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

More generally suppose we have $k$ normal populations with assumed $N\left(\mu_{i}, \sigma^{2}\right) i=$ $1, \ldots, k$ and random samples of size $n_{i}$ from each of the populations. Recall for testing $H_{0}: \mu_{1}=\mu_{2}=\cdots=\mu_{k}$ vs. $H_{1}$ : not all the $\mu_{i}$ ‘s are equal and $\sigma^{2}$ unspecified, that the F-test used to test this was
\begin{aligned} F_{k-1, n-k} &=\frac{\sum n_{i}\left(\bar{X}{i}-\bar{X}\right)^{2} /(k-1)}{s^{2}}, \quad n{i} \bar{x}{i}=\sum{j=1}^{n_{i}} X_{i j}, \quad n \bar{x}=\sum_{1}^{k} n_{i} \bar{X}{i} \ n &=\sum{1}^{k} n_{i} \end{aligned}
and
$$(n-k) s^{2}=\sum_{i=1}^{k} \sum_{j=1}^{n_{j}}\left(X_{i j}-\bar{X}{i}\right)^{2}$$ If $Y{i j}=a X_{i j}+b$ for $a \neq 0$, then
$$\frac{\sum n_{i}\left(\bar{Y}{i}-\bar{Y}\right)^{2} /(k-1)}{s{y}^{2}}=F_{k-1, n-k}$$
as previously defined. The statistic is invariant under linear transformations. Note also that $H_{0}: \mu_{1}=\cdots=\mu_{k}$ is equivalent to $H_{0}^{\prime}: a \mu_{i}+b=\cdots=a \mu_{k}+b$ and $H_{1}$ : $\Longrightarrow$ to $H_{1}^{\prime}$. So it is an invariant test and it turns out to be UMP among invariant tests so we say it is UMPI. However this test is not UMPU.

In general, for the problem of
$$H: \theta \in \Theta_{H} \quad \text { vs. } \quad K: \theta \in \Theta_{K}$$
for any transformation $g$ on $D$ which leaves the problem invariant, it is natural to restrict our attention to all tests $T(D)$ such that $T(g D)=T(D)$ for all $D \in S$. A transformation $g$ is one which essentially changes the coordinates and a test is invariant if it is independent of the particular coordinate system in which the data are expressed.
We define invariance more precisely:
Definition: If for each $D \in S$ the function $t(D)=t(g D)$ for all $g \in G, G$ a group of transformations then $t$ is invariant with respect to $G$.
Recall that a set of elements $G$ is a group if under some operation it is
(a) closed: for all $g_{1}$ and $g_{2} \in G, g_{1} g_{2} \in G$;
(b) associative: $\left(g_{1} g_{2}\right) g_{3}=g_{1}\left(g_{2} g_{3}\right)$ for all $g_{1}, g_{2}, g_{3} \in G$;
(c) has an identity element: $g_{I} g=g g_{I}=g$ where $g_{I} \in G$;
(d) has an inverse: that is, if $g \in G$ then $g^{-1} \in G$ where $g g^{-1}=g^{-1} g=g_{I}$.

## 统计代写|统计推断作业代写statistical inference代考|LOCALLY BEST TESTS

When there are no UMP tests we may sometimes restrict the alternative parameter values to cases of presumably critical interest and look for high power against these alternatives. In particular if interest is focused on alternatives close to $H_{0}$ : say $\theta=\theta_{0}$, we could define $\delta(\theta)$ as a measure of the discrepancy of a close alternative from $H_{0}$.
Definition: A level $\alpha$ test $T$ is defined as locally most powerful (LMP) if for every other test $T^{}$ there exists a $\Delta$ such that $$1-\beta_{T}(\theta) \geq 1-\beta_{T^{}}(\theta) \text { for all } \theta \text { such that } 0<\delta(\theta)<\Delta .$$

Theorem $5.8$ If among all unbiased level $\alpha$ tests $T^{*}, T$ is LMP then we say $T$ is LMPU.

1. For real $\theta$ for any $T^{}$ such that $1-\beta_{T^{}}(\theta)$ is continuously differentiable at $\theta=\theta_{0}$ with $H_{1}: \theta>\theta_{0}$ or $H_{1}: \theta<\theta_{0}$ (i.e., one sided tests) a LMP test exists and is defined such that for all level $\alpha$ tests $T^{}$ there is a unique $T$ such that $$\arg \max {T^{}}\left[\frac{d\left(1-\beta{T^{*}}(\theta)\right)}{d \theta}\right]{\theta=\theta{0}}=T .$$
2. For $\theta$ real valued and $1-\beta_{T^{}}(\theta)$ twice continuously differentiable at $\theta=\theta_{0}$ for all $T^{}$, then a LMPU level $\alpha$ test $T$ exists of $H_{0}: \theta=\theta_{0}$ vs. $H_{1}: \theta \neq \theta_{0}$ and is given by
$$\arg \max {T^{}}\left[\frac{d^{2}\left(1-\beta{T^{2}}(\theta)\right)}{d \theta^{2}}\right]{\theta=\theta{1}}=T \Longrightarrow 1-\beta_{T}(\theta) \geq 1-\beta_{T^{}}(\theta)$$
for $0<\delta(\theta)<\Delta$. All locally unbiased tests result in $$\left.\frac{d\left[1-\beta_{T^{}}(\theta)\right]}{d \theta}\right|{\theta=\theta{1}}=0$$ given the condition of being continuously differentiable. Proof of (I): For a test $T^{}$
\begin{aligned} \gamma_{T^{}}(\theta) & \equiv 1-\beta_{T^{}}(\theta)=\int T^{} f(D \mid \theta) d \mu, \ \gamma_{T^{}}^{\prime}(\theta) &=\int T^{} \frac{\partial f_{\theta}}{\partial \theta} d \mu . \end{aligned} Suppose $\gamma_{T}^{\prime}\left(\theta_{0}\right) \geq \gamma_{T^{2}}\left(\theta_{0}\right)$. Then note that \begin{aligned} &\gamma_{T}\left(\theta_{0}\right)=1-\beta_{T}\left(\theta_{0}\right)=1-\beta_{T}\left(\theta_{0}\right)=\gamma_{T^{}}\left(\theta_{0}\right) \ &\gamma_{T}^{\prime}\left(\theta_{0}\right)=\lim {\Delta \theta \rightarrow 0} \frac{\gamma{T}\left(\theta_{0}+\Delta \theta\right)-\gamma_{T}\left(\theta_{0}\right)}{\Delta \theta} \geq \lim {\Delta \theta \rightarrow 0} \frac{\gamma{T^{}}\left(\theta_{0}+\Delta \theta\right)-\gamma_{T^{}}\left(\theta_{0}\right)}{\Delta \theta}=\gamma_{T^{\prime}}^{\prime}\left(\theta_{0}\right) \ &\gamma_{T}^{\prime}\left(\theta_{0}\right)-\gamma_{T^{}}^{\prime}\left(\theta_{0}\right)=\lim {\Delta \theta \rightarrow 0} \frac{\gamma{T}\left(\theta_{0}+\Delta \theta\right)-\gamma_{T^{}}\left(\theta_{0}+\Delta \theta\right)}{\Delta \theta} \geq 0 \end{aligned}

