分类：统计推断代写

统计代写|统计推断代写Statistical inference代考|STAT6110

Posted on 2023年8月28日2023年8月28日 by statistics-lab

如果你也在怎样代写统计推断Statistical Inference 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。统计推断Statistical Inference领域，有两种主要的思想流派。每一种方法都有其支持者，但人们普遍认为，在入门课程中涵盖的所有问题上，这两种方法都是有效的，并且在应用于实际问题时得到相同的数值。传统课程只涉及其中一种方法，这使得学生无法接触到统计推断的整个领域。传统的方法，也被称为频率论或正统观点，几乎直接导致了上面的问题。另一种方法，也称为概率论作为逻辑${}^1$，直接从概率论导出所有统计推断。

统计推断Statistical Inference指的是一个研究领域，我们在面对不确定性的情况下，根据我们观察到的数据，试图推断世界的未知特性。它是一个数学框架，在许多情况下量化我们的常识所说的话，但在常识不够的情况下，它允许我们超越常识。对正确的统计推断的无知会导致错误的决策和浪费金钱。就像对其他领域的无知一样，对统计推断的无知也会让别人操纵你，让你相信一些错误的事情是正确的。

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。

统计代写|统计推断代写Statistical inference代考|STAT6110

统计代写|统计推断代写Statistical inference代考|Expectation, Variance, and Moment Generating Function of a Random Variable

The ideal situation in life would be to know with certainty what is going to happen next. This being almost never the case, the element of chance enters in all aspects of our life. A r.v. is a mathematical formulation of a random environment. Given that we have to deal with a r.v. $X$, the best thing to expect is to know the values of $X$ and the probabilities with which these values are taken on, for the case that $X$ is discrete, or the probabilities with which $X$ takes values in various subsets of the real line $\Re$ when $X$ is of the continuous type. That is, we would like to know the probability distribution of $X$. In real life, often, even this is not feasible. Instead, we are forced to settle for some numerical characteristics of the distribution of $X$. This line of arguments leads us to the concepts of the mathematical expectation and variance of a r.v., as well as to moments of higher order.
DEFINITION 1
Let $X$ be a (discrete) r.v. taking on the values $x_i$ with corresponding probabilities $f\left(x_i\right), i=1, \ldots, n$. Then the mathematical expectation of $X$ (or just expectation or mean value of $X$ or just mean of $X$ ) is denoted by $E X$ and is defined by:
$$
E X=\sum_{i=1}^n x_i f\left(x_i\right)
$$
If the r.v. $X$ takes on (countably) infinite many values $x_i$ with corresponding probabilities $f\left(x_i\right), i=1,2, \ldots$, then the expectation of $X$ is defined by:
$$
E X=\sum_{i=1}^{\infty} x_i f\left(x_i\right), \quad \text { provided } \sum_{i=1}^{\infty}\left|x_i\right| f\left(x_i\right)<\infty .
$$
Finally, if the r.v. $X$ is continuous with p.d.f. $f$, its expectation is defined by:
$$
E X=\int_{-\infty}^{\infty} x f(x) d x, \quad \text { provided this integral exists. }
$$
The alternative notations $\mu(X)$ or $\mu_X$ are also often used.

统计代写|统计推断代写Statistical inference代考|Some Probability Inequalities

If the r.v. $X$ has a known p.d.f. $f$, then, in principle, we can calculate probabilities $P(X \in B)$ for $B \subseteq \Re$. This, however, is easier said than done in practice. What one would be willing to settle for would be some suitable and computable bounds for such probabilities. This line of thought leads us to the inequalities discussed here.
(i) For any nonnegative r.v. $X$ and for any constant $c>0$, it holds:
$$
P(X \geq c) \leq E X / c .
$$
(ii) More generally, for any nonnegative function of any r.v. $X, g(X)$, and for any constant $c>0$, it holds:
$$
P[g(X) \geq c] \leq E g(X) / c .
$$
(iii) By taking $g(X)=|X-E X|$ in part (ii), the inequality reduces to the Markov inequality, namely,
$$
P(|X-E X| \geq c)=P\left(|X-E X|^r \geq c^r\right) \leq E|X-E X|^r / c^r, \quad r>0 .
$$
(iv) In particular, for $r=2$ in (15), we get the Tchebichev inequality, namely,
$$
\begin{gathered}
P(|X-E X| \geq c) \leq \frac{E(X-E X)^2}{c^2}=\frac{\sigma^2}{c^2} \quad \text { or } \
P(|X-E X|<c) \geq 1-\frac{\sigma^2}{c^2},
\end{gathered}
$$
where $\sigma^2$ stands for the $\operatorname{Var}(X)$. Furthermore, if $c=k \sigma$, where $\sigma$ is the s.d. of $X$, then:
$$
P(|X-E X| \geq k \sigma) \leq \frac{1}{k^2} \quad \text { or } \quad P(|X-E X|<k \sigma) \geq 1-\frac{1}{k^2} .
$$
REMARK 2 From the last expression, it follows that $X$ lies within $k$ s.d.’s from its mean with probability at least $1-\frac{1}{k^2}$, regardless of the distribution of $X$. It is in this sense that the s.d. is used as a yardstick of deviations of $X$ from its mean, as already mentioned elsewhere.

Thus, for example, for $k=2,3$, we obtain, respectively:
$$
P(|X-E X|<2 \sigma) \geq 0.75, \quad P(|X-E X|<3 \sigma) \geq \frac{8}{9} \simeq 0.889
$$

统计推断代考

统计代写|统计推断代写Statistical inference代考|Expectation, Variance, and Moment Generating Function of a Random Variable

生活中最理想的状态是确切地知道接下来会发生什么。这种情况几乎从未发生过，偶然的因素进入了我们生活的方方面面。rv是一个随机环境的数学公式。考虑到我们必须处理一个rv $X$，最好的期望是知道$X$的值和这些值被取值的概率，对于$X$是离散的情况，或者当$X$是连续类型时，$X$在实线$\Re$的各个子集中取值的概率。也就是说，我们想知道$X$的概率分布。在现实生活中，往往连这都是不可行的。相反，我们被迫满足于$X$分布的一些数值特征。这一系列的论证将我们引向rv的数学期望和方差的概念，以及高阶矩的概念。
定义1
设$X$是一个(离散的)rv，取相应概率$f\left(x_i\right), i=1, \ldots, n$的值$x_i$。则$X$的数学期望(或$X$的期望或平均值或$X$的平均值)用$E X$表示，定义为:
$$
E X=\sum_{i=1}^n x_i f\left(x_i\right)
$$
如果rv $X$具有(可数)无限多个值$x_i$并具有相应的概率$f\left(x_i\right), i=1,2, \ldots$，则$X$的期望定义为:
$$
E X=\sum_{i=1}^{\infty} x_i f\left(x_i\right), \quad \text { provided } \sum_{i=1}^{\infty}\left|x_i\right| f\left(x_i\right)<\infty .
$$
最后，如果r.v. $X$与p.d.f. $f$连续，则其期望定义为:
$$
E X=\int_{-\infty}^{\infty} x f(x) d x, \quad \text { provided this integral exists. }
$$
也经常使用替代符号$\mu(X)$或$\mu_X$。

统计代写|统计推断代写Statistical inference代考|Some Probability Inequalities

如果rv $X$有一个已知的p.d.f. $f$，那么，原则上，我们可以计算$B \subseteq \Re$的概率$P(X \in B)$。然而，说起来容易做起来难。人们愿意接受的是为这些概率提供一些合适的、可计算的界限。这条思路将我们引向这里讨论的不平等。
(i)对于任意非负rv $X$和任意常数$c>0$，它成立:
$$
P(X \geq c) \leq E X / c .
$$
(ii)更一般地说，对于任意rv的任意非负函数$X, g(X)$，对于任意常数$c>0$，成立:
$$
P[g(X) \geq c] \leq E g(X) / c .
$$
(iii)通过(ii)部分中的$g(X)=|X-E X|$，不等式化简为Markov不等式，即:
$$
P(|X-E X| \geq c)=P\left(|X-E X|^r \geq c^r\right) \leq E|X-E X|^r / c^r, \quad r>0 .
$$
(iv)特别地，对于(15)中的$r=2$，我们得到切比切夫不等式，即:
$$
\begin{gathered}
P(|X-E X| \geq c) \leq \frac{E(X-E X)^2}{c^2}=\frac{\sigma^2}{c^2} \quad \text { or } \
P(|X-E X|<c) \geq 1-\frac{\sigma^2}{c^2},
\end{gathered}
$$
其中$\sigma^2$代表$\operatorname{Var}(X)$。此外，如果$c=k \sigma$，其中$\sigma$是$X$的sd，则:
$$
P(|X-E X| \geq k \sigma) \leq \frac{1}{k^2} \quad \text { or } \quad P(|X-E X|<k \sigma) \geq 1-\frac{1}{k^2} .
$$
REMARK 2由上一个表达式可知，$X$位于$k$ s.d内。的均值，无论$X$的分布如何，概率至少为$1-\frac{1}{k^2}$。正是在这个意义上，标准差被用作衡量$X$与其均值偏差的尺度，这一点在其他地方已经提到过。

因此，例如，对于$k=2,3$，我们分别得到:
$$
P(|X-E X|<2 \sigma) \geq 0.75, \quad P(|X-E X|<3 \sigma) \geq \frac{8}{9} \simeq 0.889
$$

统计代写|统计推断代写Statistical inference代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|STAT7604

Posted on 2023年8月28日2023年8月28日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Independent Events and Related Results

In Example 14, it was seen that $P(A \mid B)=P(A)$. Thus, the fact that the event $B$ occurred provides no information in reevaluating the probability of $A$. Under such a circumstance, it is only fitting to say that $A$ is independent of $B$. For any two events $A$ and $B$ with $P(B)>0$, we say that $A$ is independent of $B$, if $P(A \mid B)=P(A)$. If, in addition, $P(A)>0$, then $B$ is also independent of $A$ because
$$
P(B \mid A)=\frac{P(B \cap A)}{P(A)}=\frac{P(A \cap B)}{P(A)}=\frac{P(A \mid B) P(B)}{P(A)}=\frac{P(A) P(B)}{P(A)}=P(B) .
$$
Because of this symmetry, we then say that $A$ and $B$ are independent. From the definition of either $P(A \mid B)$ or $P(B \mid A)$, it follows then that $P(A \cap B)=$ $P(A) P(B)$. We further observe that this relation is true even if one or both of $P(A), P(B)$ are equal to 0 . We take this relation as the defining relation of independence.
DEFINITION 2
Two events $A_1$ and $A_2$ are said to be independent (statistically or stochastically or in the probability sense), if $P\left(A_1 \cap A_2\right)=P\left(A_1\right) P\left(A_2\right)$. When $P\left(A_1 \cap A_2\right) \neq P\left(A_1\right) P\left(A_2\right)$ they are said to be dependent.

REMARK 2 At this point, it should be emphasized that disjointness and independence of two events are two distinct concepts; the former does not even require the concept of probability. Nevertheless, they are related in that, if $A_1 \cap A_2=\varnothing$, then they are independent if and only if at least one of $P\left(A_1\right), P\left(A_2\right)$ is equal to 0 . Thus (subject to $\left.A_1 \cap A_2=\varnothing\right), P\left(A_1\right) P\left(A_2\right)>0$ implies that $A_1$ and $A_2$ are definitely dependent.

