## 统计代写|抽样调查作业代写sampling theory of survey代考|STAT532

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## 统计代写|抽样调查作业代写sampling theory of survey代考|NONRESPONSE

To cite a simple example, suppose that unit $i$, provided it is included in a sample $s$, responds with probability $q_i, q_i$ not depending on $s$ or $Y=\left(Y_i, \ldots, Y_N\right)$. Suppose $n$ units are drawn by SRSWOR and define
$$M_i= \begin{cases}1 & \text { if unit } i \text { is sampled and responds } \ 0 & \text { otherwise }\end{cases}$$
Consider the arithmetic mean
$$\bar{y}=\frac{\sum_1^N M_i Y_i}{\sum_1^N M_i}$$
of all observations as an estimator of $\bar{Y}$. Then
$$E M_i=\frac{n}{N} q_i$$
and $E \bar{y}$ is asymptotically equal to
$$\frac{\sum q_i Y_i}{\sum q_i}$$
The bias
$$\sum\left(\frac{q_i}{\sum q_i}-\frac{1}{N}\right) Y_i$$
is negligible only if approximately
$$q_i=\frac{1}{N} \sum q_i .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|CALLBACKS

HANSEN and HURWITZ (1946) gave an elegant procedure for callbacks to tackle nonresponse problems later modified by SRINATH (1971) and J. N. K. RAO (1973), briefly described below. The population is conceptually dichotomized with $W_1\left(W_2=1-W_1\right)$ and $\widehat{Y}_1\left(\widehat{Y}_2=\left[\widehat{Y}-W_1 \widehat{Y}_1\right] / W_2\right)$ as the proportion of respondents (nonrespondents) and mean of respondents (nonrespondents) and an SRSWOR of size $n$ yields proportions $w_1=n_1 / n$ and $w_2=1-w_1=1-n_1 / n=n_2 / n$ of respondents and nonrespondents, respectively. Choosing a suitable number $K>1$ an SRSWOR of size $m_2=n_2 / K$, assumed to be an integer, is then drawn from the initial $n_2$ sample nonrespondents. Supposing that more expensive and persuasive procedures are followed in this second phase so that each of the $m_2$ units called back now responds, let $\bar{y}1$ and $\bar{y}{22}$ denote the first-phase and second-phase sample means based respectively on $n_1$ and $m_2$ respondents. Then, $\bar{Y}$ may be estimated by $\bar{y}d=w_1 \bar{y}_1+w_2 \bar{y}{22}$, and the variance
$$V\left(\bar{y}d\right)=(1-f) \frac{S^2}{n}+W_2 \frac{(K-1)}{n} S_2^2$$ by \begin{aligned} v_d= & (1-f)\left(\frac{n_1-1}{n-1}\right) w_1 \frac{s_1^2}{n_1} \ & +\frac{(N-1)\left(n_2-1\right)-(n-1)\left(m_2-1\right)}{N(n-1)} w_2 \frac{s{22}^2}{m_2} \ & +\frac{N-n}{N(n-1)}\left[w_1\left(\bar{y}1-\bar{y}_d\right)^2+w_2\left(\bar{y}{22}-\bar{y}_d\right)^2\right] . \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|NONRESPONSE

$$M_i= \begin{cases}1 & \text { if unit } i \text { is sampled and responds } \ 0 & \text { otherwise }\end{cases}$$

$$\bar{y}=\frac{\sum_1^N M_i Y_i}{\sum_1^N M_i}$$

$$E M_i=\frac{n}{N} q_i$$
$E \bar{y}$渐近等于
$$\frac{\sum q_i Y_i}{\sum q_i}$$

$$\sum\left(\frac{q_i}{\sum q_i}-\frac{1}{N}\right) Y_i$$

$$q_i=\frac{1}{N} \sum q_i .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|CALLBACKS

