# 统计代写|统计推断代写Statistical inference代考|STAT3923

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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
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## 统计代写|统计推断代写Statistical inference代考|Motivation: Election polls

Let us consider the following “practical” question.
One of $L$ candidates for an office is about to be selected by a populationcandidate. How do we predict the winner via an opinion poll?
A (naive) model of the situation could be as follows. Let us represent the preference of a particular voter by his preference vector-a basic orth $e$ in $\mathbf{R}^{L}$ with unit entry in a position $\ell$ meaning that the voter is about to vote for the $\ell$-th candidate. The entries $\mu_{\ell}$ in the average $\mu$, over the population, of these vectors are the fractions of votes in favor of the $\ell$-th candidate, and the elected candidate is the one “indexing” the largest of the $\mu_{\ell}$ ‘s. Now assume that we select at random, from the uniform distribution, a member of the population and observe his preference vector. Our observation $\omega$ is a realization of a discrete random variable taking values in the set $\Omega=\left{e_{1}, \ldots, e_{L}\right}$ of basic orths in $\mathbf{R}^{L}$, and $\mu$ is the distribution of $\omega$ (technically, the density of this distribution w.r.t. the counting measure $\Pi$ on $\Omega$ ). Selecting a small threshold $\delta$ and assuming that the true unknown to us $-\mu$ is such that the largest entry in $\mu$ is at least by $\delta$ larger than every other entry and that $\mu_{\ell} \geq \frac{1}{N}$ for all $\ell, N$ being the population size, ${ }^{13}$ we can model the population preference for the $\ell$-th candidate with
\begin{aligned} \mu \in M_{\ell} &=\left{\mu \in \mathbf{R}^{d}: \mu_{i} \geq \frac{1}{N}, \sum_{i} \mu_{i}=1, \mu_{\ell} \geq \mu_{i}+\delta \forall(i \neq \ell)\right} \ & \subset \mathcal{M}=\left{\mu \in \mathbf{R}^{d}: \mu>0, \sum_{i} \mu_{i}=1\right} \end{aligned}

## 统计代写|统计推断代写Statistical inference代考|Sequential hypothesis testing

In view of the above analysis, when predicting outcomes of “close run” elections, huge poll sizes are necessary. It, however, does not mean that nothing can be done in order to build more reasonable opinion polls. The classical related statistical idea, going back to Wald [236], is to pass to sequential tests where the observations are processed one by one, and at every instant we either accept some of our hypotheses and terminate, or conclude that the observations obtained so far are insufficient to make a reliable inference and pass to the next observation. The idea is that a properly built sequential test, while still ensuring a desired risk, will be able to make “early decisions” in the case when the distribution underlying observations is “well inside” the true hypothesis and thus is far from the alternatives. Let us show $\mathcal{C}{s}$ closeness: hypotheses in the tuple $\left{G{2 \ell-1}^{s}: \mu \in M_{\ell}, G_{2 \ell}^{s}: \mu \in M_{\ell}^{s}, 1 \leq \ell \leq 3\right}$ are not $\mathcal{C}{s}$-close to each other if the corresponding $M$-sets belong to different areas and at least one of the sets is painted dark, like $M{1}^{s}$ and $M_{2}$, but not $M_{1}$ and $M_{2}$.
how our machinery can be utilized to conceive a sequential test for the problem of predicting the outcome of $L$-candidate elections. Thus, our goal is, given a small threshold $\delta$, to decide upon $L$ hypotheses (2.94). Let us act as follows.

1. We select a factor $\theta \in(0,1)$, say, $\theta=10^{-1 / 4}$, and consider thresholds $\delta_{1}=\theta$, $\delta_{2}=\theta \delta_{1}, \delta_{3}=\theta \delta_{2}$, and so on, until for the first time we get a threshold $\leq \delta$; to save notation, we assume that this threshold is exactly $\delta$, and let the number of the thresholds be $S$.
2. We split somehow (e.g., equally) the risk $\epsilon$ which we want to guarantee into $S$ portions $\epsilon_{s}, 1 \leq s \leq S$, so that $\epsilon_{s}$ are positive and
$$\sum_{s=1}^{S} \epsilon_{s}=\epsilon .$$

# 统计推断代考

## 统计代写|统计推断代写Statistical inference代考|Motivation: Election polls

\begin{对齐} \mu \in M_{\ell} &=\left{\mu \in \mathbf{R}^{d}: \mu_{i} \geq \frac{1}{N}, \ sum_{i} \mu_{i}=1, \mu_{\ell} \geq \mu_{i}+\delta \forall(i \neq \ell)\right} \ & \subset \mathcal{M}= \left{\mu \in \mathbf{R}^{d}: \mu>0, \sum_{i} \mu_{i}=1\right} \end{aligned}\begin{aligned} \mu \in M_{\ell} &=\left{\mu \in \mathbf{R}^{d}: \mu_{i} \geq \frac{1}{N}, \sum_{i} \mu_{i}=1, \mu_{\ell} \geq \mu_{i}+\delta \forall(i \neq \ell)\right} \ & \subset \mathcal{M}=\left{\mu \in \mathbf{R}^{d}: \mu>0, \sum_{i} \mu_{i}=1\right} \end{aligned}

## 统计代写|统计推断代写Statistical inference代考|Sequential hypothesis testing

1. 㧴们以杲种方式 (例如，平等地) 分割风险㧴们要保证 $S$ 部分 $\epsilon_{s}, 1 \leq s \leq S$ ， 以便 $\epsilon_{s}$ 是积极的并且

## 有限元方法代写

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## 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。

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