The definition of independence extends to three events $A_1, A_2, A_3$, as well as to any number $n$ of events $A_1, \ldots, A_n$. Thus, three events $A_1, A_2, A_3$ for which $P\left(A_1 \cap A_2 \cap A_3\right)>0$ are said to be independent, if all conditional probabilities coincide with the respective (unconditional) probabilities:
$$
\left.\begin{array}{c}
P\left(A_1 \mid A_2\right)=P\left(A_1 \mid A_3\right)=P\left(A_1 \mid A_2 \cap A_3\right)=P\left(A_1\right) \
P\left(A_2 \mid A_1\right)=P\left(A_2 \mid A_3\right)=P\left(A_2 \mid A_1 \cap A_3\right)=P\left(A_2\right) \
P\left(A_3 \mid A_1\right)=P\left(A_3 \mid A_2\right)=P\left(A_3 \mid A_1 \cap A_2\right)=P\left(A_3\right) \
P\left(A_1 \cap A_2 \mid A_3\right)=P\left(A_1 \cap A_2\right), P\left(A_1 \cap A_3 \mid A_2\right) \
\quad=P\left(A_1 \cap A_3\right), P\left(A_2 \cap A_3 \mid A_1\right)=P\left(A_2 \cap A_3\right) .
\end{array}\right}
$$
From the definition of conditional probability, relations (1) are equivalent to:
$$
\left.\begin{array}{c}
P\left(A_1 \cap A_2\right)=P\left(A_1\right) P\left(A_2\right), P\left(A_1 \cap A_3\right)=P\left(A_1\right) P\left(A_3\right), \
P\left(A_2 \cap A_3\right)=P\left(A_2\right) P\left(A_3\right), P\left(A_1 \cap A_2 \cap A_3\right)=P\left(A_1\right) P\left(A_2\right) P\left(A_3\right) .
\end{array}\right}
$$

统计代写|统计推断代写Statistical inference代考|Basic Concepts and Results in Counting

In this brief section, some basic concepts and results are discussed regarding the way of counting the total number of outcomes of an experiment or the total number of different ways we can carry out a task. Although many readers will, undoubtedly, be familiar with parts of or the entire material in this section, it would be advisable, nevertheless, to invest some time here in introducing and adopting some notation, establishing some basic results, and then using them in computing probabilities in the classical probability framework.

Problems of counting arise in a great number of different situations. Here are some of them. In each one of these situations, we are asked to compute the number of different ways that something or other can be done. Here are a few illustrative cases.

(i) Attire yourself by selecting a T-shirt, a pair of trousers, a pair of shoes, and a cap out of $n_1$ T-shirts, $n_2$ pairs of trousers, $n_3$ pairs of shoes, and $n_4$ caps (e.g., $n_1=4, n_2=3, n_3=n_4=2$ ).
(ii) Form all $k$-digit numbers by selecting the $k$ digits out of $n$ available numbers (e.g., $k=2, n=4$ such as ${1,3,5,7}$ ).
(iii) Form all California automobile license plates by using one number, three letters and then three numbers in the prescribed order.
(iv) Form all possible codes by using a given set of symbols (e.g., form all “words” of length 10 by using the digits 0 and 1 ).
(v) Place $k$ books on the shelf of a bookcase in all possible ways.
(vi) Place the birthdays of $k$ individuals in the 365 days of a year in all possible ways.
(vii) Place $k$ letters into $k$ addressed envelopes (one letter to each envelope).
(viii) Count all possible outcomes when tossing $k$ distinct dice.
(ix) Select $k$ cards out of a standard deck of playing cards (e.g., for $k=5$, each selection is a poker hand).
(x) Form all possible $k$-member committees out of $n$ available individuals.
The calculation of the numbers asked for in situations (i) through (x) just outlined is in actuality a simple application of the so-called fundamental principle of counting, stated next in the form of a theorem.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Independent Events and Related Results

在例14中，可以看到$P(A \mid B)=P(A)$。因此，事件$B$发生的事实在重新评估$A$的概率时没有提供任何信息。在这种情况下，我们只能说$A$独立于$B$。对于含有$P(B)>0$的任意两个事件$A$和$B$，我们说$A$独立于$B$，如果$P(A \mid B)=P(A)$。另外，如果是$P(A)>0$，那么$B$也独立于$A$，因为
$$
P(B \mid A)=\frac{P(B \cap A)}{P(A)}=\frac{P(A \cap B)}{P(A)}=\frac{P(A \mid B) P(B)}{P(A)}=\frac{P(A) P(B)}{P(A)}=P(B) .
$$
由于这种对称性，我们说$A$和$B$是独立的。根据$P(A \mid B)$或$P(B \mid A)$的定义，可以得出$P(A \cap B)=$$P(A) P(B)$。我们进一步观察到，即使$P(A), P(B)$的一个或两个等于0，这个关系也是成立的。我们把这个关系作为独立性的定义关系。
定义2
两个事件$A_1$和$A_2$被认为是独立的(统计上或随机地或在概率意义上)，如果$P\left(A_1 \cap A_2\right)=P\left(A_1\right) P\left(A_2\right)$。当$P\left(A_1 \cap A_2\right) \neq P\left(A_1\right) P\left(A_2\right)$他们被认为是依赖的。

在这里，需要强调的是，两个事件的不相交性和独立性是两个不同的概念;前者甚至不需要概率的概念。然而，它们是相关的，如果$A_1 \cap A_2=\varnothing$，那么它们是独立的当且仅当$P\left(A_1\right), P\left(A_2\right)$中至少有一个等于0。因此(subject to $\left.A_1 \cap A_2=\varnothing\right), P\left(A_1\right) P\left(A_2\right)>0$)意味着$A_1$和$A_2$是绝对依赖的。

独立性的定义扩展到三个事件$A_1, A_2, A_3$，以及任意数量的$n$事件$A_1, \ldots, A_n$。因此，如果所有条件概率都与各自的(无条件)概率相一致，那么三个事件$A_1, A_2, A_3$ ($P\left(A_1 \cap A_2 \cap A_3\right)>0$)被认为是独立的:
$$
\left.\begin{array}{c}
P\left(A_1 \mid A_2\right)=P\left(A_1 \mid A_3\right)=P\left(A_1 \mid A_2 \cap A_3\right)=P\left(A_1\right) \
P\left(A_2 \mid A_1\right)=P\left(A_2 \mid A_3\right)=P\left(A_2 \mid A_1 \cap A_3\right)=P\left(A_2\right) \
P\left(A_3 \mid A_1\right)=P\left(A_3 \mid A_2\right)=P\left(A_3 \mid A_1 \cap A_2\right)=P\left(A_3\right) \
P\left(A_1 \cap A_2 \mid A_3\right)=P\left(A_1 \cap A_2\right), P\left(A_1 \cap A_3 \mid A_2\right) \
\quad=P\left(A_1 \cap A_3\right), P\left(A_2 \cap A_3 \mid A_1\right)=P\left(A_2 \cap A_3\right) .
\end{array}\right}
$$
从条件概率的定义来看，关系式(1)等价于:
$$
\left.\begin{array}{c}
P\left(A_1 \cap A_2\right)=P\left(A_1\right) P\left(A_2\right), P\left(A_1 \cap A_3\right)=P\left(A_1\right) P\left(A_3\right), \
P\left(A_2 \cap A_3\right)=P\left(A_2\right) P\left(A_3\right), P\left(A_1 \cap A_2 \cap A_3\right)=P\left(A_1\right) P\left(A_2\right) P\left(A_3\right) .
\end{array}\right}
$$

统计代写|统计推断代写Statistical inference代考|Basic Concepts and Results in Counting

在这简短的一节中，讨论了一些基本概念和结果，关于计算实验结果总数的方法或我们可以执行任务的不同方法的总数。尽管许多读者无疑会熟悉本节的部分或全部内容，但还是建议在这里花一些时间介绍和采用一些符号，建立一些基本结果，然后在经典概率框架中使用它们来计算概率。

在许多不同的情况下都会出现计数问题。下面是其中的一些。在每一种情况下，我们都被要求计算完成某件事或另一件事的不同方法的数量。这里有几个说明性的例子。

(i)从$n_1$ t恤、$n_2$裤子、$n_3$鞋子和$n_4$帽子(例如$n_1=4, n_2=3, n_3=n_4=2$)中选择一件t恤、一条裤子、一双鞋子和一顶帽子。
(ii)从$n$中选出$k$位(例如，$k=2, n=4，如${1,3,5,7}$)，形成所有$k$位的数字。
(iii)用一个数字，三个字母，然后按规定的顺序排列三个数字，形成所有加州汽车牌照。
(iv)用一组给定的符号组成所有可能的编码(例如，用数字0和1组成长度为10的所有“字”)。
(v)以各种可能的方式将$k$的书放在书架上。
(vi)以所有可能的方式将$k$个人的生日放在一年的365天中。
(vii)将$k$字母放入$k$地址信封(每个信封各放一封信)。
(viii)在掷$k$不同的骰子时计算所有可能的结果。
(ix)从一副标准扑克牌中选择$k$牌(例如，对于$k=5$，每张牌都是一手牌)。
(x)从可用的$n$个人中组成所有可能的$k$成员委员会。
在(i)到(x)的情况中所要求的数字的计算实际上是所谓的计数基本原理的简单应用，下面以定理的形式说明。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

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

Posted on 2023年8月28日2023年8月28日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Distribution of a Random Variable

For a r.v. $X$, define the set function $P_X(B)=P(X \in B)$. Then $P_X$ is a probability function because: $P_X(B) \geq 0$ for all $B, P_X(\Re)=P(X \in \Re)=1$, and, if $B_j, j=1,2, \ldots$ are pairwise disjoint then, clearly, $\left(X \in B_j\right), j \geq 1$, are also pairwise disjoint and $X \in\left(\bigcup_{j=1}^{\infty} B_j\right)=\bigcup_{j=1}^{\infty}\left(X \in B_j\right)$. Therefore
$$
\begin{aligned}
P_X\left(\bigcup_{j=1}^{\infty} B_j\right)=P\left[X \in\left(\bigcup_{j=1}^{\infty} B_j\right)\right] & =P\left[\bigcup_{j=1}^{\infty}\left(X \in B_j\right)\right] \
& =\sum_{j=1}^{\infty} P\left(X \in B_j\right)=\sum_{j=1}^{\infty} P_X\left(B_j\right) .
\end{aligned}
$$

The probability function $P_X$ is called the probability distribution of the r $v$. $X$. Its significance is extremely important because it tells us the probability that $X$ takes values in any given set $B$. Indeed, much of probability and statistics revolves around the distribution of r.v.’s in which we have an interest.

By selecting $B$ to be $(-\infty, x], x \in \Re$, we have $P_X(B)=P(X \in(-\infty, x])=$ $P(X \leq x)$. In effect, we define a point function which we denote by $F_X$; that is, $F_X(x)=P(X \leq x), x \in \Re$. The function $F_X$ is called the distribution function (d.f.) of $X$. Clearly, if we know $P_X$, then we certainly know $F_X$. Somewhat unexpectedly, the converse is also true. Namely, if we know the (relatively “few”) probabilities $F_X(x), x \in \Re$, then we can determine precisely all probabilities $P_X(B)$ for $B$ subset of $\Re$. This converse is a deep theorem in probability that we cannot deal with here. It is, nevertheless, the reason for which it is the d.f. $F_X$ we deal with, a familiar point function for which so many calculus results hold, rather than the unfamiliar set function $P_X$.

Clearly, the expressions $F_X(+\infty)$ and $F_X(-\infty)$ have no meaning because $+\infty$ and $-\infty$ are not real numbers. They are defined as follows:
$$
F_X(+\infty)=\lim {n \rightarrow \infty} F_X\left(x_n\right), x_n \uparrow \infty \quad \text { and } \quad F_X(-\infty)=\lim {n \rightarrow \infty} F_X\left(y_n\right), y_n \downarrow-\infty \text {. }
$$
These limits exist because $x<y$ implies $(-\infty, x] \subset(-\infty, y]$ and hence
$$
P_X((-\infty, x])=F_X(x) \leq F_X(y)=P_X((-\infty, y]) .
$$

统计代写|统计推断代写Statistical inference代考|Conditional Probability and Related Results

Conditional probability is a probability in its own right, as will be seen, and it is an extremely useful tool in calculating probabilities. Essentially, it amounts to suitably modifying a sample space $\mathcal{S}$, associated with a random experiment, on the evidence that a certain event has occurred. Consider the following examples, by way of motivation, before a formal definition is given.

