Posted on Categories:Statistical inference, 统计代写, 统计代考, 统计推断

# 统计代写|统计推断代考Statistical Inference代写|Sta732

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 统计代写|统计推断代考Statistical Inference代写|Sufficient Statistics

A sufficient statistic is formally defined in the following way.
Definition 6.2.1 $\quad$ A statistic $T(\mathbf{X})$ is a sufficient statistic for $\theta$ if the conditional distribution of the sample $\mathbf{X}$ given the value of $T(\mathbf{X})$ does not depend on $\theta$.

If $T(\mathbf{X})$ has a continuous distribution, then $P_\theta(T(\mathbf{X})=t)=0$ for all values of $t$. A more sophisticated notion of conditional probability than that introduced in Chapter 1 is needed to fully understand Definition 6.2.1 in this case. A discussion of this can be found in more advanced texts such as Lehmann (1986). We will do our calculations in the discrete case and will point out analogous results that are true in the continuous case.

To understand Definition 6.2.1, let $t$ be a possible value of $T(\mathbf{X})$, that is, a value such that $P_\theta(T(\mathbf{X})=t)>0$. We wish to consider the conditional probability $P_\theta(\mathbf{X}=$ $\mathbf{x} \mid T(\mathbf{X})=t)$. If $\mathbf{x}$ is a sample point such that $T(\mathbf{x}) \neq t$, then clearly $P_\theta(\mathbf{X}=\mathbf{x} \mid T(\mathbf{X})=$ $t)=0$. Thus, we are interested in $P(\mathbf{X}=\mathbf{x} \mid T(\mathbf{X})=T(\mathbf{x}))$. By the definition, if $T(\mathbf{X})$ is a sufficient statistic, this conditional probability is the same for all values of $\theta$ so we have omitted the subscript.

A sufficient statistic captures all the information about $\theta$ in this sense. Consider Experimenter 1, who observes $\mathbf{X}=\mathbf{x}$ and, of course, can compute $T(\mathbf{X})=T(\mathbf{x})$. To make an inference about $\theta$ he can use the information that $\mathbf{X}=\mathbf{x}$ and $T(\mathbf{X})=T(\mathbf{x})$. Now consider Experimenter 2, who is not told the value of $\mathbf{X}$ but only that $T(\mathbf{X})=$ $T(\mathbf{x})$. Experimenter 2 knows $P(\mathbf{X}=\mathbf{y} \mid T(\mathbf{X})=T(\mathbf{x}))$, a probability distribution on

$A_{T(\mathbf{x})}={\mathbf{y}: T(\mathbf{y})=T(\mathbf{x})}$, because this can be computed from the model without knowledge of the true value of $\theta$. Thus, Experimenter 2 can use this distribution and a randomization device, such as a random number table, to generate an observation $\mathbf{Y}$ satisfying $P(\mathbf{Y}=\mathbf{y} \mid T(\mathbf{X})=T(\mathbf{x}))=P(\mathbf{X}=\mathbf{y} \mid T(\mathbf{X})=T(\mathbf{x}))$. It turns out that, for each value of $\theta, \mathbf{X}$ and $\mathbf{Y}$ have the same unconditional probability distribution, as we shall see below. So Experimenter 1, who knows $\mathbf{X}$, and Experimenter 2, who knows $\mathbf{Y}$, have equivalent information about $\theta$. But surely the use of the random number table to generate $\mathbf{Y}$ has not added to Experimenter 2’s knowledge of $\theta$. All his knowledge about $\theta$ is contained in the knowledge that $T(\mathbf{X})=T(\mathbf{x})$. So Experimenter 2, who knows only $T(\mathbf{X})=T(\mathbf{x})$, has just as much information about $\theta$ as does Experimenter 1 , who knows the entire sample $\mathbf{X}=\mathbf{x}$.

To complete the above argument, we need to show that $\mathbf{X}$ and $\mathbf{Y}$ have the same unconditional distribution, that is, $P_\theta(\mathbf{X}=\mathbf{x})=P_\theta(\mathbf{Y}=\mathbf{x})$ for all $\mathbf{x}$ and $\theta$. Note that the events ${\mathbf{X}=\mathbf{x}}$ and ${\mathbf{Y}=\mathbf{x}}$ are both subsets of the event ${T(\mathbf{X})=T(\mathbf{x})}$. Also recall that
$$P(\mathbf{X}=\mathbf{x} \mid T(\mathbf{X})=T(\mathbf{x}))=P(\mathbf{Y}=\mathbf{x} \mid T(\mathbf{X})=T(\mathbf{x}))$$
and these conditional probabilities do not depend on $\theta$. Thus we have
\begin{aligned} P_\theta(\mathbf{X} & =\mathbf{x}) \ & =P_\theta(\mathbf{X}=\mathbf{x} \text { and } T(\mathbf{X})=T(\mathbf{x})) \ & =P(\mathbf{X}=\mathbf{x} \mid T(\mathbf{X})=T(\mathbf{x})) P_\theta(T(\mathbf{X})=T(\mathbf{x})) \quad\left(\begin{array}{c} \text { definition of } \ \text { conditional probability } \end{array}\right) \ & =P(\mathbf{Y}=\mathbf{x} \mid T(\mathbf{X})=T(\mathbf{x})) P_\theta(T(\mathbf{X})=T(\mathbf{x})) \ & =P_\theta(\mathbf{Y}=\mathbf{x} \text { and } T(\mathbf{X})=T(\mathbf{x})) \ & =P_\theta(\mathbf{Y}=\mathbf{x}) . \end{aligned}

## 统计代写|统计推断代考Statistical Inference代写|Ancillary Statistics

In the preceding sections, we considered sufficient statistics. Such statistics, in a sense, contain all the information about $\theta$ that is available in the sample. In this section we introduce a different sort of statistic, one that has a complementary purpose.

Definition 6.2.16 A statistic $S(\mathbf{X})$ whose distribution does not depend on the parameter $\theta$ is called an ancillary statistic.

Alone, an ancillary statistic contains no information about $\theta$. An ancillary statistic is an observation on a random variable whose distribution is fixed and known, unrelated to $\theta$. Paradoxically, an ancillary statistic, when used in conjunction with other statistics, sometimes does contain valuable information for inferences about $\theta$. We will investigate this behavior in the next section. For now, we just give some examples of ancillary statistics.

Example 6.2.17 (Uniform ancillary statistic) As in Example 6.2.15, let $X_1, \ldots, X_n$ be iid uniform observations on the interval $(\theta, \theta+1),-\infty<\theta<\infty$. Let $X_{(1)}<\cdots<X_{(n)}$ be the order statistics from the sample. We show below that the range statistic, $R=X_{(n)}-X_{(1)}$, is an ancillary statistic by showing that the pdf of $R$ does not depend on $\theta$. Recall that the cdf of each $X_i$ is
$$F(x \mid \theta)= \begin{cases}0 & x \leq \theta \ x-\theta & \theta<x<\theta+1 \ 1 & \theta+1 \leq x\end{cases}$$
Thus, the joint pdf of $X_{(1)}$ and $X_{(n)}$, as given by (5.5.7), is
$$g\left(x_{(1)}, \boldsymbol{x}{(n)} \mid \theta\right)= \begin{cases}n(n-1)\left(\boldsymbol{x}{(n)}-\boldsymbol{x}{(1)}\right)^{n-2} & \theta{(1)}<x_{(n)}<\theta+1 \ 0 & \text { otherwise. }\end{cases}$$

# 统计推断代写

## 统计代写|统计推断代考Statistical Inference代写|Sufficient Statistics

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## 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。