### 统计代写|统计推断作业代写statistical inference代考|MONOTONE LIKELIHOOD RATIO PROPERTY

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
• (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|MONOTONE LIKELIHOOD RATIO PROPERTY

For scalar $\theta$ we suppose $f(\theta \mid D)=L(\theta \mid D)=L(\theta \mid t(D))$ and for all $\theta^{\prime}<\theta^{\prime \prime}$,
$$Q\left(t(D) \mid \theta^{\prime}, \theta^{\prime \prime}\right)=\frac{L\left(\theta^{\prime} \mid t(D)\right)}{L\left(\theta^{\prime \prime} \mid t(D)\right)}$$
is a non-decreasing function of $t(D)$ or as $t(D) \uparrow Q \uparrow$, that is, the support for $\theta^{\prime}$ vs. $\theta^{\prime}$ does not decrease as $t$ increases and does not increase as $t$ decreases. We call this a monotone likelihood ratio (MLR) property of $L(\theta \mid t(D))$.

Theorem 4.1 Let $f(D \mid \theta)$ have the MLR property. To test $H: \theta \leq \theta_{0}$ vs. $K: \theta>\theta_{0}$ the following test $T(D)$ is UMP at level $\alpha$.
$$T(D)=\left{\begin{array}{llll} 1 & \text { if } \quad t(D)k & \text { e.g., accept } H, \end{array}\right.$$
where $k$ and $p$ are determined by
$$\alpha=E_{\theta_{0}}(T(D))=P\left(t(D)\theta_{0} so there exists a k and a p such that the test satisfies (a) and (b), that is, Q=\frac{f\left(D \mid \theta_{0}\right)}{f\left(D \mid \theta_{1}\right)}\theta^{\prime} \text { at level } \alpha^{\prime}=1-\beta\left(\theta^{\prime}\right) \text {. }$$

## 统计代写|统计推断作业代写statistical inference代考|DECISION THEORY

Suppose we now consider the testing problem from a decision theoretic point of view, that is, testing $H: \theta \leq \theta_{0}$ vs. $K: \theta>\theta_{0}$ with $d_{0}$ the decision to accept $H$ and $d_{1}$ the decision to accept $K$. Assume a loss function $l\left(\theta, d_{i}\right)$. Now it appears to be sensible to assume
$$\begin{array}{lll} l\left(\theta, d_{0}\right)=0 & \text { if } \quad \theta \leq \theta_{0} \text { and } \uparrow \text { for } \theta>\theta_{0}, \ l\left(\theta, d_{1}\right)=0 & \text { if } \quad \theta>\theta_{0} \text {, and } \uparrow \text { for } \theta \leq \theta_{0} . \end{array}$$
Hence
(1) $l\left(\theta, d_{1}\right)-l(\theta, d 0) \quad \begin{cases}>0 & \text { for } \theta \leq \theta_{0} \ <0 & \text { for } \theta>\theta_{0} .\end{cases}$
Now recall that the risk function of a test $T$ for a given $\theta$ is the average loss
\begin{aligned} &l(\theta, T)=T(D) l\left(\theta, d_{1}\right)+(1-T(D)) l\left(\theta, d_{0}\right) \ &R(\theta, T)=E_{D} l(\theta, T)=E_{D}\left[T(D) l\left(\theta, d_{1}\right)+(1-T) l\left(\theta, d_{0}\right)\right] \end{aligned}
Further suppose for tests $T(D)$ and $T^{}(D)$ that (2) $R\left(\theta, T^{}\right) \leq R(\theta, T)$ for all $\theta$,
(3) $R\left(\theta, T^{}\right)}$ dominates $T$ and we say $T$ is inadmissible. On the other hand if no such $T^{}$ exists then $T$ is admissible. A class $C$ of test (decision) procedures is complete if for any test $T$ not in $C$ there exists a test $T^{}$ in $C$ dominating it. (A complete class is minimal if it does not contain a complete subclass). A class $C$ is essentially complete if at least (2) holds. (A complete class is also essentially complete, a class is minimal essentially complete if it does not contain an essentially complete subclass).

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

Since UMP tests don’t always exist, statisticians have proceeded to find optimal tests in more restricted classes. One such restriction is unbiasedness. Another is invariance. This chapter develops the theory of uniformly most powerful unbiased (UMPU) and invariant (UMPI) tests. When it is not possible to optimize in these ways, it is still possible to make progress, at least on the mathematical front. Locally most powerful (LMP) tests are those that have greatest power in a neighborhood of the null, and locally most powerful unbiased (LMPU) tests are most powerful in a neighborhood of the null, among unbiased tests. These concepts and main results related to them are presented here. The theory is illustrated with examples and moreover, examples are given that illustrate potential flaws. The concept of a “worse than useless” test is illustrated using a commonly accepted procedure. The sequential probability ratio test is also presented.

## 统计代写|统计推断作业代写statistical inference代考|MONOTONE LIKELIHOOD RATIO PROPERTY

$$T(D)=\左{1 如果 吨(D)ķ 例如，接受 H,\对。 在H和r和ķ一种ndp一种r和d和吨和r米一世n和db是 \alpha=E_{\theta_{0}}(T(D))=P\left(t(D)\theta_{0}s这吨H和r和和X一世s吨s一种ķ一种nd一种ps在CH吨H一种吨吨H和吨和s吨s一种吨一世sF一世和s(一种)一种nd(b),吨H一种吨一世s,Q=\frac{f\left(D \mid \theta_{0}\right)}{f\left(D \mid \theta_{1}\right)}\theta^{\prime} \text { 在级别} \alpha^{\prime}=1-\beta\left(\theta^{\prime}\right) \text {. }$$

## 统计代写|统计推断作业代写statistical inference代考|DECISION THEORY

l(θ,d0)=0 如果 θ≤θ0 和 ↑ 为了 θ>θ0, l(θ,d1)=0 如果 θ>θ0， 和 ↑ 为了 θ≤θ0.

(1)l(θ,d1)−l(θ,d0){>0 为了 θ≤θ0 <0 为了 θ>θ0.

l(θ,吨)=吨(D)l(θ,d1)+(1−吨(D))l(θ,d0) R(θ,吨)=和Dl(θ,吨)=和D[吨(D)l(θ,d1)+(1−吨)l(θ,d0)]

(3) R\left(\theta, T^{}\right)}R\left(\theta, T^{}\right)}占主导地位吨我们说吨是不可接受的。另一方面，如果没有这样吨那么存在吨是可以接受的。一类C的测试（决定）程序是完整的，如果对于任何测试吨不在C存在一个测试吨在C支配它。（如果一个完整的类不包含一个完整的子类，那么它就是最小的）。一类C如果至少 (2) 成立，则基本上是完整的。（一个完整的类也是本质上完整的，如果一个类不包含本质上完整的子类，那么它就是最小的本质上完整的类）。

## 广义线性模型代考

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## MATLAB代写

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