In tossing three distinct coins once (Example 26 in Chapter 1), consider the events $B=$ “exactly 2 heads occur” $={H H T, H T H, T H H}, A=” 2$ specified coins (e.g., coins #1 and #2) show heads” $={H H H, H H T}$. Then $P(B)=\frac{3}{8}$ and $P(A)=\frac{2}{8}=\frac{1}{4}$. Now, suppose we are told that event $B$ has occurred and we are asked to evaluate the probability of $A$ on the basis of this evidence. Clearly, what really matters here is the event $B$, and, given that $B$ has occurred, the event $A$ occurs only if the sample point $H H T$ appeared; that is, the event ${H H T}=A \cap B$ occurred. The required probability is then $\frac{1}{3}=\frac{1 / 8}{3 / 8}=\frac{P(A \cap B)}{P(B)}$, and the notation employed is $P(A \mid B)$ (probability of $A$, given that $B$ has occurred or, just, given $B)$. Thus, $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$. Observe that $P(A \mid B)=$ $\frac{1}{3}>\frac{1}{4}=P(A)$.

In rolling two distinct dice once (Example 27 in Chapter 1), consider the event $B$ defined by: $B=$ “the sum of numbers on the upper face is 5 “, so that $B=$ ${(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)}$, and let $A=$ “the sum of numbers on the upper faces is $\geq 4$.” Then $A^c=$ “the sum of numbers on the upper faces is $\leq 3 “={(1,1),(1,2),(2,1)}$, so that $P(B)=\frac{10}{36}=\frac{5}{18}$ and $P(A)=1-P\left(A^c\right)=1-\frac{3}{36}=\frac{33}{36}=\frac{11}{12}$. Next, if we are told that $B$ has occurred, then the only way that $A$ occurs is if $A \cap B$ occurs, where $A \cap B=$ “the sum of numbers on the upper faces is both $\geq 4$ and $\leq 5$ (i.e., either 4 or 5$) “={(1,3),(1,4),(2,2),(2,3),(3,1),(3,2),(4,1)}$. Thus, $P(A \mid B)=\frac{7}{10}=$ $\frac{7 / 36}{10 / 36}=\frac{P(A \cap B)}{P(B)}$, and observe that $P(A \mid B)=\frac{7}{10}<\frac{11}{12}=P(A)$.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Distribution of a Random Variable

对于rv $X$，定义set函数$P_X(B)=P(X \in B)$。那么$P_X$是一个概率函数，因为:$P_X(B) \geq 0$对于所有的$B, P_X(\Re)=P(X \in \Re)=1$，如果$B_j, j=1,2, \ldots$是两两不相交的，那么显然，$\left(X \in B_j\right), j \geq 1$也是两两不相交的和$X \in\left(\bigcup_{j=1}^{\infty} B_j\right)=\bigcup_{j=1}^{\infty}\left(X \in B_j\right)$。因此
$$
\begin{aligned}
P_X\left(\bigcup_{j=1}^{\infty} B_j\right)=P\left[X \in\left(\bigcup_{j=1}^{\infty} B_j\right)\right] & =P\left[\bigcup_{j=1}^{\infty}\left(X \in B_j\right)\right] \
& =\sum_{j=1}^{\infty} P\left(X \in B_j\right)=\sum_{j=1}^{\infty} P_X\left(B_j\right) .
\end{aligned}
$$

概率函数$P_X$称为r的概率分布$v$。$X$。它的重要性非常重要，因为它告诉我们$X$在任意给定集合$B$中取值的概率。事实上，很多概率和统计都是围绕着rv的分布展开的。是我们感兴趣的。

通过选择$B$为$(-\infty, x], x \in \Re$，我们得到$P_X(B)=P(X \in(-\infty, x])=$$P(X \leq x)$。实际上，我们定义了一个点函数，用$F_X$表示;也就是$F_X(x)=P(X \leq x), x \in \Re$。函数$F_X$称为$X$的分布函数(d.f.)。显然，如果我们知道$P_X$，那么我们当然知道$F_X$。出乎意料的是，反之亦然。也就是说，如果我们知道(相对“少数”)概率$F_X(x), x \in \Re$，那么我们就可以精确地确定$\Re$子集的$B$的所有概率$P_X(B)$。这个逆是概率论中的一个深奥定理，我们在这里无法处理。然而，这是我们处理d.f. $F_X$的原因，这是一个熟悉的点函数，许多微积分结果都适用，而不是不熟悉的集合函数$P_X$。

显然，表达式$F_X(+\infty)$和$F_X(-\infty)$没有意义，因为$+\infty$和$-\infty$不是实数。它们的定义如下:
$$
F_X(+\infty)=\lim {n \rightarrow \infty} F_X\left(x_n\right), x_n \uparrow \infty \quad \text { and } \quad F_X(-\infty)=\lim {n \rightarrow \infty} F_X\left(y_n\right), y_n \downarrow-\infty \text {. }
$$
这些限制的存在是因为$x<y$意味着$(-\infty, x] \subset(-\infty, y]$，因此
$$
P_X((-\infty, x])=F_X(x) \leq F_X(y)=P_X((-\infty, y]) .
$$

统计代写|统计推断代写Statistical inference代考|Conditional Probability and Related Results

正如我们将看到的，条件概率本身就是一种概率，它是计算概率时非常有用的工具。从本质上讲，它相当于适当地修改与随机实验相关的样本空间$\mathcal{S}$，以证明某个事件已经发生。在给出正式定义之前，考虑下面的例子。

在一次抛三枚不同的硬币时(第1章中的例26)，考虑事件$B=$“恰好有2个正面出现”$={H H T, H T H, T H H}, A=” 2$指定的硬币(例如，硬币＃1和＃2)显示正面“$={H H H, H H T}$。然后是$P(B)=\frac{3}{8}$和$P(A)=\frac{2}{8}=\frac{1}{4}$。现在，假设我们被告知事件$B$已经发生，我们被要求根据这个证据评估$A$的概率。显然，这里真正重要的是事件$B$，并且，假设$B$已经发生，事件$A$只有在样本点$H H T$出现时才会发生;也就是说，发生了事件${H H T}=A \cap B$。所需的概率是$\frac{1}{3}=\frac{1 / 8}{3 / 8}=\frac{P(A \cap B)}{P(B)}$，使用的符号是$P(A \mid B)$(如果发生了$B$，那么$A$的概率就是$B)$)。因此，$P(A \mid B)=\frac{P(A \cap B)}{P(B)}$。观察$P(A \mid B)=$$\frac{1}{3}>\frac{1}{4}=P(A)$。

在一次掷两个不同的骰子(第1章例27)中，考虑事件 $B$ 定义如下: $B=$ “上面的数字的总和是5”，所以 $B=$ ${(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1)}$，让 $A=$ 上面的数字的总和是 $\geq 4$然后 $A^c=$ 上面的数字的总和是 $\leq 3 “={(1,1),(1,2),(2,1)}$，所以 $P(B)=\frac{10}{36}=\frac{5}{18}$ 和 $P(A)=1-P\left(A^c\right)=1-\frac{3}{36}=\frac{33}{36}=\frac{11}{12}$．接下来，如果我们被告知 $B$ 已经发生了，那么唯一的办法就是 $A$ 发生的是if $A \cap B$ 发生，其中 $A \cap B=$ “上面的数字之和是两者 $\geq 4$ 和 $\leq 5$ (即，不是4就是5$) “={(1,3),(1,4),(2,2),(2,3),(3,1),(3,2),(4,1)}$．因此， $P(A \mid B)=\frac{7}{10}=$ $\frac{7 / 36}{10 / 36}=\frac{P(A \cap B)}{P(B)}$观察一下 $P(A \mid B)=\frac{7}{10}<\frac{11}{12}=P(A)$．

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|STAT434

Posted on 2023年8月16日2023年8月28日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Fieller’s Theorem

Fieller’s Theorem (Fieller 1954) is a clever argument to get an exact confidence set on a ratio of normal means.

Given a random sample $\left(X_1, Y_1\right), \ldots,\left(X_n, Y_n\right)$ from a bivariate normal distribution with parameters $\left(\mu_X, \mu_Y, \sigma_X^2, \sigma_Y^2, \rho\right)$, a confidence set on $\theta=\mu_Y / \mu_X$ can be formed in the following way. For $i=1, \ldots, n$, define $Z_{\theta i}=Y_i-\theta X_i$ and $\bar{Z}\theta=\bar{Y}-\theta \bar{X}$. It can be shown that $\bar{Z}\theta$ is normal with mean 0 and variance
$$
V_\theta=\frac{1}{n}\left(\sigma_Y^2-2 \theta \rho \sigma_Y \sigma_X+\theta^2 \sigma_X^2\right)
$$
$V_\theta$ can be estimated with $\hat{V}\theta$, given by $$ \begin{aligned} \hat{V}\theta & =\frac{1}{n(n-1)} \sum_{i=1}^n\left(Z_{\theta i}-\bar{Z}\theta\right)^2 \ & =\frac{1}{n-1}\left(S_Y^2-2 \theta S{Y X}+\theta^2 S_X^2\right),
\end{aligned}
$$
where
$$
S_Y^2=\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2, \quad S_X^2=\frac{1}{n} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2, \quad S_{Y X}=\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)\left(X_i-\bar{X}\right)
$$

统计代写|统计推断代写Statistical inference代考|What About Other Intervals?

Vardeman (1992) asks the question in the title of this Miscellanea, arguing that mainstream statistics should spend more time on intervals other than two-sided confidence intervals. In particular, he lists (a) one-sided intervals, (b) distributionfree intervals, (c) prediction intervals, and (d) tolerance intervals.
We have seen one-sided intervals, and distribution-free intervals are intervals whose probability guarantee holds with little (or no) assumption on the underlying cdf (see Exercise 9.58). The other two interval definitions, together with the usual confidence interval, provide use with a hierarchy of inferences, each more stringent than the previous.
If $X_1, X_2, \ldots, X_n$ are iid from a population with $\operatorname{cdf} F(x \mid \theta)$, and $C(\mathbf{x})=[l(\mathbf{x}), u(\mathbf{x})]$ is an interval, for a specified value $1-\alpha$ it is a
(i) confidence interval if $P_\theta[l(\mathbf{X}) \leq \theta \leq u(\mathbf{X})] \geq 1-\alpha$;
(ii) prediction interval if $P_\theta\left[l(\mathbf{X}) \leq X_{n+1} \leq u(\mathbf{X})\right] \geq 1-\alpha$;
(iii) tolerance interval if, for a specified value $p, P_\theta[F(u(\mathbf{X}) \mid \theta)-F(l(\mathbf{X}) \mid \theta) \geq p] \geq$ $1-\alpha$.

So a confidence interval covers a mean, a prediction interval covers a new random variable, and a tolerance interval covers a proportion of the population. Thus, each gives a different inference, with the appropriate one being dictated by the problem at hand.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Fieller’s Theorem

费勒定理(Fieller’s Theorem, 1954)是一个很聪明的论点，可以得到正态均值之比的精确置信集。

给定一个参数为$\left(\mu_X, \mu_Y, \sigma_X^2, \sigma_Y^2, \rho\right)$的二元正态分布的随机样本$\left(X_1, Y_1\right), \ldots,\left(X_n, Y_n\right)$，可以通过以下方式形成$\theta=\mu_Y / \mu_X$上的置信集。对于$i=1, \ldots, n$，定义$Z_{\theta i}=Y_i-\theta X_i$和$\bar{Z}\theta=\bar{Y}-\theta \bar{X}$。可以看出$\bar{Z}\theta$是均值为0，方差为0的正态分布
$$
V_\theta=\frac{1}{n}\left(\sigma_Y^2-2 \theta \rho \sigma_Y \sigma_X+\theta^2 \sigma_X^2\right)
$$
$V_\theta$可以用$\hat{V}\theta$来估计，由$$ \begin{aligned} \hat{V}\theta & =\frac{1}{n(n-1)} \sum_{i=1}^n\left(Z_{\theta i}-\bar{Z}\theta\right)^2 \ & =\frac{1}{n-1}\left(S_Y^2-2 \theta S{Y X}+\theta^2 S_X^2\right),
\end{aligned}
$$给出
在哪里
$$
S_Y^2=\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2, \quad S_X^2=\frac{1}{n} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2, \quad S_{Y X}=\frac{1}{n} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)\left(X_i-\bar{X}\right)
$$

统计代写|统计推断代写Statistical inference代考|What About Other Intervals?