HANSEN和HURWITZ(1946)给出了一个优雅的回调程序来处理无响应问题，后来由SRINATH(1971)和j.n. K. RAO(1973)修改，下面简要介绍。人口在概念上分为$W_1\left(W_2=1-W_1\right)$和$\widehat{Y}_1\left(\widehat{Y}_2=\left[\widehat{Y}-W_1 \widehat{Y}_1\right] / W_2\right)$作为受访者的比例(非受访者)和受访者的平均值(非受访者)，大小为$n$的SRSWOR分别产生受访者和非受访者的比例$w_1=n_1 / n$和$w_2=1-w_1=1-n_1 / n=n_2 / n$。选择一个合适的数字$K>1$，然后从初始的$n_2$样本非受访者中提取大小为$m_2=n_2 / K$的SRSWOR(假设为整数)。假设在第二阶段中遵循更昂贵和更有说服力的程序，以便每个回调的$m_2$单位现在都做出响应，让$\bar{y}1$和$\bar{y}{22}$分别表示基于$n_1$和$m_2$受访者的第一阶段和第二阶段样本平均值。然后，$\bar{Y}$可以用$\bar{y}d=w_1 \bar{y}_1+w_2 \bar{y}{22}$来估计，方差
$$V\left(\bar{y}d\right)=(1-f) \frac{S^2}{n}+W_2 \frac{(K-1)}{n} S_2^2$$ by \begin{aligned} v_d= & (1-f)\left(\frac{n_1-1}{n-1}\right) w_1 \frac{s_1^2}{n_1} \ & +\frac{(N-1)\left(n_2-1\right)-(n-1)\left(m_2-1\right)}{N(n-1)} w_2 \frac{s{22}^2}{m_2} \ & +\frac{N-n}{N(n-1)}\left[w_1\left(\bar{y}1-\bar{y}_d\right)^2+w_2\left(\bar{y}{22}-\bar{y}_d\right)^2\right] . \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|MATH4060

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey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。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

## 统计代写|抽样调查作业代写sampling theory of survey代考|Linear Unbiased Estimators

Let a sensitive variable $y$ be defined on a finite population $U=(1, \ldots, N)$ with values $Y_i, i=1, \ldots, N$, which are supposed to be unavailable through a DR survey. Suppose a sample $s$ of size $n$ is chosen according to a design $p$ with a selection probability $p(s)$. In order to estimate $Y=\sum_1^N Y_i$, let an RR as a value $Z_i$ be available on request from each sampled person labeled $i$ included in a sample. Before describing how a $Z_i$ may be generated, let us note the properties required of it. We will denote by $E_R\left(V_R, C_R\right)$ the operator for expectation (variance, covariance) with respect to the randomized procedure of generating RR. The basic RRs $Z_i$ should allow derivation by a simple transformation reduced RRs as $R_i$ ‘s satisfying the conditions
(a) $E_R\left(R_i\right)=Y_i$
(b) $V_R\left(R_i\right)=\alpha_i Y_i^2+\beta_i Y_i+\theta_i$ with $\alpha_i(>0), \beta_i, \theta_i$ ‘s as known constants
(c) $C_R\left(R_i, R_j\right)=0$ for $i \neq j$
(d) estimators $v_i=a_i R_i^2+b_i R_i+C_i$ exist, $a_i, b_i, c_i$ known constants, such that $E_R\left(v_i\right)=V_R\left(R_i\right)=V_i$, say, for all $i$.

We will illustrate only two possible ways of obtaining $Z_i$ ‘s from a sampled individual $i$ on request. First, let two vectors $A=$ $\left(A_1, \ldots, A_T\right)^{\prime}$ and $B=\left(B_1, \ldots, B_L\right)^{\prime}$ of suitable real numbers be chosen with means $\bar{A} \neq 0, \bar{B}$ and variances $\sigma_A^2, \sigma_B^2$. A sample person $i$ is requested to independently choose at random $a_i$ out of $A$ and $b_i$ out of $B$, and report the value $Z_i=a_i Y_i+b_i$. Then, it follows that $E_R\left(Z_i\right)=\bar{A} Y_i+\bar{B}$, giving
$$R_i=\left(Z_i-\bar{B}\right) / \bar{A}$$
such that
\begin{aligned} E_R\left(R_i\right) & =Y_i, \ V_R\left(R_i\right) & =\left(Y_i^2 \sigma_A^2+\sigma_B^2\right) /(\bar{A})^2=V_i, \ C_R\left(R_i, R_J\right) & =0, \quad i \neq j \end{aligned}

and
$$v_i=\left(\sigma_A^2 R_i^2+\sigma_B^2\right) /\left(\sigma_A^2+\bar{A}^2\right)$$
has
$$E_R\left(v_i\right)=V_i .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|A Few Specific Strategies

