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## 统计代写|抽样调查代考Survey sampling代写|DOMAIN ESTIMATION

Let $D$ be a domain of interest within a population $U=(1, \ldots$, $i, \ldots, N)$. Let $N_D$ be the unknown size of $D$. Let a sample $s$ of size $n$ be drawn from $U$ with a probability $p(s)$ according to a design $p$ admitting positive inclusion probabilities $\pi_i, \pi_{i j}$. Let for $i=1,2, \ldots, N$
\begin{aligned} & I_{D i}=1(0) \quad \text { if } \quad i \in D(i \notin D) \ & Y_{D i}=Y_i(0) \text { if } \quad i \in D(i \notin D) \text {. } \ & \end{aligned}
Then the unknown domain size, total, and mean are, respectively,
$$N_D=\sum_1^N I_{D i}, T_D=\sum_1^N Y_{D i} \quad \text { and } \quad \bar{T}D=\frac{T_D}{N_D}$$ In analogy to $\underline{Y}=\left(Y_1, \ldots, Y_i, \ldots, Y_N\right)^{\prime}$ we write $\underline{I}_D=\left(I{D 1}, \ldots\right.$, $\left.I_{D i}, \ldots, I_{D N}\right)^{\prime}$ and $\underline{Y}D=\left(Y{D 1}, \ldots, Y_{D i}, \ldots, Y_{D N}\right)^{\prime}$. Then, corresponding to any estimator $t=t(s, \underline{Y})=\hat{Y}$, for $Y=\Sigma_1^N Y_i$ we may immediately choose estimators for $N_D$ and $T_D$, respectively,
$$\widehat{N}_D=t\left(s, \underline{I}_D\right) \quad \text { and } \quad \widehat{T}_D=t\left(s, \underline{Y}_D\right) .$$
It may then be a natural step to take the estimator $\widehat{T}_D$ for $\bar{T}_D$ as
$$\widehat{T}_D=\frac{\widehat{T}_D}{\widehat{N}_D}$$

## 统计代写|抽样调查代考Survey sampling代写|POSTSTRATIFICATION

Suppose a finite population $U=(1, \ldots, i, \ldots, N)$ of $N$ units consists of $L$ post-strata of known sizes $N_h, h=1, \ldots, L$ but unknown compositions with respective post-strata totals $Y_h=$ $\sum_i^{N_h} Y_{h i}$ and means $\bar{Y}h=Y_h / N_h, h=1, \ldots, L$. Let a simple random sample $s$ of size $n$ have been drawn from $U$ yielding the sample configuration $\underline{n}=\left(n_1, \ldots, n_h, \ldots, n_L\right)$ where $n_h(\geq 0)$ is the number of units of $s$ coming from the $h$ th post-stratum, $h=1, \ldots, L, \sum{h=1}^L n_h=n$. In order to estimate $\bar{Y}=\Sigma W_h \bar{Y}h$, writing $W_h=\frac{N_h}{N}, h=1, \ldots, L$ we proceed as follows. Let $I_h=1(0)$ if $n_h>0\left(n_h=0\right)$. Then, $$E\left(I_h\right)=\operatorname{Prob}\left(I_h=1\right)=1-\left(\begin{array}{c} N-N_h \ n \end{array}\right) /\left(\begin{array}{c} N \ n \end{array}\right), h=1, \ldots, L .$$ For $\bar{Y}$ a reasonable estimator may be taken as $$t{p s t}=t_{p s t}(\underline{Y})=\frac{\sum W_h \bar{y}_h I_h / E\left(I_h\right)}{\sum W_h I_h / E\left(I_h\right)}$$
writing $\bar{y}_h$ as the mean of the $n_h$ units in the sample consisting of members of the $h$ th post-stratum, if $n_h>0$; if $n_h=0$, then $\bar{y}_h$ is taken as $\bar{Y}_h$. It follows that $x=\sum W_h \bar{y}_h I_h / E\left(I_h\right)$ is an unbiased estimator for $\bar{Y}$ and $b=\sum W_h I_h / E\left(I_h\right)$ an unbiased estimator for 1. Yet, instead of taking just a as an unbiased estimator for $\bar{Y}$, this biased estimator of the ratio form $\frac{x}{b}$ is proposed by DOSS, HARTLEY and SOMAYAJULU (1979) because it has the following linear invariance property not shared by itself:

Assume $Y_i=\alpha+\beta Z_i$; then $\bar{y}h=\alpha+\beta \bar{z}_h$ and $t{p s t}(\underline{Y})=$ $\alpha+\beta t_{p s t}(\underline{Z})$, with obvious notations. Further properties of $t_{p s t}$ have been investigated by Doss et al. (1979) but are too complicated to merit further discussion here.

