### 金融代写|风险理论代写Risk theory代考|MATH4128

statistics-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|Bayes and Empirical Bayes

Let $\boldsymbol{X}=\left(X_1, \ldots, X_n\right)$ be a vector of r.v.s describing the outcome of a statistical experiment. For example, in the insurance context, $n$ can be the number of persons insured for losses due to accidents in the previous year, and $X_i$ the payment made to the $i$ th.

A traditional (frequentists’) model is to assume the $X_i$ to be i.i.d. with a common distribution $F_\theta$ where $\theta$ is an unknown parameter (possibly multidimensional). F.g. in the accident insurance example, one could let $b$ denote the probability that a person has an accident within one year, $b=\mathbb{P}\left(X_i>0\right)$, and one could assume that the cost of the accident has a $\operatorname{gamma}(\alpha, \lambda)$ distribution. Thus the density of $X_i$ is
$$f_{b, \alpha, \lambda}(x)=b \mathbb{1}{x=0}+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} \mathbb{1}{x>0}$$
w.r.t. the measure defined as Lebesgue measure $\mathrm{d} x$ on $(0, \infty)$ with an added atom of unit size at $x=0$. Then $\theta=(b, \alpha, \lambda)$, and the conventional statistical procedure would be to compute estimates $\widehat{b}, \widehat{\alpha}, \widehat{\lambda}$ of $b, \alpha, \lambda$. These estimates could then be used as basis for computing first the expectation
$$\mathbb{E}{\widehat{\theta}} X=\mathbb{E}{\widehat{b}, \widehat{\alpha}, \widehat{\lambda}} X=(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$
of $X$ under the estimated parameters, and next one could use $\mathbb{E}_{\widehat{\theta}} X$ as the net premium and add a loading corresponding to one of the premium rules discussed in Sect. I.3. For example, the expected value principle would lead to the premium
$$p=(1+\eta)(1-\widehat{b}) \widehat{\alpha} / \widehat{\lambda}$$

We now turn to the general implementation of Bayesian ideas in insurance. Here one considers an insured with risk parameter $Z^1$ and an r.v. with distribution $\pi^{(0)}(\cdot)$, with observable past claims $X_1, \ldots, X_n$ and an unobservable claim amount $X_{n+1}$ for year $n+1$. The aim is to assert which (net) premium the insured is to pay in year $n+1$

For a fixed $\zeta$, let $\mu(\zeta)=\mathbb{E}\zeta X{n+1}$, where $\mathbb{E}\zeta[\cdot]=\mathbb{E}[\cdot \mid Z=\zeta]$. The (net) collective premium $H{\mathrm{Coll}}$ is $\mathbb{E} \mu(\boldsymbol{Z})=\mathbb{E} X_{n+1}$. This is the premium we would charge without prior statistics $X_1, \ldots, X_n$ on the insured. The individual premium is $H_{\text {Ind }}=\mathbb{E}\left[X_{n+1} \mid \boldsymbol{Z}\right]=\mu(\boldsymbol{Z})$. This is the ideal net premium in the sense of supplying the complete relevant prior information on the customer. The Bayes premium $H_{\text {Bayes }}$ is defined as $\mathbb{E}\left[\mu(\boldsymbol{Z}) \mid X_1, \ldots, X_n\right]$. That is, $H_{\text {Bayes }}$ is the expected value of $X_{n+1}$ in the posterior distribution.

Note that the individual premium is unobservable because $\boldsymbol{Z}$ is so; the Bayes premium is ‘our best guess of $H_{\text {Ind }}$ based upon the observations’. To make this precise, let $H^$ be another premium rule, that is, a function of $X_1, \ldots, X_n$ and the prior parameters. We then define its loss as $$\ell_{H^}=\mathbb{E}\left[\mu(\boldsymbol{Z})-H^\right]^2=\left|\mu(\boldsymbol{Z})-H^\right|^2$$
where $|X|=\left(\mathbb{E} X^2\right)^{1 / 2}$ is the $L_2$-norm (in obvious notation, we write $\ell_{\text {Coll }}=\ell_{H_{\text {Coll }}}$ etc). In mathematical terms, the optimality property of the Bayes premium is then that it minimizes the quadratic loss:
Theorem 1.3 For any $H^, \ell_{\text {Bayes }} \leq \ell_{H^}$. That is,
$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$

## 金融代写|风险理论代写Risk theory代考|Bayes and Empirical Bayes

$$f_{b, \alpha, \lambda}(x)=b 1 x=0+(1-b) \frac{\lambda^\alpha x^{\alpha-1}}{\Gamma(\alpha)} \mathrm{e}^{-\lambda x_1} 1 x>0$$
wrt 定义为 Lebesgue 度量的度量 $\mathrm{d} x$ 上 $(0, \infty)$ 添加一个单位大小的原子 $x=0$. 然后 $\theta=(b, \alpha, \lambda)$ ，而传统的统 计程序是计算估计值 $\hat{b}, \widehat{\alpha}, \widehat{\lambda}$ 的 $b, \alpha, \lambda$. 然后可以将这些估计值用作首先计算期望值的基础
$$\mathbb{E} \hat{\theta} X=\mathbb{E} \hat{b}, \widehat{\alpha}, \hat{\lambda} X=(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$

$$p=(1+\eta)(1-\hat{b}) \widehat{\alpha} / \widehat{\lambda}$$

$$\mathbb{E}\left(H_{\text {Bayes }}-H_{\text {Ind }}\right)^2 \leq \mathbb{E}\left(H^*-H_{\text {Ind }}\right)^2$$

## 广义线性模型代考

statistics-lab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。