Let us illustrate a few familiar specific cases. Corresponding to the HTE $\bar{t}=\bar{t}(s, Y)=\sum_i \frac{Y_i}{\pi_i} I_{s i}$, we have the derived estimator $e=(s, R)=\sum_i \frac{R_i}{\pi_i} I_{s i}$ for which
$$M=-\sum_{i<j} \sum_j\left(\pi_i \pi_j-\pi_{i j}\right)\left(Y_i / \pi_i-Y_j / \pi_j\right)^2+\sum_i \frac{V_i}{\pi_i}$$
and
To LAHIRI’s (1951) ratio estimator $t_L=Y_i / \sum_s P_i$ based on LAHIRI-MIDZUNO-SEN (LMS, 1951, 1952, 1953) scheme corresponds the estimator
$$e_L=\sum_s R_i / \sum_s P_i$$
$\left(0<P_i<1, \Sigma_1^N P_i=1\right)$ for which
$$M=\sum_{i<j} \sum_{i j}\left(1-\frac{1}{C_1} \sum_s \frac{I_{s i j}}{P_s}\right)+\sum V_i E_p\left(I_{s i} / P_s^2\right),$$
where
\begin{aligned} C_r & =\left(\begin{array}{c} N-r \ n-r \end{array}\right), r=0,1,2, \ldots, P_s=\sum_s P_i, a_{i j} \ & =P_i P_j\left(Y_i / P_i-Y_j / P_j\right)^2 \end{aligned}

\begin{aligned} m= & \sum \sum P_i P_j I_{s i} I_{s i j}\left(\frac{N-1}{n-1}-\frac{1}{P_s}\right) / \ & P_s\left[\left(\frac{R_i}{P_i}-\frac{R_j}{P_j}\right)^2-\left(\frac{v_i}{p_i^2}+\frac{v_j}{p_j^2}\right)\right]+\sum v_i I_{s i} / P_s^2 \end{aligned}
is unbiased for $M$. If $t_L$ and $e_L$ above are based on SRSWOR in $n$ draws, then, $M$ equals
\begin{aligned} M^{\prime}= & -\frac{1}{C_0}\left[\sum_{i<j} \sum_{i j} \sum_s\left(\frac{I_{s i j}}{p_s^2}-\frac{I_{s j}}{P_s}-\frac{I_{s j}}{P_s}+1\right)\right. \ & \left.-\sum_i V_i\left(\sum_s I_{s i} / P_s^2\right)\right] \end{aligned}
and
\begin{aligned} m^{\prime}= & -\frac{N(N-1)}{n(n-1) C_0} \sum_{i<j} \sum_{\hat{a}{i j}} I{s i j} \sum_s\left(\frac{I_{s i j}}{p_s^2}-\frac{I_{s i}}{P_s}-\frac{I_{s j}}{P_s}+1\right) \ & +\frac{1}{C_0} \frac{N}{n} \sum v_i I_{s i}\left(\sum_s I_{s i} / P_s^2\right) \end{aligned}
writing
$$\left.\widehat{a}_{i j}=\left{\left(\frac{R_i}{P_i}-\frac{R_j}{P_j}\right)^2-\frac{v_j}{P_i^2}+\frac{v_j}{P_j^2}\right)\right} P_i P_j$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Linear Unbiased Estimators

(a) $E_R\left(R_i\right)=Y_i$
(b) $V_R\left(R_i\right)=\alpha_i Y_i^2+\beta_i Y_i+\theta_i$，其中$\alpha_i(>0), \beta_i, \theta_i$为已知常数
(c) $i \neq j$为$C_R\left(R_i, R_j\right)=0$
(d)估计量$v_i=a_i R_i^2+b_i R_i+C_i$存在，$a_i, b_i, c_i$已知常数，使得$E_R\left(v_i\right)=V_R\left(R_i\right)=V_i$对所有$i$都成立。

$$e=\frac{N}{n} \sum_{i \in s} M_i \bar{y}_i$$

ARNAB(1988)认为限制SRSWOR既不必要也不可取，丢弃$s_i$或$s_i^{\prime}$中的ssus既不可取也不必要，并进一步概括了SRINATH和HIDIROGLOU(1980)的基本思想。根据DES RAJ(1968)的一般策略，他建议从估计器开始
$$e_D=\sum_s b_{s i} I_{s i} T_i$$

