes premium can be characterized as the orthogonal projection of $\mu(Z)$ onto the space $L$ of all square integrable $\sigma\left(X_1, \ldots, X_n\right)$-measurable r.v.s, that is, of all (measurable) functions $\psi\left(X_1, \ldots, X_n\right)$ of $X_1, \ldots, X_n$ such that $\mathbb{E} \psi\left(X_1, \ldots, X_n\right)^2<\infty$

We now define $L_{\text {aff }} \subset L$ as the set of all affine combinations of $X_1, \ldots, X_n$, that is,
$$L_{\text {aff }}=\left{a_0+a_1 X_1+\cdots+a_n X_n: a_0, a_1, \ldots, a_n \in \mathbb{R}\right}$$
The credibility premium $H_{\text {Cred }}$ is defined as the orthogonal projection of $\mu(Z)$ on $L_{\text {aff. In formulas, }}$
$$H_{\text {Bayes }}=P_L \mu(Z), \quad H_{\text {Cred }}=P_{L_{\text {aff }}} \mu(Z) .$$
Since $L_{\text {aff }} \subset L$, we have $P_{L_{\text {aff }}}=P_{L_{\text {aff }}} \circ P_L$. If $H_{\text {Bayes }}=P_L \mu(Z)$ is an affine combination of the $X_i$, i.e. an affine combination of the $X_i$, it therefore equals $P_{L_{\text {aff }}} \mu(Z)=H_{\text {Cred. }}$. This form of the Bayes premium was precisely what was found in the exponential family setting of Sect. 2, cf. the examples and Proposition 2.5. Therefore the Bayes premium and the credibility premium are the same thing there.

## 金融代写|风险理论代写Risk theory代考|The Bühlmann Model

The Bühlmann model considers a population of $M$ risks, characterized by their risk parameters $Z_1, \ldots, Z_M$. For risk $m=1, \ldots, M$, a vector $X_m=\left(X_{1 m} \ldots X_{n_m m}\right)$ (typically claim sizes or claim numbers in consecutive years) is observed, such that given $Z_m=\zeta$, the $X_{k m}$ are i.i.d. with distribution $F_\zeta$. The $Z_m$ are assumed i.i.d.

and we define $\mu(\zeta), \sigma^2(\zeta)$ etc. as above. Similarly, $H_{\text {Coll }}=\mathbb{E} \mu\left(Z_m\right)$. The question is what the credibility premium $H_{\text {Cred }}^{(m)}$ looks like for each $m$, that is, which affine combination minimizes
$$\mathbb{E}\left(\mu\left(Z_m\right)-a_{0 m}-\sum_{m=1}^M \sum_{k=1}^{n_m} a_{k m} X_{k m}\right)^2 .$$
The answer is simple and follows directly from Corollary $3.5$ : one can just do the calculations for risk $m$ in isolation as if no statistics on risks $m^{\prime} \neq m$ were available, that is, just use the formulas of the preceding section:
Proposition 3.6 Define
$$c_m=n_m /\left(n_m+\sigma^2 / \tau^2\right), \quad \bar{X}{\cdot m}=\left(X{1 m}+\cdots+X_{n_m m}\right) / n_m .$$
Then the mean square error (3.10) is minimized by taking $a_{0 m}=\left(1-c_m\right) H_{\text {Coll, }}$, $a_{k m}=1 / n_m, a_{k m^{\prime}}=0$ for $m^{\prime} \neq m$. Thus,
$$H_{\mathrm{Cred}}^{(m)}=\left(1-c_m\right) H_{\mathrm{Coll}}+c_m \bar{X}{\bullet} .$$ Again, there is an alternative proof via the multivariate normal distribution: use the reasoning of Remark 3.3, according to which we can replace the $\mu\left(Z_m\right), X{k m}$ with normal r.v.s $V_m^, X_{k m}^$ such that all means and variances/covariances remain the same. The task is then to show that
$$\mathbb{E}\left[V_m^* \mid \boldsymbol{X}^\right]=\left(1-c_m\right) H_{\text {Coll }}+c_m \bar{V}{\bullet m}^,$$
where $\boldsymbol{X}^-\left(\boldsymbol{X}_1^ \ldots \boldsymbol{X}_M^\right)$. Ilowever, the $\boldsymbol{X}{m^{\prime}}$ with $m^{\prime}+m$ are independent of $\boldsymbol{X}m$, and therefore the $\boldsymbol{X}{m^{\prime}}^$ with $m^{\prime} \neq m$ are independent of $\boldsymbol{X}_m^*$.

# 风险理论代考

## 金融代写|风险理论代写Risk theory代考|The Simple Credibility Model

$$H_{\text {Bayes }}=\mathbb{E}\left[\mu(Z) \mid X_1, \ldots, X_n\right]$$

㧴们现在定义 $L_{\text {aff }} \subset L$ 作为所有仿射组合的集合 $X_1, \ldots, X_n$ ，那是，
$\backslash 1 \mathrm{eft}$ 的分隔符缺失或无法识别

$$H_{\text {Bayes }}=P_L \mu(Z), \quad H_{\text {Cred }}=P_{L_{\text {aff }}} \mu(Z) .$$

## 金融代写|风险理论代写Risk theory代考|The Bühlmann Model

Bühlmann 模型考虑了人口 $M$ 风险，以其风险参数为特征 $Z_1, \ldots, Z_M$. 对于风险 $m=1, \ldots, M$, 一个向量 $X_m=\left(X_{1 m} \ldots X_{n m m}\right)($ 通常是连续几年的索赔规模或 索赔数量），因此给定 $Z_m=\zeta ，$ 这 $X_{k m}$ 是独立同分布的 $F_\zeta \cdot$ 这 $Z_m$ 假设 iid
㧴们定义 $\mu(\zeta), \sigma^2(\zeta)$ 等如上。相似地， $H_{\text {Coll }}=\mathbb{E} \mu\left(Z_m\right)$. 问题是信誉溢价是多少 $H_{\text {Cred }}^{(m)}$ 看起来每个 $m$ ，即哪个仿射组合最小化
$$\mathbb{E}\left(\mu\left(Z_m\right)-a_{0 m}-\sum_{m=1}^M \sum_{k=1}^{n_m} a_{k m} X_{k m}\right)^2$$

$$c_m=n_m /\left(n_m+\sigma^2 / \tau^2\right), \quad \bar{X} \cdot m=\left(X 1 m+\cdots+X_{n_m m}\right) / n_m .$$

$$H_{\text {Cred }}^{(m)}=\left(1-c_m\right) H_{\text {Coll }}+c_m \bar{X} \bullet .$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师