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We have seen in Section $2.5$ that in almost all practical situations, the UMVUE for a finite population total does not exist. The criterion of admissibility is used to guard against the selection of a bad estimator.
An estimator $T$ is said to be admissible in the class $C$ of estimators for a given sampling design $p$ if there does not exist any other estimator in the class $C$ better than $T$. In other words, there does not exist an alternative estimator $T^(\neq T) \in C$, for which following inequalities hold. (i) $V_p\left(T^\right) \leq V_p(T) \quad \forall T^(\neq T) \in C$ and $\mathbf{y} \in R^N$ and (ii) $V_p\left(T^\right)<V_p(T)$ for at least one $\mathbf{y} \in R^N$
Theorem 2.6.1
In the class of linear homogeneous unbiased estimators $C_{l h}$, the HTE $\widehat{Y}{h t}$ based on a sampling design $p$ with $\pi_i>0 \forall i=1, \ldots, N$ is admissible for estimating the population total $Y$. Proof The proof is immediate from Theorem 2.5.2. Since $\widehat{Y}{h t}$ is the UMVUE when $\mathbf{y} \in R_0$, we cannot find an estimator $\forall T^*\left(\neq \widehat{Y}{h t}\right) \in C{l h}$ for which (2.6.1) holds.

## 统计代写|抽样理论代考Sampling Theory代写|SUFFICIENCY IN FINITE POPULATION

An estimator $e(s, \mathbf{y})$ is said to be inadmissible in a class of estimators $C$ if there exists an estimator $e^*(s, \gamma)(\in C)$ better than $e(s, \gamma)$. Hence it is natural to question how an inadmissible estimator could be improved. The method of improvement of an inadmissible estimator with the aid of sufficient statistics is known as Rao-Blackwellization. The concept of sufficient statistics in survey sampling was introduced by Basu (1958), while the concepts of linear sufficiency, distribution-free sufficient statistics, and Bayesian sufficiency were also introduced by Godambe (1966, 1968). Details have been given by Cassel et al. (1977), Chaudhuri and Stenger (1992), and Thompson and Seber (1996).

Let $s=\left(i_1, \ldots, i_k, \ldots, i_{n_s}\right)$ be an ordered sample of size $n_s$ selected from a population $U$ with probability $p(s)$ using a sampling design $p$, where the unit $i_k$ is selected at the $k$ th draw. After selection of sample $s$, the responses $y_{i_1}, \ldots, y_{i_{n s}}$ were obtained from sampled units $i_1, \ldots, i_{n_s}$, respectively. The ordered data based on the ordered sample $s$ are denoted by $d=\left{\left(i_1, y_{i_1}\right), \ldots,\left(i_k, y_{i_{n_s}}\right)\right}=\left(i_k, y_{i_k} ; i_k \in s\right)$.

Let $\widetilde{s}=\left(j_1, j_2, \ldots, j_{v_s}\right)$ with $j_1<j_2<\ldots<j_{v_s}$ be the unordered sample obtained from $s$ by taking distinct units of $s$ and arranging them in ascending order of their labels. Let us denote the operator $r$, which transforms the ordered sample $s$ to the unordered sample $\widetilde{s}$, i.e., $r(s)=\widetilde{s}$. The unordered data are denoted by $\tilde{d}=\left{\left(j_1, y_{j_1}\right), \ldots,\left(j_{v_s}, \gamma_{j_{v_s}}\right)\right}=\left(j, \gamma_j ; j \in \widetilde{s}\right)$. We define the operator $R$ to get unordered data $\tilde{d}$ from ordered data $d$, i.e., $R(d)=\tilde{d}$.
Example 2.7.1
Let $U=(1,2,3,4,5), \quad \mathbf{y}=(10,15,15,20,10), \quad$ and $\quad s=(5,2,5)$. Here $y_1=10, \quad y_2=15, \quad y_3=15, \quad y_4=20, \quad$ and $\quad y_5=10 ; \quad r(s)=\widetilde{s}=(2,5)$, $d={(5,10),(2,15),(5,10)}$ and $R(d)=\widetilde{d}={(2,15),(5,10)}$.

# 抽样理论代写

## 统计代写|抽样理论代考Sampling Theory代写|SUFFICIENCY IN FINITE POPULATION

Basu（1958）引入，而线性充分性、无分布充分统计和贝叶斯充分性的概念也由Godambe（1966、1968）引入。Cassel 等人 给出了详细信息。(1977)、Chaudhuri 和 Stenger (1992)，以及 Thompson 和 Seber (1996)。

(left 劰分或无法识别的分隔符 示运算符 $r$, 转换有序样本 $s$ 到无序样本 $\tilde{s}$ ，那是， $r(s)=\tilde{s}$. 无序数据表示为 $\backslash l$ left 缺少或无法识别的分隔符 我 们定义运算符 $R$ 获取无序数据 $\tilde{d}$ 来自有序数据 $d$ ，那是， $R(d)=\tilde{d}$.

$y_1=10, \quad y_2=15, \quad y_3=15, \quad y_4=20, \quad$ 和 $\quad y_5=10 ; \quad r(s)=\tilde{s}=(2,5), d=(5,10),(2,15),(5,10)$ 和 $R(d)=\widetilde{d}=(2,15),(5,10)$

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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。