Vardeman(1992)在《杂记》的标题中提出了这个问题，他认为主流统计学应该花更多的时间研究双侧置信区间以外的区间。特别地，他列出了(a)单侧区间，(b)无分布区间，(c)预测区间，(d)容忍区间。
我们已经看到了单侧区间，而无分布区间是在对底层cdf进行很少(或没有)假设的情况下概率保证成立的区间(参见练习9.58)。另外两个区间定义，加上通常的置信区间，提供了推理层次结构的使用，每一个都比前一个更严格。
如果$X_1, X_2, \ldots, X_n$是从具有$\operatorname{cdf} F(x \mid \theta)$的总体中抽取的，并且$C(\mathbf{x})=[l(\mathbf{x}), u(\mathbf{x})]$是一个区间，对于指定值$1-\alpha$，它是一个
(i)置信区间为$P_\theta[l(\mathbf{X}) \leq \theta \leq u(\mathbf{X})] \geq 1-\alpha$;
(ii) $P_\theta\left[l(\mathbf{X}) \leq X_{n+1} \leq u(\mathbf{X})\right] \geq 1-\alpha$的预测区间;
(iii)公差区间，如果为规定值$p, P_\theta[F(u(\mathbf{X}) \mid \theta)-F(l(\mathbf{X}) \mid \theta) \geq p] \geq$$1-\alpha$。

置信区间覆盖均值，预测区间覆盖新随机变量，容差区间覆盖总体的一部分。因此，每个人都给出了不同的推断，适当的推断取决于手头的问题。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|STA2023

Posted on 2023年8月16日2023年8月28日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Bayesian Optimality

The goal of obtaining a smallest confidence set with a specified coverage probability can also be attained using Bayesian criteria. If we have a posterior distribution $\pi(\theta \mid \mathbf{x})$, the posterior distribution of $\theta$ given $\mathbf{X}=\mathbf{x}$, we would like to find the set $C(\mathbf{x})$ that satisfies
(i) $\int_{C(\mathbf{x})} \pi(\theta \mid \mathbf{x}) d \mathbf{x}=1-\alpha$
(ii) $\quad$ Size $(C(\mathbf{x})) \leq \operatorname{Size}\left(C^{\prime}(\mathbf{x})\right)$
for any set $C^{\prime}(\mathbf{x})$ satisfying $\int_{C^{\prime}(\mathbf{x})} \pi(\theta \mid \mathbf{x}) d \mathbf{x} \geq 1-\alpha$.
If we take our measure of size to be length, then we can apply Theorem 9.3.2 and obtain the following result.

Corollary 9.3.10 If the posterior density $\pi(\theta \mid \mathbf{x})$ is unimodal, then for a given value of $\alpha$, the shortest credible interval for $\theta$ is given by
$$
{\theta: \pi(\theta \mid \mathbf{x}) \geq k} \quad \text { where } \int_{{\theta: \pi(\theta \mid \mathbf{x}) \geq k}} \pi(\theta \mid \mathbf{x}) d \theta=1-\alpha .
$$
The credible set described in Corollary 9.3.10 is called a highest posterior density (HPD) region, as it consists of the values of the parameter for which the posterior density is highest. Notice the similarity in form between the HPD region and the likelihood region.

Example 9.3.11 (Poisson HPD region) In Example 9.2.16 we derived a $1-\alpha$ credible set for a Poisson parameter. We now construct an HPD region. By Corollary 9.3.10, this region is given by $\left{\lambda: \pi\left(\lambda \mid \sum x\right) \geq k\right}$, where $k$ is chosen so that
$$
1-\alpha=\int_{{\lambda: \pi(\lambda \mid \Sigma x) \geq k}} \pi\left(\lambda \mid \sum x\right) d \lambda .
$$
Recall that the posterior pdf of $\lambda$ is $\operatorname{gamma}\left(a+\sum x,[n+(1 / b)]^{-1}\right)$, so we need to find $\lambda_L$ and $\lambda_U$ such that
$$
\pi\left(\lambda_L \mid \sum x\right)=\pi\left(\lambda_U \mid \sum x\right) \text { and } \int_{\lambda_L}^{\lambda_U} \pi\left(\lambda \mid \sum x\right) d \lambda=1-\alpha
$$

统计代写|统计推断代写Statistical inference代考|Loss Function Optimality

In the previous two sections we looked at optimality of interval estimators by first requiring them to have a minimum coverage probability and then looking for the shortest interval. However, it is possible to put these requirements together in one loss function and use decision theory to search for an optimal estimator. In interval estimation, the action space $\mathcal{A}$ will consist of subsets of the parameter space $\Theta$ and, more formally, we might talk of “set estimation,” since an optimal rule may not necessarily be an interval. However, practical considerations lead us to mainly consider set estimators that are intervals and, happily, many optimal procedures turn out to be intervals.

We use $C$ (for confidence interval) to denote elements of $\mathcal{A}$, with the meaning of the action $C$ being that the interval estimate ” $\theta \in C$ ” is made. A decision rule $\delta(\mathbf{x})$ simply specifies, for each $\mathbf{x} \in \mathcal{X}$, which set $C \in \mathcal{A}$ will be used as an estimate of $\theta$ if $\mathbf{X}=\mathbf{x}$ is observed. Thus we will use the notation $C(\mathbf{x})$, as before.

The loss function in an interval estimation problem usually includes two quantities: a measure of whether the set estimate correctly includes the true value $\theta$ and a measure of the size of the set estimate. We will, for the most part, consider only sets $C$ that are intervals, so a natural measure of size is Length $(C)=$ length of $C$. To express the correctness measure, it is common to use
$$
I_C(\theta)= \begin{cases}1 & \theta \in C \ 0 & \theta \notin C .\end{cases}
$$

统计推断代考

统计代写|统计推断代写Statistical inference代考|Bayesian Optimality

使用贝叶斯准则也可以获得具有指定覆盖概率的最小置信集的目标。如果我们有一个后验分布$\pi(\theta \mid \mathbf{x})$, $\theta$的后验分布已知$\mathbf{X}=\mathbf{x}$，我们想找到一个集合$C(\mathbf{x})$满足
(i) $\int_{C(\mathbf{x})} \pi(\theta \mid \mathbf{x}) d \mathbf{x}=1-\alpha$
(ii) $\quad$尺寸$(C(\mathbf{x})) \leq \operatorname{Size}\left(C^{\prime}(\mathbf{x})\right)$
对于任意集$C^{\prime}(\mathbf{x})$满足$\int_{C^{\prime}(\mathbf{x})} \pi(\theta \mid \mathbf{x}) d \mathbf{x} \geq 1-\alpha$。
如果我们将大小的度量取为长度，那么我们可以应用定理9.3.2并得到如下结果。

如果后验密度$\pi(\theta \mid \mathbf{x})$是单峰的，那么对于$\alpha$的给定值，$\theta$的最短可信区间为
$$
{\theta: \pi(\theta \mid \mathbf{x}) \geq k} \quad \text { where } \int_{{\theta: \pi(\theta \mid \mathbf{x}) \geq k}} \pi(\theta \mid \mathbf{x}) d \theta=1-\alpha .
$$
在推论9.3.10中描述的可信集称为最高后验密度(HPD)区域，因为它由后验密度最高的参数值组成。注意HPD区域和似然区域在形式上的相似性。

在例9.2.16中，我们为泊松参数导出了一个$1-\alpha$可信集。现在我们构建一个火奴鲁鲁警局区域。根据推论9.3.10，该区域由$\left{\lambda: \pi\left(\lambda \mid \sum x\right) \geq k\right}$给出，其中选择$k$以便
$$
1-\alpha=\int_{{\lambda: \pi(\lambda \mid \Sigma x) \geq k}} \pi\left(\lambda \mid \sum x\right) d \lambda .
$$
回想一下$\lambda$的后验pdf是$\operatorname{gamma}\left(a+\sum x,[n+(1 / b)]^{-1}\right)$，所以我们需要找到$\lambda_L$和$\lambda_U$这样
$$
\pi\left(\lambda_L \mid \sum x\right)=\pi\left(\lambda_U \mid \sum x\right) \text { and } \int_{\lambda_L}^{\lambda_U} \pi\left(\lambda \mid \sum x\right) d \lambda=1-\alpha
$$

统计代写|统计推断代写Statistical inference代考|Loss Function Optimality

在前两节中，我们通过首先要求区间估计器具有最小覆盖概率，然后寻找最短间隔来研究区间估计器的最优性。然而，有可能将这些需求放在一个损失函数中，并使用决策理论来搜索最优估计量。在区间估计中，动作空间$\mathcal{A}$将由参数空间$\Theta$的子集组成，更正式地说，我们可以称之为“集合估计”，因为最优规则不一定是区间。然而，实际的考虑使我们主要考虑区间的集合估计量，令人高兴的是，许多最优过程结果是区间。

我们使用$C$(置信区间)来表示$\mathcal{A}$的元素，动作$C$的含义是进行区间估计“$\theta \in C$”。决策规则$\delta(\mathbf{x})$简单地为每个$\mathbf{x} \in \mathcal{X}$指定，如果观察到$\mathbf{X}=\mathbf{x}$，那么将使用$C \in \mathcal{A}$作为$\theta$的估定值。因此，我们将像以前一样使用$C(\mathbf{x})$符号。

区间估计问题中的损失函数通常包括两个量:集估计是否正确包含真值$\theta$的度量和集估计的大小的度量。在大多数情况下，我们将只考虑作为间隔的集合$C$，因此大小的自然度量是Length $(C)=$$C$的长度。表示正确性度量，通常使用
$$
I_C(\theta)= \begin{cases}1 & \theta \in C \ 0 & \theta \notin C .\end{cases}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|STAT5160

Posted on 2023年8月16日2023年8月28日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|L088 Function Optimality

A decision theoretic analysis, as in Section 7.3.4, may be used to compare hypothesis tests, rather than just comparing them via their power functions. To carry out this kind of analysis, we must specify the action space and loss function for our hypothesis testing problem.

In a hypothesis testing problem, only two actions are allowable, “accept $H_0$ ” or “reject $H_0$.” These two actions might be denoted $a_0$ and $a_1$, respectively. The action space in hypothesis testing is the two-point set $\mathcal{A}=\left{a_0, a_1\right}$. A decision rule $\delta(\mathbf{x})$ (a hypothesis test) is a function on $\mathcal{X}$ that takes on only two values, $a_0$ and $a_1$. The set $\left{\mathbf{x}: \delta(\mathbf{x})=a_0\right}$ is the acceptance region for the test, and the set $\left{\mathbf{x}: \delta(\mathbf{x})=a_1\right}$ is the rejection region, just as in Definition 8.1.3.

The loss function in a hypothesis testing problem should reflect the fact that, if $\theta \in \Theta_0$ and decision $a_1$ is made, or if $\theta \in \Theta_0^c$ and decision $a_0$ is made, a mistake has been made. But in the other two possible cases, the correct decision has been made. Since there are only two possible actions, the loss function $L(\theta, a)$ in a hypothesis testing problem is composed of only two parts. The function $L\left(\theta, a_0\right)$ is the loss incurred for various values of $\theta$ if the decision to accept $H_0$ is made, and $L\left(\theta, a_1\right)$ is the loss incurred for various values of $\theta$ if the decision to reject $H_0$ is made.