# 抽样调查代写

## 统计代写|抽样调查代考Survey sampling代写|DOMAIN ESTIMATION

\begin{aligned} & I_{D i}=1(0) \quad \text { if } \quad i \in D(i \notin D) \ & Y_{D i}=Y_i(0) \text { if } \quad i \in D(i \notin D) \text {. } \ & \end{aligned}

$$N_D=\sum_1^N I_{D i}, T_D=\sum_1^N Y_{D i} \quad \text { and } \quad \bar{T}D=\frac{T_D}{N_D}$$与$\underline{Y}=\left(Y_1, \ldots, Y_i, \ldots, Y_N\right)^{\prime}$类似，我们写$\underline{I}D=\left(I{D 1}, \ldots\right.$, $\left.I{D i}, \ldots, I_{D N}\right)^{\prime}$和$\underline{Y}D=\left(Y{D 1}, \ldots, Y_{D i}, \ldots, Y_{D N}\right)^{\prime}$。然后，对应于任意估计量$t=t(s, \underline{Y})=\hat{Y}$，对于$Y=\Sigma_1^N Y_i$，我们可以立即分别选择$N_D$和$T_D$的估计量，
$$\widehat{N}_D=t\left(s, \underline{I}_D\right) \quad \text { and } \quad \widehat{T}_D=t\left(s, \underline{Y}_D\right) .$$

$$\widehat{T}_D=\frac{\widehat{T}_D}{\widehat{N}_D}$$

## 统计代写|抽样调查代考Survey sampling代写|POSTSTRATIFICATION

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。

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## 统计代写|抽样调查代考Survey sampling代写|Unbiased Estimation of $Y$

Let $E_1, V_1$ denote expectation variance operators for the sampling design in the first stage and $E_L, V_L$ those in the later stages. Let $R_i$ be independent variables satisfying
(a) $E_L\left(R_i\right)=Y_i$,
(b) $V_L\left(R_i\right)=V_i$ or
(c) $V_L\left(R_i\right)=V_{s i}$
and let there exist (b) $)^{\prime}$ random variables $v_i$ such that $E_L\left(v_i\right)=$ $V_i$ or (c)’ random variables $v_{s i}$ such that $E_L\left(v_{s i}\right)=V_{s i}$.

Let $E=E_1 E_L=E_L E_1$ be the overall expectation and $V=$ $E_1 V_L+V_1 E_L=E_L V_1+V_L E_1$ the overall variance operators. CHAUDHURI, ADHIKARI and DIHIDAR (2000a, 2000b) have illustrated how these commutativity assumptions may be valid in the context of survey sampling.
Let
\begin{aligned} t_b & =\sum b_{s i} I_{s i} Y_i, \ M_1\left(t_b\right) & =E_1\left(t_b-Y\right)^2=\sum \sum d_{i j} y_i y_j, \ d_{i j} & =E_1\left(b_{s i} I_{s i}-1\right)\left(b_{s j} I_{s j}-1\right), \end{aligned}

$d_{s i j}$ be constants free of $Y$ such that
$$E_1\left(d_{s i j} I_{s i j}\right)=d_{i j} \forall_{i, j} \text { in } U .$$
Let $w_i$ ‘s be certain non-zero constants. Then, one gets
\begin{aligned} M_1\left(t_b\right)= & -\sum \sum_{i<j} d_{i j} w_i w_j\left(\frac{Y_i}{w_i}-\frac{Y_j}{w_j}\right)^2 \ & +\sum \beta_i \frac{Y_i^2}{w_i} \text { when } \beta_i=\sum_{j=1}^N d_{i j} w_j . \end{aligned}

## 统计代写|抽样调查代考Survey sampling代写|PPSWR Sampling of First-Stage Units

First, from DES RAJ (1968) we note the following. Suppose a PPSWR sample of fsus is chosen in $n$ draws from $U$ using normed size measures $P_i\left(0<P_i<i, \Sigma P_i=1\right)$. Writing $y_r\left(p_r\right)$ for the $Y_i\left(p_i\right)$ value for the unit chosen on the $r$ th draw, $(r=$ $1, \ldots, n)$ the HANSEN-HURWITZ estimator
$$t_{H H}=\frac{1}{n} \sum_{n=1}^n \frac{y_r}{p_r}$$

