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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|概率论代写Probability theory代考|Almost Sure and Measure Convergence

In the following, $(\Omega, \mathcal{A}, \mu)$ will be a $\sigma$-finite measure space. We first define in metric spaces almost sure convergence and convergence in measure and then compare both concepts. To this end, we need two lemmas that ensure that the distance function associated with two measurable maps is again measurable. In the following, let $(E, d)$ be a separable metric space with Borel $\sigma$-algebra $\mathcal{B}(E)$. “Separable” means that there exists a countable dense subset. For $x \in E$ and $r>0$, denote by $B_r(x)={y \in E: d(x, y)<r}$ the ball with radius $r$ centered at $x$.
Lemma 6.1 Let $f, g: \Omega \rightarrow E$ be measurable with respect to $\mathcal{A}-\mathcal{B}(E)$. Then the map $H: \Omega \rightarrow[0, \infty), \omega \mapsto d(f(\omega), g(\omega))$ is $\mathcal{A}-\mathcal{B}([0, \infty))$-measurable.

Proof Let $F \subset E$ be countable and dense. By the triangle inequality, $d(x, z)+$ $d(z, y) \geq d(x, y)$ for all $x, y \in E$ and $z \in F$. Let $\left(z_n\right){n \in \mathbb{N}}$ be a sequence in $F$ with $z_n \stackrel{n \rightarrow \infty}{\longrightarrow} x$. Since $d$ is continuous, we have $d\left(x, z_n\right)+d\left(z_n, y\right) \stackrel{n \rightarrow \infty}{\longrightarrow} d(x, y)$. Putting things together, we infer $\inf {z \in F}(d(x, z)+d(z, y))=d(x, y)$. Since $x \mapsto$ $d(x, z)$ is continuous and hence measurable, the maps $f_z, g_z: \Omega \rightarrow[0, \infty)$ with $f_z(\omega)=d(f(\omega), z)$ and $g_z(\omega)=d(g(\omega), z)$ are also measurable. Thus $f_z+g_z$ and $H=\inf _{z \in F}\left(f_z+g_z\right)$ are measurable.
(A somewhat more systematic proof is based on the fact that $(f, g)$ is $\mathcal{A}-\mathcal{B}(E \times$ $E$ )-measurable (this will follow from Theorem 14.8) and that $d: E \times E \rightarrow[0, \infty$ ) is continuous and hence $\mathcal{B}(E \times E)-\mathcal{B}([0, \infty))$-measurable. As a composition of measurable maps, $\omega \mapsto d(f(\omega), g(\omega))$ is measurable.)
Let $f, f_1, f_2, \ldots: \Omega \rightarrow E$ be measurable with respect to $\mathcal{A}-\mathcal{B}(E)$.

统计代写|概率论代写Probability theory代考|Uniform Integrability

From the preceding section, we can conclude that convergence in measure plus existence of $L^1$ limit points implies $L^1$-convergence. Hence convergence in measure plus relative sequential compactness in $L^1$ yields convergence in $L^1$. In this section, we study a criterion for relative sequential compactness in $L^1$, the so-called uniform integrability.
Definition 6.16 A family $\mathcal{F} \subset \mathcal{L}^1(\mu)$ is called uniformly integrable if
$$\inf {0 \leq g \in \mathcal{L}^1(\mu)} \sup {f \in \mathcal{F}} \int(|f|-g)^{+} d \mu=0 .$$
Theorem 6.17 The family $\mathcal{F} \subset \mathcal{L}^1(\mu)$ is uniformly integrable if and only if
$$\inf {0 \leq \tilde{g} \in \mathcal{L}^1(\mu)} \sup {f \in \mathcal{F}} \int_{|f|>\tilde{g} \mid}|f| d \mu=0 .$$
If $\mu(\Omega)<\infty$, then uniform integrability is equivalent to either of the following two conditions: (i) $\inf {a \in[0, \infty)} \sup {f \in \mathcal{F}} \int(|f|-a)^{+} d \mu=0$, (ii) $\inf {a \in[0, \infty)} \sup {f \in \mathcal{F}} \int_{{|f|>a}}|f| d \mu=0$.
Proof Clearly, $(|f|-g)^{+} \leq|f| \cdot \mathbb{1}{{|f|>g}}$; hence (6.3) implies uniform integrability. Now assume (6.2). For $\varepsilon>0$, choose $g{\varepsilon} \in \mathcal{L}^1(\mu)$ such that
$$\sup {f \in \mathcal{F}} \int\left(|f|-g{\varepsilon}\right)^{+} d \mu \leq \varepsilon .$$
Define $\widetilde{g_{\varepsilon}}=2 g_{\varepsilon / 2}$. Then, for $f \in \mathcal{F}$,
$$\int_{\left{|f|>\widetilde{\left.g_{\varepsilon}\right}}\right.}|f| d \mu \leq \int_{\left{|f|>\widetilde{\left.g_{\varepsilon}\right}}\right.}\left(|f|-g_{\varepsilon / 2}\right)^{+} d \mu+\int_{\left{|f|>\widetilde{g_{\varepsilon}}\right}} g_{\varepsilon / 2} d \mu$$

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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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