The simplest kind of loss in a testing problem is called $0-1$ loss and is defined by
$$
L\left(\theta, a_0\right)=\left{\begin{array}{ll}
0 & \theta \in \Theta_0 \
1 & \theta \in \Theta_0^{\mathrm{c}}
\end{array} \quad \text { and } \quad L\left(\theta, a_1\right)= \begin{cases}1 & \theta \in \Theta_0 \
0 & \theta \in \Theta_0^{\mathrm{c}} .\end{cases}\right.
$$
With $0-1$ loss, the value 0 is lost if a correct decision is made and the value 1 is lost if an incorrect decision is made. This is a particularly simple situation in which both types of error have the same consequence. A slightly more realistic loss, one that gives different costs to the two types of error, is generalized 0-1 loss,
$$
L\left(\theta, a_0\right)=\left{\begin{array}{cc}
0 & \theta \in \Theta_0 \
c_{\mathrm{II}} & \theta \in \Theta_0^{\mathrm{c}}
\end{array} \quad \text { and } \quad L\left(\theta, a_1\right)=\left{\begin{array}{cc}
c_{\mathrm{I}} & \theta \in \Theta_0 \
0 & \theta \in \Theta_0^{\mathrm{c}}
\end{array}\right.\right.
$$

统计代写|统计推断代写Statistical inference代考|Monotonic Power FUnction

In this chapter we used the property of MLR quite extensively, particularly in relation to properties of power functions of tests. The concept of stochastic ordering can also be used to obtain properties of power functions. (Recall that stochastic ordering has already been encountered in previous chapters, for example, in Exercises $1.49,3.41-3.43$, and 5.19 . A cdf $F$ is stochastically greater than a cdf $G$ if $F(x) \leq G(x)$ for all $x$, with strict inequality for some $x$, which implies that if $X \sim F, Y \sim G$, then $P(X>x) \geq P(Y>x)$ for all $x$, with strict inequality for some $x$. In other words, $F$ gives more probability to greater values.)

In terms of hypothesis testing, it is of ten the case that the distribution under the alternative is stochastically greater than under the null distribution. For example, if we have a random sample from a $\mathrm{n}\left(\theta, \sigma^2\right)$ population and are interested in testing $H_0: \theta \leq \theta_0$ versus $H_1: \theta>\theta_0$, it is true that all the distributions in the alternative are stochastically greater than all those in the null. Gilat (1977) uses the property of stochastic ordering, rather than MLR, to prove monotonicity of power functions under general conditions.

The likelihood ratio $L\left(\theta_1 \mid \mathbf{x}\right) / L\left(\theta_0 \mid \mathbf{x}\right)=f\left(\mathbf{x} \mid \theta_1\right) / f\left(\mathbf{x} \mid \theta_0\right)$ plays an important role in the testing of $H_0: \theta=\theta_0$ versus $H_1: \theta=\theta_1$. This ratio is equal to the LRT statistic $\lambda(\mathbf{x})$ for values of $\mathbf{x}$ that yield small values of $\lambda$. Also, the Neyman-Pearson Lemma says that the UMP level $\alpha$ test of $H_0$ versus $H_1$ can be defined in terms of this ratio. This likelihood ratio also has an important Bayesian interpretation. Suppose $\pi_0$ and $\pi_1$ are our prior probabilities for $\theta_0$ and $\theta_1$. Then, the posterior odds in favor of $\theta_1$ are
$$
\frac{P\left(\theta=\theta_1 \mid \mathbf{x}\right)}{P\left(\theta=\theta_0 \mid \mathbf{x}\right)}=\frac{f\left(\mathbf{x} \mid \theta_1\right) \pi_1 / m(\mathbf{x})}{f\left(\mathbf{x} \mid \theta_0\right) \pi_0 / m(\mathbf{x})}=\frac{f\left(\mathbf{x} \mid \theta_1\right)}{f\left(\mathbf{x} \mid \theta_0\right)} \cdot \frac{\pi_1}{\pi_0}
$$
$\pi_1 / \pi_0$ are the prior odds in favor of $\theta_1$. The likelihood ratio is the amount these prior odds should be adjusted, having observed the data $\mathbf{X}=\mathbf{x}$, to obtain the posterior odds. If the likelihood ratio equals 2 , then the prior odds are doubled. The likelihood ratio does not depend on the prior probabilities. Thus, it is interpreted as the evidence in the data favoring $H_1$ over $H_0$. This kind of interpretation is discussed by Royall (1997).

统计推断代考

统计代写|统计推断代写Statistical inference代考|L088 Function Optimality

如第7.3.4节所述，决策理论分析可以用于比较假设检验，而不仅仅是通过它们的幂函数进行比较。为了进行这种分析，我们必须为假设检验问题指定动作空间和损失函数。

在假设检验问题中，只允许两个动作，“接受$H_0$”或“拒绝$H_0$”，这两个动作可以分别表示为$a_0$和$a_1$。假设检验中的动作空间是两点集$\mathcal{A}=\left{a_0, a_1\right}$。决策规则$\delta(\mathbf{x})$(假设检验)是$\mathcal{X}$上的一个函数，它只接受两个值$a_0$和$a_1$。集合$\left{\mathbf{x}: \delta(\mathbf{x})=a_0\right}$为测试的接受区域，集合$\left{\mathbf{x}: \delta(\mathbf{x})=a_1\right}$为拒绝区域，定义8.1.3。

假设检验问题中的损失函数应该反映这样一个事实:如果做出了$\theta \in \Theta_0$和$a_1$决策，或者如果做出了$\theta \in \Theta_0^c$和$a_0$决策，那么就犯了一个错误。但在另外两种可能的情况下，已经做出了正确的决定。由于只有两种可能的行为，假设检验问题中的损失函数$L(\theta, a)$仅由两部分组成。如果决定接受$H_0$，则函数$L\left(\theta, a_0\right)$是对$\theta$的各种值造成的损失，如果决定拒绝$H_0$，则函数$L\left(\theta, a_1\right)$是对$\theta$的各种值造成的损失。

测试问题中最简单的一种损耗称为$0-1$损耗，定义为
$$
L\left(\theta, a_0\right)=\left{\begin{array}{ll}
0 & \theta \in \Theta_0 \
1 & \theta \in \Theta_0^{\mathrm{c}}
\end{array} \quad \text { and } \quad L\left(\theta, a_1\right)= \begin{cases}1 & \theta \in \Theta_0 \
0 & \theta \in \Theta_0^{\mathrm{c}} .\end{cases}\right.
$$
对于$0-1$ loss，如果做出正确的决策，则丢失值0，如果做出错误的决策，则丢失值1。这是一种特别简单的情况，其中两种类型的错误具有相同的结果。一种更现实的损失是广义的0-1损失，它会给两种错误带来不同的代价，
$$
L\left(\theta, a_0\right)=\left{\begin{array}{cc}
0 & \theta \in \Theta_0 \
c_{\mathrm{II}} & \theta \in \Theta_0^{\mathrm{c}}
\end{array} \quad \text { and } \quad L\left(\theta, a_1\right)=\left{\begin{array}{cc}
c_{\mathrm{I}} & \theta \in \Theta_0 \
0 & \theta \in \Theta_0^{\mathrm{c}}
\end{array}\right.\right.
$$

统计代写|统计推断代写Statistical inference代考|Monotonic Power FUnction

这一章中，我们相当广泛地使用了MLR的性质，特别是与幂函数的性质有关的测试。随机排序的概念也可以用来得到幂函数的性质。(回想一下，随机排序已经在前面的章节中遇到过，例如，在练习中 $1.49,3.41-3.43$5.19。A cdf $F$ 随机大于CDF $G$ 如果 $F(x) \leq G(x)$ 对所有人 $x$，对一些人来说是严格的不平等 $x$，这意味着如果 $X \sim F, Y \sim G$那么， $P(X>x) \geq P(Y>x)$ 对所有人 $x$，对一些人来说是严格的不平等 $x$．换句话说， $F$ 给更大的值更多的概率。)

在假设检验方面，备选分布下的分布随机地大于零分布下的分布。例如，如果我们有一个来自$\mathrm{n}\left(\theta, \sigma^2\right)$总体的随机样本，并且对$H_0: \theta \leq \theta_0$和$H_1: \theta>\theta_0$的测试感兴趣，那么替代中的所有分布都随机地大于null中的所有分布。Gilat(1977)利用随机排序的性质，而不是MLR，证明了一般条件下幂函数的单调性。

似然比$L\left(\theta_1 \mid \mathbf{x}\right) / L\left(\theta_0 \mid \mathbf{x}\right)=f\left(\mathbf{x} \mid \theta_1\right) / f\left(\mathbf{x} \mid \theta_0\right)$在$H_0: \theta=\theta_0$对$H_1: \theta=\theta_1$的检验中起着重要作用。对于产生较小的$\lambda$值的$\mathbf{x}$，该比率等于LRT统计值$\lambda(\mathbf{x})$。此外，内曼-皮尔逊引理表明，$H_0$与$H_1$的UMP水平$\alpha$测试可以根据该比率来定义。这个似然比也有一个重要的贝叶斯解释。假设$\pi_0$和$\pi_1$是$\theta_0$和$\theta_1$的先验概率。那么，支持$\theta_1$的后验概率为
$$
\frac{P\left(\theta=\theta_1 \mid \mathbf{x}\right)}{P\left(\theta=\theta_0 \mid \mathbf{x}\right)}=\frac{f\left(\mathbf{x} \mid \theta_1\right) \pi_1 / m(\mathbf{x})}{f\left(\mathbf{x} \mid \theta_0\right) \pi_0 / m(\mathbf{x})}=\frac{f\left(\mathbf{x} \mid \theta_1\right)}{f\left(\mathbf{x} \mid \theta_0\right)} \cdot \frac{\pi_1}{\pi_0}
$$
$\pi_1 / \pi_0$是支持$\theta_1$的优先赔率。似然比是观察数据$\mathbf{X}=\mathbf{x}$后，调整这些先验几率以获得后验几率的数量。如果似然比等于2，则先前的概率加倍。似然比不依赖于先验概率。因此，它被解释为数据中的证据有利于$H_1$而不是$H_0$。Royall(1997)讨论了这种解释。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|ECON3130

Posted on 2023年7月21日2023年8月24日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Likelihood Ratio Tests

The likelihood ratio method of hypothesis testing is related to the maximum likelihood estimators discussed in Section 7.2.2, and likelihood ratio tests are as widely applicable as maximum likelihood estimation. Recall that if $X_1, \ldots, X_n$ is a random sample from a population with pdf or $\operatorname{pmf} f(x \mid \theta)(\theta$ may be a vector), the likelihood function is defined as
$$
L\left(\theta \mid x_1, \ldots, x_n\right)=L(\theta \mid \mathbf{x})=f(\mathbf{x} \mid \theta)=\prod_{i=1}^n f\left(x_i \mid \theta\right) .
$$
Let $\Theta$ denote the entire parameter space. Likelihood ratio tests are defined as follows.
Definition 8.2.1 The likelihood ratio test statistic for testing $H_0: \theta \in \Theta_0$ versus $H_1: \theta \in \Theta_0^c$ is
$$
\lambda(\mathbf{x})=\frac{\sup {\Theta_0} L(\theta \mid \mathbf{x})}{\sup {\Theta} L(\theta \mid \mathbf{x})} .
$$
A likelihood ratio test (LRT) is any test that has a rejection region of the form ${\mathbf{x}: \lambda(\mathbf{x})$ $\leq c}$, where $c$ is any number satisfying $0 \leq c \leq 1$.

The rationale behind LRTs may best be understood in the situation in which $f(x \mid \theta)$ is the pmf of a discrete random variable. In this case, the numerator of $\lambda(\mathbf{x})$ is the maximum probability of the observed sample, the maximum being computed over parameters in the null hypothesis. (See Exercise 8.4.) The denominator of $\lambda(\mathbf{x})$ is the maximum probability of the observed sample over all possible parameters. The ratio of these two maxima is small if there are parameter points in the alternative hypothesis for which the observed sample is much more likely than for any parameter point in the null hypothesis. In this situation, the LRT criterion says $H_0$ should be rejected and $H_1$ accepted as true. Methods for selecting the number $c$ are discussed in Section 8.3.