might be used to estimate $Y$ because $E_p\left(t_{H H}\right)=Y$ if $Y_i$ could be ascertained. But since $Y_i$ ‘s are not ascertainable, suppose that each time an fsu $i$ appears in one of the $n$ independent draws by PPSWR method, an independent subsample of elements is selected in subsequent stages in such a manner that estimators $\hat{y}r$ for $y_r$ are available such that $E_L\left(\hat{y}_r\right)=y_r$ and $V_L\left(\hat{y}_r\right)=\sigma_r^2$ with uncorrelated $y_1, y_2, \ldots, y_n$. Then, DAS RAJ’s (1968) proposed estimator for $Y$ is $$e_H=\frac{1}{n} \sum{r=1}^n \frac{\hat{y}r}{p_r}$$ for which the variance is \begin{aligned} V\left(e_H\right) & =V_p\left(t{H H}\right)+E_p\left[\frac{1}{n^2} \sum_{r=1}^n \frac{\sigma_r^2}{p_r^2}\right] \ & =\frac{1}{n} \sum P_i\left(\frac{Y_i}{P_i}-Y\right)^2+\frac{1}{n} \sum_1^N \frac{\sigma_i^2}{P_i} \ & =V_H, \text { say. } \end{aligned}
It follows that
\begin{aligned} & v_H=\frac{1}{2 n^2(n-1)} \sum_{\substack{r=1 r^{\prime}=1 \ r \neq r^{\prime}}}^n\left(\frac{\hat{y}{r^{\prime}}}{p{r^{\prime}}}-\frac{\hat{y}r}{p_r}\right)^2 \ & r \neq r^{\prime} \ & \end{aligned} is an unbiased estimator for $V_H$ because \begin{aligned} E_l\left(v_H\right) & =\frac{1}{2 n^2(n-1)} \sum{r \neq r^{\prime}}\left[\frac{y_r^2}{p_r^2}+\frac{y_{r^{\prime}}^2}{p_{r^{\prime}}^2}+\frac{\sigma_r^2}{p_r^2}+\frac{\sigma_{r^{\prime}}^2}{p_{r^{\prime}}^2}-2 \frac{y_r}{p_r} \frac{y_{r^{\prime}}}{p_{r^{\prime}}}\right] \ E v_H & =E_p E_L\left(v_H\right)=\frac{1}{n}\left(\sum \frac{Y_i^2}{P_i}-Y^2\right)+\frac{1}{n} \sum \frac{\sigma_i^2}{P_i} \ & =\frac{1}{n} \sum P_i\left(\frac{Y_i}{P_i}-Y\right)^2+\frac{1}{n} \sum \frac{\sigma_i^2}{P_i}=V\left(e_H\right) . \end{aligned}

# 抽样调查代写

## 统计代写|抽样调查代考Survey sampling代写|Unbiased Estimation of $Y$

(a) $E_L\left(R_i\right)=Y_i$;
(b) $V_L\left(R_i\right)=V_i$或
(c) $V_L\left(R_i\right)=V_{s i}$

\begin{aligned} t_b & =\sum b_{s i} I_{s i} Y_i, \ M_1\left(t_b\right) & =E_1\left(t_b-Y\right)^2=\sum \sum d_{i j} y_i y_j, \ d_{i j} & =E_1\left(b_{s i} I_{s i}-1\right)\left(b_{s j} I_{s j}-1\right), \end{aligned}

$d_{s i j}$ 使用不含$Y$的常量
$$E_1\left(d_{s i j} I_{s i j}\right)=d_{i j} \forall_{i, j} \text { in } U .$$

\begin{aligned} M_1\left(t_b\right)= & -\sum \sum_{i<j} d_{i j} w_i w_j\left(\frac{Y_i}{w_i}-\frac{Y_j}{w_j}\right)^2 \ & +\sum \beta_i \frac{Y_i^2}{w_i} \text { when } \beta_i=\sum_{j=1}^N d_{i j} w_j . \end{aligned}

## 统计代写|抽样调查代考Survey sampling代写|PPSWR Sampling of First-Stage Units

$$Z_i=\frac{Y_i}{X_i}$$

$$M_p(t)=\sum \sum Z_i Z_j\left(X_i X_j d_{i j}\right)$$

$$\sum_i \sum_j X_i X_j d_{i j}=0$$

$$M_p(t)=\sum \sum Z_i Z_j a_{i j}$$
$Z_i ; i=1, \ldots, N$中的非负二次型是否服从
$$\sum_i \sum_j a_{i j}=0$$

$$\sum_j a_{i j}=0 .$$

\begin{aligned} M_p(t) & =-\sum_{i<j} \sum_i\left(Z_i-Z_j\right)^2 a_{i j} \ & =-\sum_{i<j} \sum_i\left(\frac{Y_i}{X_i}-\frac{Y_j}{X_j}\right)^2 X_i X_j d_{i j} . \end{aligned}

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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。

Posted on Categories:Survey sampling, 抽样调查, 统计代写, 统计代考

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## 统计代写|抽样调查代考Survey sampling代写|Use of Chebyshev Inequality

Here we determine sample size by keeping permissible error to a certain level with probability exceeding a certain preassigned value $(1-\alpha)$. Let $t$ be an unbiased estimator for $\bar{Y}$. Using the Chebyshev Inequality, we have
$$\operatorname{Prob}[|t-\bar{Y}| \leq d] \geq 1-\frac{V(t)}{d^2}$$
The sample size $n$ is determined from the relation $1-\frac{V(t)}{d^2}=1-\alpha$ which is equivalent to
$$\frac{V(t)}{d^2}=\alpha$$

