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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|Inequalities and the Fischer–Riesz Theorem

We present one of the most important inequalities of probability theory, Jensen’s inequality for convex functions, and indicate how to derive from it Hölder’s inequality and Minkowski’s inequality. They in turn yield the triangle inequality for $|\cdot|_p$ and help in determining the dual space of $L^p(\mu)$. However, for the formal proofs of the latter inequalities, we will follow a different route.

Before stating Jensen’s inequality, we give a primer on the basics of convexity of sets and functions.

Definition 7.4 A subset $G$ of a vector space (or of an affine linear space) is called convex if, for any two points $x, y \in G$ and any $\lambda \in[0,1]$, we have $\lambda x+(1-$ i) $y \in G$.

(i) The convex subsets of $\mathbb{R}$ are the intervals.
(ii) A linear subspace of a vector space is convex.
(iii) The set of all probability measures on a measurable space is a convex set. $\diamond$
Definition 7.6 Let $G$ be a convex set. A map $\varphi: G \rightarrow \mathbb{R}$ is called convex if for any two points $x, y \in G$ and any $\lambda \in[0,1]$, we have
$$\varphi(\lambda x+(1-\lambda) y) \leq \lambda \varphi(x)+(1-\lambda) \varphi(y) .$$
$\varphi$ is called concave if $(-\varphi)$ is convex.
Let $I \subset \mathbb{R}$ be an interval. Let $\varphi: I \rightarrow \mathbb{R}$ be continuous and in the interior $I^{\circ}$ twice continuously differentiable with second derivative $\varphi^{\prime \prime}$. Then $\varphi$ is convex if and only if $\varphi^{\prime \prime}(x) \geq 0$ for all $x \in I^{\circ}$. To put it differently, the first derivative $\varphi^{\prime}$ of a convex function is a monotone increasing function. In the next theorem, we will see that this is still true even if $\varphi$ is not twice continuously differentiable when we pass to the right-sided derivative $D^{+} \varphi$ (or to the left-sided derivative), which we show always exists.

统计代写|概率论代写Probability theory代考|Lebesgue’s Decomposition Theorem

In this section, we employ the properties of Hilhert spaces that we derived in the last section in order to decompose a measure into a singular part and a part that is absolutely continuous, both with respect to a second given measure. Furthermore, we show that the absolutely continuous part has a density. Let $\mu$ and $v$ be measures on $(\Omega, \mathcal{A})$. By Definition 4.13, a measurable function $f: \Omega \rightarrow[0, \infty)$ is called a density of $v$ with respect to $\mu$ if
$$v(A):=\int f \mathbb{1}A d \mu \quad \text { for all } A \in \mathcal{A} .$$ On the other hand, for any measurable $f: \Omega \rightarrow[0, \infty)$, equation (7.3) defines a measure $v$ on $(\Omega, \mathcal{A})$. In this case, we also write $$v=f \mu \quad \text { and } \quad f=\frac{d v}{d \mu} .$$ For example, the normal distribution $v=\mathcal{N}{0,1}$ has the density $f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}$ with respect to the Lebesgue measure $\mu=\lambda$ on $\mathbb{R}$.
If $g: \Omega \rightarrow[0, \infty]$ is measurable, then (by Theorem 4.15)
$$\int g d v=\int g f d \mu$$
Hence $g \in \mathcal{L}^1(\nu)$ if and only if $g f \in \mathcal{L}^1(\mu)$, and in this case (7.5) holds.
If $v=f \mu$, then $v(A)=0$ for all $A \in \mathcal{A}$ with $\mu(A)=0$. The situation is quite the opposite for, e.g., the Poisson distribution $\mu=\operatorname{Poi}{\varrho}$ with parameter $\varrho>0$ and $v=\mathcal{N}{0,1}$. Here $\mathbb{N}_0 \subset \mathbb{R}$ is a $v$-null set with $\mu\left(\mathbb{R} \backslash \mathbb{N}_0\right)=0$. We say that $v$ is singular to $\mu$.

The main goal of this chapter is to show that an arbitrary $\sigma$-finite measure $v$ on a measurable space $(\Omega, \mathcal{A})$ can be decomposed into a part that is singular to the $\sigma$-finite measure $\mu$ and a part that has a density with respect to $\mu$ (Lebesgue’s decomposition theorem, Theorem 7.33).

概率论代考

统计代写|概率论代写概率论代考|不等式和fisher -riesz定理

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7.4向量空间(或仿射线性空间)的子集$G$称为凸，如果对于任意两点$x, y \in G$和任意两点$\lambda \in[0,1]$，我们有$\lambda x+(1-$ i) $y \in G$

(i) $\mathbb{R}$的凸子集是区间。
(ii)向量空间的线性子空间是凸的。
(iii)在可测空间上的所有概率度量的集合是凸集。$\diamond$

$$\varphi(\lambda x+(1-\lambda) y) \leq \lambda \varphi(x)+(1-\lambda) \varphi(y) .$$
$\varphi$称为凹映射。

统计代写|概率论代写概率论代考|勒贝格分解定理

$$v(A):=\int f \mathbb{1}A d \mu \quad \text { for all } A \in \mathcal{A} .$$ 另一方面，对于任何可测量的 $f: \Omega \rightarrow[0, \infty)$，式(7.3)定义了一个测度 $v$ 在 $(\Omega, \mathcal{A})$。在这种情况下，我们也写 $$v=f \mu \quad \text { and } \quad f=\frac{d v}{d \mu} .$$ 例如，正态分布 $v=\mathcal{N}{0,1}$ 有密度 $f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2}$ 关于勒贝格测度 $\mu=\lambda$ 在 $\mathbb{R}$.

$$\int g d v=\int g f d \mu$$

有限元方法代写

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MATLAB代写

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