If we think of doing the maximization over both the entire parameter space (unrestricted maximization) and a subset of the parameter space (restricted maximization), then the correspondence between LRTs and MLEs becomes more clear. Suppose $\hat{\theta}$, an MLE of $\theta$, exists; $\hat{\theta}$ is obtained by doing an unrestricted maximization of $L(\theta \mid \mathbf{x})$. We can also consider the MLE of $\theta$, call it $\hat{\theta}_0$, obtained by doing a restricted maximization, assuming $\Theta_0$ is the parameter space. That is, $\hat{\theta}_0=\hat{\theta}_0(\mathbf{x})$ is the value of $\theta \in \Theta_0$ that maximizes $L(\theta \mid \mathbf{x})$. Then, the LRT statistic is
$$
\lambda(\mathbf{x})=\frac{L\left(\hat{\theta}_0 \mid \mathbf{x}\right)}{L(\hat{\theta} \mid \mathbf{x})}
$$

统计代写|统计推断代写Statistical inference代考|Bayesian Tests

Hypothesis testing problems may also be formulated in a Bayesian model. Recall from Section 7.2.3 that a Bayesian model includes not only the sampling distribution $f(\mathbf{x} \mid \theta)$ but also the prior distribution $\pi(\theta)$, with the prior distribution reflecting the experimenter’s opinion about the parameter $\theta$ prior to sampling.

The Bayesian paradigm prescribes that the sample information be combined with the prior information using Bayes’ Theorem to obtain the posterior distribution $\pi(\theta \mid \mathbf{x})$. All inferences about $\theta$ are now based on the posterior distribution.

In a hypothesis testing problem, the posterior distribution may be used to calculate the probabilities that $H_0$ and $H_1$ are true. Remember, $\pi(\theta \mid \mathbf{x})$ is a probability distribution for a random variable. Hence, the posterior probabilities $P\left(\theta \in \Theta_0 \mid \mathbf{x}\right)=P\left(H_0\right.$ is true $\mid \mathbf{x})$ and $P\left(\theta \in \Theta_0^{\mathbf{c}} \mid \mathbf{x}\right)=P\left(H_1\right.$ is true $\left.\mid \mathbf{x}\right)$ may be computed.

The probabilities $P\left(H_0\right.$ is true $\left.\mid \mathbf{x}\right)$ and $P\left(H_1\right.$ is true $\left.\mid \mathbf{x}\right)$ are not meaningful to the classical statistician. The classical statistician considers $\theta$ to be a fixed number. Consequently, a hypothesis is either true or false. If $\theta \in \Theta_0, P\left(H_0\right.$ is true $\left.\mid \mathbf{x}\right)=1$ and $P\left(H_1\right.$ is true $\left.\mathbf{x}\right)=0$ for all values of $\mathbf{x}$. If $\theta \in \Theta_0^{\mathbf{c}}$, these values are reversed. Since these probabilities are unknown (since $\theta$ is unknown) and do not depend on the sample $\mathbf{x}$, they are not used by the classical statistician. In a Bayesian formulation of a hypothesis testing problem, these probabilities depend on the sample $\mathbf{x}$ and can give useful information about the veracity of $H_0$ and $H_1$.

One way a Bayesian hypothesis tester may choose to use the posterior distribution is to decide to accept $H_0$ as true if $P\left(\theta \in \Theta_0 \mid \mathbf{X}\right) \geq P\left(\theta \in \Theta_0^{\mathbf{c}} \mid \mathbf{X}\right)$ and to reject $H_0$ otherwise. In the terminology of the previous sections, the test statistic, a function of the sample, is $P\left(\theta \in \Theta_0^c \mid \mathbf{X}\right)$ and the rejection region is $\left{\mathbf{x}: P\left(\theta \in \Theta_0^c \mid \mathbf{x}\right)>\frac{1}{2}\right}$. Alternatively, if the Bayesian hypothesis tester wishes to guard against falsely rejecting $H_0$, he may decide to reject $H_0$ only if $P\left(\theta \in \Theta_0^{\mathrm{c}} \mid \mathbf{X}\right)$ is greater than some large number, .99 for example.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Likelihood Ratio Tests

假设检验的似然比方法与7.2.2节中讨论的极大似然估计量有关，似然比检验与极大似然估计一样广泛适用。回想一下，如果$X_1, \ldots, X_n$是总体中的随机样本(pdf或$\operatorname{pmf} f(x \mid \theta)(\theta$可能是矢量)，则似然函数定义为
$$
L\left(\theta \mid x_1, \ldots, x_n\right)=L(\theta \mid \mathbf{x})=f(\mathbf{x} \mid \theta)=\prod_{i=1}^n f\left(x_i \mid \theta\right) .
$$
设$\Theta$表示整个参数空间。似然比检验的定义如下。
8.2.1检验$H_0: \theta \in \Theta_0$与$H_1: \theta \in \Theta_0^c$的似然比检验统计量为
$$
\lambda(\mathbf{x})=\frac{\sup {\Theta_0} L(\theta \mid \mathbf{x})}{\sup {\Theta} L(\theta \mid \mathbf{x})} .
$$
似然比检验(LRT)是具有${\mathbf{x}: \lambda(\mathbf{x})$$\leq c}$形式的拒绝区域的任何检验，其中$c$是满足$0 \leq c \leq 1$的任何数字。

在$f(x \mid \theta)$是离散随机变量的pmf的情况下，LRTs背后的基本原理可能得到最好的理解。在这种情况下，$\lambda(\mathbf{x})$的分子是观察到的样本的最大概率，最大值是根据零假设中的参数计算的。(参见练习8.4。)$\lambda(\mathbf{x})$的分母是观测样本在所有可能参数上的最大概率。如果在备选假设中存在参数点，观察到的样本比零假设中的任何参数点更有可能出现，那么这两个最大值的比值就很小。在这种情况下，LRT准则说$H_0$应该被拒绝，$H_1$应该被接受为真。选择编号$c$的方法将在8.3节中讨论。

如果我们考虑在整个参数空间(无限制最大化)和参数空间的子集(受限最大化)上进行最大化，那么lrt和mle之间的对应关系就会变得更加清晰。假设存在一个最大似然值$\theta$$\hat{\theta}$;$\hat{\theta}$是通过无限制地最大化$L(\theta \mid \mathbf{x})$得到的。我们也可以考虑$\theta$的最大似然值(MLE)，称其为$\hat{\theta}_0$，通过做一个受限的最大化得到，假设$\Theta_0$是参数空间。即$\hat{\theta}_0=\hat{\theta}_0(\mathbf{x})$是使$L(\theta \mid \mathbf{x})$最大化的$\theta \in \Theta_0$值。那么，轻轨统计数据为
$$
\lambda(\mathbf{x})=\frac{L\left(\hat{\theta}_0 \mid \mathbf{x}\right)}{L(\hat{\theta} \mid \mathbf{x})}
$$

统计代写|统计推断代写Statistical inference代考|Bayesian Tests

假设检验问题也可以用贝叶斯模型来表述。回想一下，在第7.2.3节中，贝叶斯模型不仅包括抽样分布$f(\mathbf{x} \mid \theta)$，还包括先验分布$\pi(\theta)$，其中先验分布反映了在抽样之前实验者对参数$\theta$的看法。

贝叶斯范式规定使用贝叶斯定理将样本信息与先验信息结合得到后验分布$\pi(\theta \mid \mathbf{x})$。所有关于$\theta$的推论现在都基于后验分布。

在假设检验问题中，后验分布可以用来计算$H_0$和$H_1$为真的概率。记住，$\pi(\theta \mid \mathbf{x})$是随机变量的概率分布。因此，后验概率$P\left(\theta \in \Theta_0 \mid \mathbf{x}\right)=P\left(H_0\right.$为真$\mid \mathbf{x})$和$P\left(\theta \in \Theta_0^{\mathbf{c}} \mid \mathbf{x}\right)=P\left(H_1\right.$为真$\left.\mid \mathbf{x}\right)$可以计算出来。

概率$P\left(H_0\right.$为真$\left.\mid \mathbf{x}\right)$和$P\left(H_1\right.$为真$\left.\mid \mathbf{x}\right)$对古典统计学家来说是没有意义的。古典统计学家认为$\theta$是一个固定的数字。因此，假设不是对就是错。如果$\theta \in \Theta_0, P\left(H_0\right.$为真$\left.\mid \mathbf{x}\right)=1$和$P\left(H_1\right.$为真$\left.\mathbf{x}\right)=0$对于$\mathbf{x}$的所有值。如果是$\theta \in \Theta_0^{\mathbf{c}}$，则这些值相反。由于这些概率是未知的(因为$\theta$是未知的)，并且不依赖于样本$\mathbf{x}$，所以经典统计学家不使用它们。在假设检验问题的贝叶斯公式中，这些概率取决于样本$\mathbf{x}$，并且可以提供关于$H_0$和$H_1$的准确性的有用信息。

贝叶斯假设检验者可能选择使用后验分布的一种方法是，如果$P\left(\theta \in \Theta_0 \mid \mathbf{X}\right) \geq P\left(\theta \in \Theta_0^{\mathbf{c}} \mid \mathbf{X}\right)$，决定接受$H_0$为真，否则拒绝$H_0$。在前几节的术语中，检验统计量(样本的函数)为$P\left(\theta \in \Theta_0^c \mid \mathbf{X}\right)$，拒绝区域为$\left{\mathbf{x}: P\left(\theta \in \Theta_0^c \mid \mathbf{x}\right)>\frac{1}{2}\right}$。或者，如果贝叶斯假设检验员希望防止错误地拒绝$H_0$，他可能只在$P\left(\theta \in \Theta_0^{\mathrm{c}} \mid \mathbf{X}\right)$大于某个较大的数字(例如0.99)时才决定拒绝$H_0$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|ST502

Posted on 2023年7月21日2023年8月24日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|Sufficiency and Unbiasedness

In the previous section, the concept of sufficiency was not used in our search for unbiased estimates. We will now see that consideration of sufficiency is a powerful tool, indeed.

The main theorem of this section, which relates sufficient statistics to unbiased estimates, is, as in the case of the Cramér-Rao Theorem, another clever application of some well-known theorems. Recall from Chapter 4 that if $X$ and $Y$ are any two random variables, then, provided the expectations exist, we have
$$
\begin{aligned}
\mathrm{E} X & =\mathrm{E}[\mathrm{E}(X \mid Y)], \
\operatorname{Var} X & =\operatorname{Var}[\mathrm{E}(X \mid Y)]+\mathrm{E}[\operatorname{Var}(X \mid Y)] .
\end{aligned}
$$
Using these tools we can prove the following theorem.
Theorem 7.3.17 (Rao-Blackwell) Let $W$ be any unbiased estimator of $\tau(\theta)$, and let $T$ be a sufficient statistic for $\theta$. Define $\phi(T)=\mathrm{E}(W \mid T)$. Then $\mathrm{E}\theta \phi(T)=\tau(\theta)$ and $\operatorname{Var}\theta \phi(T) \leq \operatorname{Var}\theta W$ for all $\theta$; that is, $\phi(T)$ is a uniformly better unbiased estimator of $\tau(\theta)$. Proof: From (7.3.13) we have $$ \tau(\theta)=\mathrm{E}\theta W=\mathrm{E}\theta[\mathrm{E}(W \mid T)]=\mathrm{E}\theta \phi(T),
$$
so $\phi(T)$ is unbiased for $\tau(\theta)$. Also,
$$
\begin{array}{rlr}
\operatorname{Var}\theta W & =\operatorname{Var}\theta[\mathrm{E}(W \mid T)]+\mathrm{E}\theta[\operatorname{Var}(W \mid T)] & \ & =\operatorname{Var}\theta \phi(T)+\mathrm{E}\theta[\operatorname{Var}(W \mid T)] \ & \geq \operatorname{Var}\theta \phi(T) . & (\operatorname{Var}(W ; T) \geq 0)
\end{array}
$$

统计代写|统计推断代写Statistical inference代考|Loss Function Optimality

Our evaluations of point estimators have been based on their mean squared error performance. Mean squared error is a special case of a function called a loss function. The study of the performance, and the optimality, of estimators evaluated through loss functions is a branch of decision theory.