## 统计代写|统计推断作业代写statistical inference代考|TEST CONSTRUCTION

So far N-P theory has not really given a principle for constructing a test. It has indicated how we should compare tests that is, for a given size the test with the larger power is superior, or for sample space $S$, we want $T(D)$ to be such that for all tests $T^{}(D)$ $$E\left(T^{} \mid H\right) \leq \alpha$$
choose $T(D)$ such that
$$E\left(T^{*} \mid K\right) \leq E(T \mid K)$$
and as this doesn’t always happen we go on to other criteria, unbiasedness, invariance, and so forth.

There are several test construction methods that are not dependent on the N-P approach but are often evaluated by the properties inherent in that approach. The most popular one is the Likelihood Ratio Test (LRT) criterion.

Specifically the Likelihood Ratio Test (LRT) criterion statistic for a set of parameters $\theta$ to test $H_{\theta}$ vs. $K_{\theta}$ is defined as
$$\frac{\sup {\theta \in H{\theta}} L(\theta \mid D)}{\sup {\theta \in\left(H{\theta} \cup K_{\theta}\right)} L(\theta \mid D)}=\lambda(D) \leq 1$$
with critical region defined as $\lambda<k_{\alpha}$ reject $H_{\theta}$. This yields
$$P\left{\lambda(D)<k_{\alpha}\right} \leq \alpha$$

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

Fķ−1,n−ķ=∑n一世(X¯一世−X¯)2/(ķ−1)s2,n一世X¯一世=∑j=1n一世X一世j,nX¯=∑1ķn一世X¯一世 n=∑1ķn一世

(n−ķ)s2=∑一世=1ķ∑j=1nj(X一世j−X¯一世)2如果是一世j=一种X一世j+b为了一种≠0， 然后
∑n一世(是¯一世−是¯)2/(ķ−1)s是2=Fķ−1,n−ķ

H:θ∈θH 对比 ķ:θ∈θķ

(a) 封闭的：对于所有G1和G2∈G,G1G2∈G;
(b) 联想：(G1G2)G3=G1(G2G3)对全部G1,G2,G3∈G;
(c) 具有标识元素：G一世G=GG一世=G在哪里G一世∈G;
(d) 有一个逆：即，如果G∈G然后G−1∈G在哪里GG−1=G−1G=G一世.

## 统计代写|统计推断作业代写statistical inference代考|LOCALLY BEST TESTS

1. 真的θ对于任何吨这样1−b吨(θ)是连续可微的θ=θ0和H1:θ>θ0或者H1:θ<θ0（即，单面测试）存在 LMP 测试并定义为，对于所有级别一种测试吨有一个独特的吨这样参数⁡最大限度吨[d(1−b吨∗(θ))dθ]θ=θ0=吨.
2. 为了θ真正有价值和1−b吨(θ)两次连续可微θ=θ0对全部吨，然后是LMPU级别一种测试吨存在的H0:θ=θ0对比H1:θ≠θ0并且由
参数⁡最大限度吨[d2(1−b吨2(θ))dθ2]θ=θ1=吨⟹1−b吨(θ)≥1−b吨(θ)
为了0<d(θ)<Δ. 所有局部无偏测试结果d[1−b吨(θ)]dθ|θ=θ1=0给定连续可微的条件。(I) 的证明：用于测试吨
C吨(θ)≡1−b吨(θ)=∫吨F(D∣θ)dμ, C吨′(θ)=∫吨∂Fθ∂θdμ.认为C吨′(θ0)≥C吨2(θ0). 然后注意C吨(θ0)=1−b吨(θ0)=1−b吨(θ0)=C吨(θ0) C吨′(θ0)=林Δθ→0C吨(θ0+Δθ)−C吨(θ0)Δθ≥林Δθ→0C吨(θ0+Δθ)−C吨(θ0)Δθ=C吨′′(θ0) C吨′(θ0)−C吨′(θ0)=林Δθ→0C吨(θ0+Δθ)−C吨(θ0+Δθ)Δθ≥0

## 统计代写|统计推断作业代写statistical inference代考|TEST CONSTRUCTION

P\left{\lambda(D)<k_{\alpha}\right} \leq \alpha

## 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。