For an SRSWOR design with $t=\bar{\gamma}(s)$, Eq. (3.5.5) yields
\begin{aligned} n & =\left[\frac{1}{N}+\alpha \frac{N-1}{N}\left(\frac{d}{\sigma_\gamma}\right)^2\right]^{-1} \ & =N\left[1+\gamma^2 \alpha(N-1)\right]^{-1} \end{aligned}
where $d=\gamma \quad \sigma_\gamma$
For an SRSWR design with $t=\bar{\gamma}\left(s_o\right)$, Eq. (3.5.6) yields
$$n=\frac{\sigma_\gamma^2}{\alpha d^2}=\frac{1}{\alpha \gamma^2}$$
Substituting $\alpha=0.05$ and $\gamma=1$ in (3.5.8), we get $n=20$, i.e., selection of $n=20$ ensures
$$\operatorname{Prob}\left[\left|\bar{Y}\left(s_o\right)-\bar{Y}\right| \leq \sigma_\gamma\right] \geq 0.95$$

## 统计代写|抽样调查代考Survey sampling代写|Simple Random Sampling Without Replacement

Let us suppose that a population consists of $N$ units of which $N_A(=N \pi)$ units possess certain rare characteristics $A$ and that the remaining $N_B=N-N_A$ do not possess this characteristic. Let $X$ be the number of units required to be drawn to get $m$ units that possess characteristic $A$. Here $X$ is a random variable whose probability distribution depends on $m$ and $N_A$. The probability distribution of $X$ is given by
\begin{aligned} P(X=x) & =f\left(x \mid N_A, m\right) \ & =\text { Probability of getting }(m-1) \text { units bearing characteristic } \end{aligned}
$A$ in the first $x-1$ draws and at the $x$ th draw one unit is selected from the group $A$.
\begin{aligned} & =\frac{\left(\begin{array}{c} N_A \ m-1 \end{array}\right)\left(\begin{array}{c} N_B \ (x-1)-(m-1) \end{array}\right)}{\left(\begin{array}{c} N \ x-1 \end{array}\right)} \frac{N_A-(m-1)}{N-(x-1)} ; \ & x \geq m, m+1, \ldots \ & \end{aligned}

Theorem 3.6.1
(i) An unbiased estimator of $\pi$ is
$$\widehat{\pi}=\frac{m-1}{x-1}$$
(ii) An unbiased estimator for the variance of $\widehat{\pi}$ is
$$\widehat{V}(\widehat{\pi})=\frac{\widehat{\pi}(1-\widehat{\pi})}{x-2}\left(1-\frac{x-1}{N}\right)$$

# 抽样调查代写

## 统计代写|抽样调查代考Survey sampling代写|Use of Chebyshev Inequality

$$\operatorname{Prob}[|t-\bar{Y}| \leq d] \geq 1-\frac{V(t)}{d^2}$$

$$\frac{V(t)}{d^2}=\alpha$$

$$n=\left[\frac{1}{N}+\alpha \frac{N-1}{N}\left(\frac{d}{\sigma_\gamma}\right)^2\right]^{-1}=N\left[1+\gamma^2 \alpha(N-1)\right]^{-1}$$

$$n=\frac{\sigma_\gamma^2}{\alpha d^2}=\frac{1}{\alpha \gamma^2}$$

$$\operatorname{Prob}\left[\left|\bar{Y}\left(s_o\right)-\bar{Y}\right| \leq \sigma_\gamma\right] \geq 0.95$$

## 统计代写|抽样调查代考Survey sampling代写|Simple Random Sampling Without Replacement

$N_B=N-N_A$ 不具备这个特性。让 $X$ 是需要抽取的单位数 $m$ 具有特征的单位 $A$. 这

$P(X=x)=f\left(x \mid N_A, m\right) \quad=$ Probability of getting $(m-1)$ units bearing characteristic
$A$ 在第一 $x-1$ 绘制并在 $x$ th抽取一个单位从组中选择 $A$.
$$=\frac{\left(N_A m-1\right)\left(N_B(x-1)-(m-1)\right)}{(N x-1)} \frac{N_A-(m-1)}{N-(x-1)} ; \quad x \geq m, m+1, \ldots$$

(i) 的无偏估计量 $\pi$ 是
$$\widehat{\pi}=\frac{m-1}{x-1}$$
(ii) 方差的无偏估计量 $\widehat{\pi}$ 是
$$\widehat{V}(\widehat{\pi})=\frac{\widehat{\pi}(1-\widehat{\pi})}{x-2}\left(1-\frac{x-1}{N}\right)$$

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。