After the data $\mathbf{X}=\mathbf{x}$ are observed, where $X \sim f(\mathbf{x} \mid \theta), \theta \in \Theta$, a decision regarding $\theta$ is made. The set of allowable decisions is the action space, denoted by $\mathcal{A}$. Often in point estimation problems $\mathcal{A}$ is equal to $\Theta$, the parameter space, but this will change in other problems (such as hypothesis testing-see Section 8.3.5).

The loss function in a point estimation problem reflects the fact that if an action $a$ is close to $\theta$, then the decision $a$ is reasonable and little loss is incurred. If $a$ is far from $\theta$, then a large loss is incurred. The loss function is a nonnegative function that generally increases as the distance between $a$ and $\theta$ increases. If $\theta$ is real-valued, two commonly used loss functions are
absolute error loss, $\quad L(\theta, a)=|a-\theta|$,
and
squared error loss, $L(\theta, a)=(a-\theta)^2$.
Both of these loss functions increase as the distance between $\theta$ and $a$ increases, with minimum value $L(\theta, \theta)=0$. That is, the loss is minimum if the action is correct. Squared error loss gives relatively more penalty for large discrepancies, and absolute error loss gives relatively more penalty for small discrepancies. A variation of squared error loss, one that penalizes overestimation more than underestimation, is
$$
L(\theta, a)= \begin{cases}(a-\theta)^2 & \text { if } a<\theta \ 10(a-\theta)^2 & \text { if } a \geq \theta\end{cases}
$$
A loss that penalizes errors in estimation more if $\theta$ is near 0 than if $|\theta|$ is large, a relative squared error loss, is
$$
L(\theta, a)=\frac{(a-\theta)^2}{|\theta|+1}
$$

统计推断代考

统计代写|统计推断代写Statistical inference代考|Sufficiency and Unbiasedness

在前一节中，我们在寻找无偏估计时没有使用充分性的概念。我们现在将看到，考虑是否足够确实是一个强有力的工具。

本节的主要定理将充分统计量与无偏估计联系起来，与cram – rao定理一样，它是对一些著名定理的另一个巧妙应用。回想第4章，如果$X$和$Y$是任意两个随机变量，那么，假设期望存在，我们有
$$
\begin{aligned}
\mathrm{E} X & =\mathrm{E}[\mathrm{E}(X \mid Y)], \
\operatorname{Var} X & =\operatorname{Var}[\mathrm{E}(X \mid Y)]+\mathrm{E}[\operatorname{Var}(X \mid Y)] .
\end{aligned}
$$
利用这些工具，我们可以证明以下定理。
定理7.3.17 (Rao-Blackwell)设$W$为$\tau(\theta)$的任意无偏估计量，且设$T$为$\theta$的充分统计量。定义$\phi(T)=\mathrm{E}(W \mid T)$。然后$\mathrm{E}\theta \phi(T)=\tau(\theta)$和$\operatorname{Var}\theta \phi(T) \leq \operatorname{Var}\theta W$表示所有的$\theta$;也就是说，$\phi(T)$是$\tau(\theta)$的一致更好的无偏估计量。证明:从(7.3.13)我们得到$$ \tau(\theta)=\mathrm{E}\theta W=\mathrm{E}\theta[\mathrm{E}(W \mid T)]=\mathrm{E}\theta \phi(T),
$$
所以$\phi(T)$对$\tau(\theta)$是无偏的。还有，
$$
\begin{array}{rlr}
\operatorname{Var}\theta W & =\operatorname{Var}\theta[\mathrm{E}(W \mid T)]+\mathrm{E}\theta[\operatorname{Var}(W \mid T)] & \ & =\operatorname{Var}\theta \phi(T)+\mathrm{E}\theta[\operatorname{Var}(W \mid T)] \ & \geq \operatorname{Var}\theta \phi(T) . & (\operatorname{Var}(W ; T) \geq 0)
\end{array}
$$

统计代写|统计推断代写Statistical inference代考|Loss Function Optimality

我们对点估计器的评价是基于它们的均方误差性能。均方误差是一种叫做损失函数的特殊情况。通过损失函数评估估计器的性能和最优性的研究是决策理论的一个分支。

在观察到数据$\mathbf{X}=\mathbf{x}$(其中$X \sim f(\mathbf{x} \mid \theta), \theta \in \Theta$)之后，就会做出关于$\theta$的决策。允许的决策集合是动作空间，用$\mathcal{A}$表示。通常在点估计问题$\mathcal{A}$等于$\Theta$，参数空间，但这将改变在其他问题(如假设检验-见8.3.5节)。

点估计问题中的损失函数反映了一个事实，即如果一个动作$a$接近$\theta$，则该决策$a$是合理的，损失很小。如果$a$离$\theta$很远，那么就会产生很大的损失。损失函数是非负函数，通常随着$a$和$\theta$之间距离的增加而增加。如果$\theta$为实值，则两种常用的损失函数为
绝对误差损失，$\quad L(\theta, a)=|a-\theta|$，
和
平方误差损失，$L(\theta, a)=(a-\theta)^2$。
这两个损失函数都随着$\theta$和$a$之间距离的增加而增加，最小值为$L(\theta, \theta)=0$。也就是说，如果行动正确，损失是最小的。对于较大的差异，平方误差损失会给相对较多的惩罚，对于较小的差异，绝对误差损失会给相对较多的惩罚。平方误差损失的变化，对高估的惩罚大于对低估的惩罚，是
$$
L(\theta, a)= \begin{cases}(a-\theta)^2 & \text { if } a<\theta \ 10(a-\theta)^2 & \text { if } a \geq \theta\end{cases}
$$
如果$\theta$接近于0，相对平方误差损失(相对平方误差损失)比$|\theta|$较大时对估计误差的惩罚更大
$$
L(\theta, a)=\frac{(a-\theta)^2}{|\theta|+1}
$$

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|Sta732

Posted on 2023年7月21日2023年8月24日 by statistics-lab

统计代写|统计推断代写Statistical inference代考|The Likelihood Function

Let $f(\mathbf{x} \mid \theta)$ denote the joint pdf or pmf of the sample $\mathbf{X}=$ $\left(X_1, \ldots, X_n\right)$. Then, given that $\mathbf{X}=\mathbf{x}$ is observed, the function of $\theta$ defined by
$$
L(\theta \mid \mathbf{x})=f(\mathbf{x} \mid \theta)
$$
is called the likelihood function.
If $\mathbf{X}$ is a discrete random vector, then $L(\theta \mid \mathbf{x})=P_\theta(\mathbf{X}=\mathbf{x})$. If we compare the likelihood function at two parameter points and find that
$$
P_{\theta_1}(\mathbf{X}=\mathbf{x})=L\left(\theta_1 \mid \mathbf{x}\right)>L\left(\theta_2 \mid \mathbf{x}\right)=P_{\theta_2}(\mathbf{X}=\mathbf{x}),
$$
then the sample we actually observed is more likely to have occurred if $\theta=\theta_1$ than if $\theta=\theta_2$, which can be interpreted as saying that $\theta_1$ is a more plausible value for the true value of $\theta$ than is $\theta_2$. Many different ways have been proposed to use this information, but certainly it seems reasonable to examine the probability of the sample we actually observed under various possible values of $\theta$. This is the information provided by the likelihood function.

If $X$ is a continuous, real-valued random variable and if the pdf of $X$ is continuous in $x$, then, for small $\epsilon, P_\theta(x-\epsilon<X<x+\epsilon)$ is approximately $2 \epsilon f(x \mid \theta)=2 \epsilon L(\theta \mid x)$ (this follows from the definition of a derivative). Thus,
$$
\frac{P_{\theta_1}(x-\epsilon<X<x+\epsilon)}{P_{\theta_2}(x-\epsilon<X<x+\epsilon)} \approx \frac{L\left(\theta_1 \mid x\right)}{L\left(\theta_2 \mid x\right)},
$$
and comparison of the likelihood function at two parameter values again gives an approximate comparison of the probability of the observed sample value, $\mathbf{x}$.

Definition 6.3.1 almost sems to be defining the likelihood function to be the same as the pdf or pmf. The only distinction between these two functions is which variable is considered fixed and which is varying. When we consider the pdf or $\operatorname{pmf} f(\mathbf{x} \mid \theta)$, we are considering $\theta$ as fixed and $\mathbf{x}$ as the variable; when we consider the likelihood function $L(\theta \mid \mathbf{x})$, we are considering $\mathbf{x}$ to be the observed sample point and $\theta$ to be varying over all possible parameter values.

统计代写|统计推断代写Statistical inference代考|The Formal Likelihood Principle

For discrete distributions, the Likelihood Principle can be derived from two intuitively simpler ideas. This is also true, with some qualifications, for continuous distributions. In this subsection we will deal only with discrete distributions. Berger and Wolpert (1984) provide a thorough discussion of the Likelihood Principle in both the discrete and continuous cases. These results were first proved by Birnbaum (1962) in a landmark paper, but our presentation more closely follows that of Berger and Wolpert.
Formally, we define an experiment $E$ to be a triple $(\mathbf{X}, \theta,{f(\mathbf{x} \mid \theta)})$, where $\mathbf{X}$ is a random vector with $\operatorname{pmf} f(\mathbf{x} \mid \theta)$ for some $\theta$ in the parameter space $\Theta$. An experimenter, knowing what experiment $E$ was performed and having observed a particular sample $\mathbf{X}=\mathbf{x}$, will make some inference or draw some conclusion about $\theta$. This conclusion we denote by $\operatorname{Ev}(E, \mathbf{x})$, which stands for the evidence about $\theta$ arising from $E$ and $\mathbf{x}$.
Example 6.3.4 (Evidence function) Let $E$ be the experiment consisting of observing $X_1, \ldots, X_n$ iid $\mathrm{n}\left(\mu, \sigma^2\right), \sigma^2$ known. Since the sample mean, $\bar{X}$, is a sufficient statistic for $\mu$ and $\mathrm{E} \bar{X}=\mu$, we might use the observed value $\bar{X}=\bar{x}$ as an estimate of $\mu$. To give a measure of the accuracy of this estimate, it is common to report the standard deviation of $\bar{X}, \sigma / \sqrt{n}$. Thus we could define $\operatorname{Ev}(E, \mathbf{x})=(\bar{x}, \sigma / \sqrt{n})$. Here we see that the $\bar{x}$ coordinate depends on the observed sample $\mathbf{x}$, while the $\sigma / \sqrt{n}$ coordinate depends on the knowledge of $E$.

To relate the concept of an evidence function to something familiar we now restate the Sufficiency Principle of Section 6.2 in terms of these concepts.

FORMAL SUFFICIENCY PRINCIPLE: Consider experiment $E=(\mathbf{X}, \theta,{f(\mathbf{x} \mid \theta)})$ and suppose $T(\mathbf{X})$ is a sufficient statistic for $\theta$. If $\mathbf{x}$ and $\mathbf{y}$ are sample points satisfying $T(\mathbf{x})=T(\mathbf{y})$, then $\operatorname{Ev}(E, \mathbf{x})=\operatorname{Ev}(E, \mathbf{y})$.

