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

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• 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

## 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

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