统计代写|统计推断作业代写statistical inference代考|Unbiased and Invariant Tests

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断statistical inference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。我们的专家在代写统计推断statistical inference代写方面经验极为丰富，各种代写统计推断statistical inference相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

统计代写|统计推断作业代写statistical inference代考|UNBIASED TESTS

Whenever a UMP test exists at level $\alpha$, we have shown that $\alpha \leq 1-\beta_{K}$, that is, the power is at least as large as the size. If this were not so then there would be a test $T^{} \equiv \alpha$ which did not use the data and had greater power. Further, such a test is sometimes termed as “worse than useless” since it has smaller power than the useless test $T^{} \equiv \alpha$.

Now UMP tests do not always exist except in fairly restricted situations. Therefore N-P theory proposes that in the absence of a UMP test, a test should at least be unbiased, that is, $1-\beta_{K} \geq \alpha$. Further, if among all unbiased tests at level $\alpha$, there exists one that is UMP then this is to be favored and termed UMPU. So for a class of problems for which UMP tests do not exist there may exist UMPU tests.

统计代写|统计推断作业代写statistical inference代考|ADMISSIBILITY AND TESTS SIMILAR ON THE BOUNDARY

A test $T$ at level $\alpha$ is called admissible if there exists no other test that has power no less than $T$ for all alternatives while actually exceeding it for some alternatives, that is, no other test has power that dominates the power of test $T$. Hence if $T$ is UMPU then it is admissible, that is, if a test $T^{*}$ dominates $T$ then it also must be unbiased and hence $T$ is not UMPU which contradicts the assumption of $T$ being UMPU. Hence $T$ is admissible.
Recall a test $T$ which satisfies
$$\begin{array}{ll} 1-\beta_{T}(\theta) \leq \alpha & \text { for } \theta \in \Theta_{H} \ 1-\beta_{T}(\theta) \geq \alpha & \text { for } \theta \in \Theta_{K} \end{array}$$
is an unbiased test. If $T$ is unbiased and $1-\beta_{T}(\theta)$ is a continuous function of $\theta$ there is a boundary of values $\theta \in w$ for which $1-\beta_{T}(\theta)=\alpha$. Such a level $\alpha$ test with boundary $w$ is said to be similar on the boundary.

Theorem $5.1$ If $f(D \mid \theta)$ is such that for every test $T$ at level $\alpha$ the power function $1-\beta_{T}(\theta)$ is continuous and a test $T^{\prime}$ is UMP among all level $\alpha$ tests similar on the boundary then $T^{\prime}$ is UMPU.

Proof: The class of tests similar on the boundary contains the class of unbiased tests, that is, if $1-\beta_{T}(\theta)$ is continuous then every unbiased test is similar on the boundary. Further if $T^{\prime}$ is UMP among similar tests it is at least as powerful as any unbiased test and hence as $T^{*} \equiv \alpha$ so that it is unbiased and hence UMPU.
Finding a UMP similar test is often easier than finding a UMPU test directly so that this may provide a way of finding UMPU tests.

统计代写|统计推断作业代写statistical inference代考|NEYMAN STRUCTURE AND COMPLETENESS

Now if there is a sufficient statistic $t(D)$ for $\theta$ then any test $T$ such that
$$E_{D \mid t}(T(D) \mid t(D) ; \quad \theta \in w)=\alpha$$
is called a test of Neyman structure (NS) with respect to $t(D)$. This is a level $\alpha$ test since
$$E_{\theta}(T(D) \mid \theta \in w)=E_{t} E_{D \mid t}(T(D) \mid t(D) ; \theta \in w)=E_{t}(\alpha)=\alpha .$$
Now it is often easy to obtain a MP test among tests having Neyman structure termed UMPNS. If every similar test has Neyman structure, this test would be MP among similar tests or UMPS. A sufficient condition for a similar test to have Neyman structure is bounded completeness of the family $f(t \mid \theta)$ of sufficient statistics.
A family $\mathcal{F}$ of probability functions is complete if for any $g(x)$ satisfying
$$E(g(x))=0 \quad \text { for all } f \in \mathcal{F}$$
then $g(x)=0$.
A slightly weaker condition is “boundedly complete” in which the above holds for all bounded functions $g$.

Theorem $5.3$ Suppose for $f(D \mid \theta), t(D)$ is a sufficient statistic. Then a necessary and sufficient condition for all similar tests to have NS with respect to $t(D)$ is that $f(t)$ be boundedly complete.
Proof of bounded completeness implies NS given similar tests:
Assume $f(t)$ is boundedly complete and let $T(D)$ be similar. Now $E(T(D)-\alpha)=0$ for $f(D \mid \theta)$ and if $E(T(D)-\alpha \mid t(D)=t, \theta \in w)=g(t)$ such that
$$\left.0=E_{D}(T(D)-\alpha)=E_{t} E_{D \mid t}(T(D)-\alpha) \mid t(D)=t, \theta \in w\right)=E_{t} g(t)=0$$
Since $T(D)-\alpha$ is bounded then so is $g(t)$ and from the bounded completeness of $f(t), E_{t}(g(t))=0$ implies $g(t)=0$ such that
$$E(T(D)-\alpha \mid t(D)=t, \quad \theta \in w)=0 \quad \text { or } \quad E(T(D) \mid t(D)=t, \quad \theta \in w)=\alpha \text { or NS. }$$

统计代写|统计推断作业代写statistical inference代考|ADMISSIBILITY AND TESTS SIMILAR ON THE BOUNDARY

1−b吨(θ)≤一种 为了 θ∈θH 1−b吨(θ)≥一种 为了 θ∈θķ

统计代写|统计推断作业代写statistical inference代考|NEYMAN STRUCTURE AND COMPLETENESS

0=和D(吨(D)−一种)=和吨和D∣吨(吨(D)−一种)∣吨(D)=吨,θ∈在)=和吨G(吨)=0

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。