统计推断代考

统计代写|统计推断代写Statistical inference代考|The Likelihood Function

设$f(\mathbf{x} \mid \theta)$为样本的联合pdf或pmf $\mathbf{X}=$$\left(X_1, \ldots, X_n\right)$。然后，假设观察到$\mathbf{X}=\mathbf{x}$，则$\theta$的函数定义为
$$
L(\theta \mid \mathbf{x})=f(\mathbf{x} \mid \theta)
$$
称为似然函数。
如果$\mathbf{X}$是离散随机向量，则$L(\theta \mid \mathbf{x})=P_\theta(\mathbf{X}=\mathbf{x})$。如果我们比较两个参数点的似然函数，我们会发现
$$
P_{\theta_1}(\mathbf{X}=\mathbf{x})=L\left(\theta_1 \mid \mathbf{x}\right)>L\left(\theta_2 \mid \mathbf{x}\right)=P_{\theta_2}(\mathbf{X}=\mathbf{x}),
$$
那么我们实际观察到的样本更有可能发生在$\theta=\theta_1$而不是$\theta=\theta_2$的情况下，这可以解释为$\theta_1$比$\theta_2$更有可能是$\theta$的真实值。已经提出了许多不同的方法来使用这些信息，但当然，在$\theta$的各种可能值下检查我们实际观察到的样本的概率似乎是合理的。这是由似然函数提供的信息。

如果$X$是一个连续的实值随机变量，如果$X$的pdf在$x$中是连续的，那么，对于小的$\epsilon, P_\theta(x-\epsilon<X<x+\epsilon)$近似为$2 \epsilon f(x \mid \theta)=2 \epsilon L(\theta \mid x)$(这是由导数的定义得出的)。因此，
$$
\frac{P_{\theta_1}(x-\epsilon<X<x+\epsilon)}{P_{\theta_2}(x-\epsilon<X<x+\epsilon)} \approx \frac{L\left(\theta_1 \mid x\right)}{L\left(\theta_2 \mid x\right)},
$$
在两个参数值处的似然函数的比较再次给出了观测样本值的概率的近似比较，$\mathbf{x}$。

定义6.3.1似乎将似然函数定义为与pdf或pmf相同。这两个函数之间的唯一区别是哪个变量是固定的，哪个是变化的。当我们考虑pdf或$\operatorname{pmf} f(\mathbf{x} \mid \theta)$时，我们认为$\theta$是固定的，$\mathbf{x}$是变量;当我们考虑似然函数$L(\theta \mid \mathbf{x})$时，我们认为$\mathbf{x}$是观测样本点，$\theta$在所有可能的参数值上都是变化的。

统计代写|统计推断代写Statistical inference代考|The Formal Likelihood Principle

对于离散分布，似然原理可以从两个直观上更简单的概念推导出来。这对于连续分布也是成立的，但有一些限定条件。在本小节中，我们只处理离散分布。Berger和Wolpert(1984)对离散和连续情况下的似然原理进行了深入的讨论。这些结果首先由Birnbaum(1962)在一篇具有里程碑意义的论文中证明，但我们的介绍更接近Berger和Wolpert。
形式上，我们将一个实验$E$定义为一个三元$(\mathbf{X}, \theta,{f(\mathbf{x} \mid \theta)})$，其中$\mathbf{X}$是一个随机向量，对于参数空间$\Theta$中的某个$\theta$来说是一个随机向量$\operatorname{pmf} f(\mathbf{x} \mid \theta)$。实验者，知道做了什么实验$E$，观察了一个特定的样本$\mathbf{X}=\mathbf{x}$，就会对$\theta$做出一些推断或得出一些结论。我们用$\operatorname{Ev}(E, \mathbf{x})$表示这个结论，它代表了从$E$和$\mathbf{x}$产生的关于$\theta$的证据。
例6.3.4(证据函数)设$E$为观察$X_1, \ldots, X_n$ iid $\mathrm{n}\left(\mu, \sigma^2\right), \sigma^2$的已知实验。由于样本均值$\bar{X}$是$\mu$和$\mathrm{E} \bar{X}=\mu$的充分统计量，我们可以使用观测值$\bar{X}=\bar{x}$作为$\mu$的估计值。为了给出这个估计的准确性，通常报告$\bar{X}, \sigma / \sqrt{n}$的标准偏差。因此我们可以定义$\operatorname{Ev}(E, \mathbf{x})=(\bar{x}, \sigma / \sqrt{n})$。在这里，我们看到$\bar{x}$坐标取决于观察到的样本$\mathbf{x}$，而$\sigma / \sqrt{n}$坐标取决于$E$的知识。

为了将证据功能的概念与我们熟悉的东西联系起来，我们现在根据这些概念重申第6.2节的充分性原则。

形式充分性原则:考虑实验$E=(\mathbf{X}, \theta,{f(\mathbf{x} \mid \theta)})$，假设$T(\mathbf{X})$是$\theta$的充分统计量。如果$\mathbf{x}$和$\mathbf{y}$是满足$T(\mathbf{x})=T(\mathbf{y})$的样本点，则$\operatorname{Ev}(E, \mathbf{x})=\operatorname{Ev}(E, \mathbf{y})$。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

统计代写|统计推断代写Statistical inference代考|Almost Sure Convergence

Posted on 2023年7月1日2023年10月17日 by statistics-lab

如果你也在怎样代写统计推断Statistical Inference 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。统计推断Statistical Inference是利用数据分析来推断概率基础分布的属性的过程。推断性统计分析推断人口的属性，例如通过测试假设和得出估计值。假设观察到的数据集是从一个更大的群体中抽出的。

统计推断Statistical Inference（可以与描述性统计进行对比。描述性统计只关注观察到的数据的属性，它并不依赖于数据来自一个更大的群体的假设。在机器学习中，推理一词有时被用来代替 “通过评估一个已经训练好的模型来进行预测”；在这种情况下，推断模型的属性被称为训练或学习（而不是推理），而使用模型进行预测被称为推理（而不是预测）；另见预测推理。

统计代写|统计推断代写Statistical inference代考|Almost Sure Convergence

A type of convergence that is stronger than convergence in probability is almost sure convergence (sometimes confusingly known as convergence with probability 1). This type of convergence is similar to pointwise convergence of a sequence of functions, except that the convergence need not occur on a set with probability 0 (hence the “almost” sure).

Definition 5.5.6 A sequence of random variables, $X_1, X_2, \ldots$, converges almost surely to a random variable $X$ if, for every $\epsilon>0$,
$$
P\left(\lim _{n \rightarrow \infty}\left|X_n-X\right|<\epsilon\right)=1
$$
Notice the similarity in the statements of Definitions 5.5.1 and 5.5.6. Although they look similar, they are very different statements, with Definition 5.5.6 much stronger. To understand almost sure convergence, we must recall the basic definition of a random variable as given in Definition 1.4.1. A random variable is a real-valued function defined on a sample space $S$. If a sample space $S$ has elements denoted by $s$, then $X_n(s)$ and $X(s)$ are all functions defined on $S$. Definition 5.5.6 states that $X_n$ converges to $X$ almost surely if the functions $X_n(s)$ converge to $X(s)$ for all $s \in S$ except perhaps for $s \in N$, where $N \subset S$ and $P(N)=0$. Example 5.5.7 illustrates almost sure convergence. Example 5.5.8 illustrates the difference between convergence in probability and almost sure convergence.

统计代写|统计推断代写Statistical inference代考|Convergence in Distribution

We have already encountered the idea of convergence in distribution in Chapter 2. Remember the properties of moment generating functions (mgfs) and how their convergence implies convergence in distribution (Theorem 2.3.12).

Definition 5.5.10 A sequence of random variables, $X_1, X_2, \ldots$, converges in distribution to a random variable $X$ if
$$
\lim {n \rightarrow \infty} F{X_n}(x)=F_X(x)
$$
at all points $x$ where $F_X(x)$ is continuous.
Example 5.5.11 (Maximum of uniforms) If $X_1, X_2, \ldots$ are iid uniform $(0,1)$ and $X_{(n)}=\max {1 \leq i \leq n} X_i$, let us examine if (and to where) $X{(n)}$ converges in distribution.
As $n \rightarrow \infty$, we expect $X_{(n)}$ to get close to 1 and, as $X_{(n)}$ must necessarily be less than 1 , we have for any $\varepsilon>0$,
$$
\begin{aligned}
P\left(\left|X_{(n)}-1\right| \geq \varepsilon\right) & =P\left(X_{(n)} \geq 1+\varepsilon\right)+P\left(X_{(n)} \leq 1-\varepsilon\right) \
& =0+P\left(X_{(n)} \leq 1-\varepsilon\right) .
\end{aligned}
$$
Next using the fact that we have an iid sample, we can write
$$
P\left(X_{(n)} \leq 1-\varepsilon\right)=P\left(X_i \leq 1-\varepsilon, i=1, \ldots n\right)=(1-\varepsilon)^n
$$

which goes to 0 . So we have proved that $X_{(n)}$ converges to 1 in probability. However, if we take $\varepsilon=t / n$, we then have
$$
P\left(X_{(n)} \leq 1-t / n\right)=(1-t / n)^n \rightarrow e^{-t},
$$
which, upon rearranging, yields
$$
P\left(n\left(1-X_{(n)}\right) \leq t\right) \rightarrow 1-e^{-t}
$$
that is, the random variable $n\left(1-X_{(n)}\right)$ converges in distribution to an exponential(1) random variable.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Almost Sure Convergence

一种比概率收敛更强的收敛类型是几乎确定收敛(有时被混淆地称为概率为1的收敛)。这种类型的收敛类似于函数序列的点向收敛，除了收敛不必发生在概率为0的集合上(因此是“几乎”确定)。

5.5.6一个随机变量序列$X_1, X_2, \ldots$几乎必然收敛于随机变量$X$，如果对每一个$\epsilon>0$，
$$
P\left(\lim _{n \rightarrow \infty}\left|X_n-X\right|<\epsilon\right)=1
$$
注意定义5.5.1和5.5.6语句中的相似之处。尽管它们看起来很相似，但它们是非常不同的语句，定义5.5.6要强得多。为了理解几乎肯定的收敛性，我们必须回顾定义1.4.1中给出的随机变量的基本定义。随机变量是定义在样本空间$S$上的实值函数。如果示例空间$S$包含以$s$表示的元素，则$X_n(s)$和$X(s)$都是在$S$上定义的函数。定义5.5.6指出，如果函数$X_n(s)$对所有$s \in S$都收敛于$X(s)$，那么$X_n$几乎肯定收敛于$X$，除了$s \in N$，其中$N \subset S$和$P(N)=0$。例5.5.7说明了几乎肯定的收敛性。例5.5.8说明了概率收敛和几乎确定收敛之间的区别。

统计代写|统计推断代写Statistical inference代考|Convergence in Distribution

我们已经在第2章中遇到了分布收敛的概念。记住矩生成函数(mgfs)的性质，以及它们的收敛如何意味着分布的收敛(定理2.3.12)。

5.5.10一个随机变量序列$X_1, X_2, \ldots$在分布上收敛于一个随机变量$X$ if
$$
\lim {n \rightarrow \infty} F{X_n}(x)=F_X(x)
$$
在$F_X(x)$连续的所有点$x$上。
例5.5.11(制服的最大值)如果$X_1, X_2, \ldots$是统一的$(0,1)$和$X_{(n)}=\max {1 \leq i \leq n} X_i$，让我们检查$X{(n)}$是否(以及到哪里)在分布上收敛。
对于$n \rightarrow \infty$，我们期望$X_{(n)}$接近于1，因为$X_{(n)}$必须小于1，对于任何$\varepsilon>0$，
$$
\begin{aligned}
P\left(\left|X_{(n)}-1\right| \geq \varepsilon\right) & =P\left(X_{(n)} \geq 1+\varepsilon\right)+P\left(X_{(n)} \leq 1-\varepsilon\right) \
& =0+P\left(X_{(n)} \leq 1-\varepsilon\right) .
\end{aligned}
$$
接下来利用我们有一个iid样本，我们可以写
$$
P\left(X_{(n)} \leq 1-\varepsilon\right)=P\left(X_i \leq 1-\varepsilon, i=1, \ldots n\right)=(1-\varepsilon)^n
$$

它趋于0。我们已经证明了$X_{(n)}$在概率上收敛于1。但是，如果我们取$\varepsilon=t / n$，我们就有
$$
P\left(X_{(n)} \leq 1-t / n\right)=(1-t / n)^n \rightarrow e^{-t},
$$
重新排列后会产生什么
$$
P\left(n\left(1-X_{(n)}\right) \leq t\right) \rightarrow 1-e^{-t}
$$
即随机变量$n\left(1-X_{(n)}\right)$在分布上收敛于指数型(1)随机变量。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写