\begin{aligned} V\left(e_D\right) & =\sum Y_i^2\left(\alpha_i-1\right)+\sum_{i \neq j} Y_i Y_j\left(\alpha_{i j}-1\right)+\sum \alpha_i \sigma_i^2 \ V_L\left(T_i\right) & =\sigma_i^2 \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimation of Y

$$\bar{Y}=\sum_1^N Y_i / \sum_1^N M_i$$

$$e_R=\frac{\sum_s T_i}{\sum_s M_i},$$
$\bar{Y}$的有偏估计，使用$T_i$作为$Y_i$的无偏估计，基于从fsu $i$采样的后期阶段的样本，使得$E_L\left(T_i\right)=Y_i$与$V_L\left(T_i\right)$等于$V_{s i}$或$\sigma_i^2$分别承认无偏估计$\hat{V}{s i}$或$\hat{\sigma}i^2$，使得$E_L\left(\hat{V}{s i}\right)=V{s i}$或$E_L\left(\hat{\sigma}_i^2\right)=\sigma_i^2$。

$$t=\sum_s b_{s i} I_{s i} Y_i$$

$$M=E_p(t-Y)^2=\sum \sum Y_i Y_j d_{i j}$$

$$E_p\left(b_{s i} I_{s i}-1\right)\left(b_{s j} I_{s j}-1\right)=d_{i j} .$$

$$M_p\left(t_L\right)=E_p\left(t_L-Y\right)^2=V_p\left(t_L\right)+B_p^2\left(t_L\right) .$$

## 广义线性模型代考

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## MATLAB代写

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## 统计代写|抽样调查作业代写sampling theory of survey代考|BREWER’S ASYMPTOTIC APPROACH

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey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。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

## 统计代写|抽样调查作业代写sampling theory of survey代考|BREWER’S ASYMPTOTIC APPROACH

Looking for properties of a strategy as population and sample sizes increase presumes some relation between $p_1, p_2, \ldots$ on one hand and between $t_1, t_2, \ldots$ on the other hand.

In this and the next section relations on the design and estimator sequence, respectively, are introduced.

Consistency of an estimator $t_T$ is easy to decide on if Assumption $\mathbf{A}$ is true and $p_T$ satisfies a special condition considered by BREWER (1979):

Assumption B: Using Assumption $\boldsymbol{A}$ and starting with an arbitrary design $p_1$ of fixed size $n_1$ for $\mathcal{U}(1)$, then $p_T$ is as follows: Apply $p_1$ not only to $\mathcal{U}(1)$ but also, independently, to $\mathcal{U}(2)$, $\ldots, \mathcal{U}(T)$ and amalgamate the corresponding samples
$$s(1), s(2), \ldots, s(T)$$

to form
$$s_T=s(1) \cup s(2) \cup \cdots \cup s(T) .$$
A design satisfying Assumption $\mathbf{B}$ to give the selection probability for $s_T$ is appreciably limited in scope and application.

Some authors have considered such restrictive designs, notably HANSEN, MADOW and TEPPING (1983). However, interesting results have been derived under less restrictive assumptions as well as by alternative approaches.

We mention ISAKI and FULLER (1982) proving the consistency of the HT estimator under rather general conditions on $p_T$. In fact, they even drop Assumption $\mathbf{A}$, a condition that seems quite rational to us.

BREWER’s approach should be adequate where it is advisable to partition a large population $\mathcal{U}_T$ into subsets of similar size and structure and to use these subsets as strata in the selection procedure. This is acceptable only if there is no loss in efficiency. But it is doubtful that this may always be the case.
We plan to enlarge BREWER’s class of designs and obtain a class containing the designs in common use and with the same technical amenities as BREWER’s class.

## 统计代写|抽样调查作业代写sampling theory of survey代考|MOMENT-TYPE ESTIMATORS

To establish meaningful results of asymptotic unbiasedness and consistency, the estimators $t_1, t_2, \ldots$ of a sequence to be considered must be somehow related to each other. Subsequently, a relation is assumed that is based on the concept of a moment estimator we define as follows: Let $A_i, B_i, C_i, \ldots$ be values associated with $i \in U$. Then, for $s \subset U$ with $n(s)=n$
$$\frac{1}{n} \sum_s A_i, \quad \frac{1}{n} \sum_s A_i B_i, \quad \frac{1}{n} \sum_s A_i B_i C_i$$
are sample moments. Examples are
$$\frac{1}{n} \sum_s \frac{Y_i}{\pi_i}, \quad \frac{1}{n} \sum_s X_{i 1} Y_i, \quad \frac{1}{n} \sum_s \frac{X_{i 1} X_{i 2}}{\pi_i}$$
where $Y_i, X_{i 1}, X_{i 2}$ are values of variables $y, x_1, x_2$, respectively, and $\pi_i$ inclusion probabilities defined by a design for $i \in U$.
$$\frac{1}{N} \sum_1^N A_i, \quad \frac{1}{N} \sum_1^N A_i B_i, \quad \frac{1}{N} \sum_1^N A_i B_i C_i$$
are population moments corresponding to the sampling moments Eq. (5.7).

A moment estimator $t$ is an estimator that may be written as a function of sample moments $m^{(1)}, m^{(2)}, \ldots, m^{(v)}$ :
$$t=f\left(m^{(1)}, m^{(2)}, \ldots, m^{(\nu)}\right) .$$

Obvious examples of moment estimators are the sample mean, the HT-estimator, the $\mathrm{HH}$-estimator, and the ratio estimator.
Now, let $t_1$ be a moment estimator, that is,
$$t_1=f\left(m_1^{(1)}, \ldots, m_1^{(\nu)}\right)$$
where $m_1^{(1)}, \ldots, m_1^{(v)}$ are sample moments for $s_1$.
Then, $t_T$ may be defined in a natural way:
$$t_T=f\left(m_T^{(1)}, m_T^{(2)}, \ldots, m_T^{(\nu)}\right)$$
where $m_T^{(j)}$ is the sample moment for $s_T$ corresponding to $m_1^{(j)}$, $j=1,2, \ldots, v$. As an example, we mention the ratio estimator
$$t_1=\frac{\sum_{s_1} Y_i}{\sum_{s_1} X_i} \bar{X}$$
for which
$$t_T=\frac{\sum_{s_T} Y_i}{\sum_{s_T} X_i} \bar{X}$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|BREWER’S ASYMPTOTIC APPROACH

$$s(1), s(2), \ldots, s(T)$$

$$s_T=s(1) \cup s(2) \cup \cdots \cup s(T) .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|MOMENT-TYPE ESTIMATORS

$$\frac{1}{n} \sum_s A_i, \quad \frac{1}{n} \sum_s A_i B_i, \quad \frac{1}{n} \sum_s A_i B_i C_i$$

$$\frac{1}{n} \sum_s \frac{Y_i}{\pi_i}, \quad \frac{1}{n} \sum_s X_{i 1} Y_i, \quad \frac{1}{n} \sum_s \frac{X_{i 1} X_{i 2}}{\pi_i}$$

$$\frac{1}{N} \sum_1^N A_i, \quad \frac{1}{N} \sum_1^N A_i B_i, \quad \frac{1}{N} \sum_1^N A_i B_i C_i$$

$$t=f\left(m^{(1)}, m^{(2)}, \ldots, m^{(\nu)}\right) .$$

$$t_1=f\left(m_1^{(1)}, \ldots, m_1^{(\nu)}\right)$$

$$t_T=f\left(m_T^{(1)}, m_T^{(2)}, \ldots, m_T^{(\nu)}\right)$$

$$t_1=\frac{\sum_{s_1} Y_i}{\sum_{s_1} X_i} \bar{X}$$

$$t_T=\frac{\sum_{s_T} Y_i}{\sum_{s_T} X_i} \bar{X}$$

## 广义线性模型代考

statistics-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。

## 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。

## 统计代写|抽样调查作业代写sampling theory of survey代考|Balancing for Polynomial Models

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey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。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

## 统计代写|抽样调查作业代写sampling theory of survey代考|Balancing for Polynomial Models

We return to the model $\mathcal{M}{10}^{\prime}$ of 4.1.2 and consider an extension $\mathcal{M}_k$ defined as follows: \begin{aligned} Y_i & =\sum{j=0}^k \beta_j X_i^j+\varepsilon_i \ E_m\left(\varepsilon_i\right) & =0, V_m\left(\varepsilon_i\right)=\sigma^2, C_m\left(\varepsilon_i, \varepsilon_j\right)=0, \text { for } i \neq j \end{aligned}
where $i, j=1,2, \ldots, N$. By generalizing the developments of section 4.1.2, we derive.

RESULT 4.2 Let $\mathcal{M}_k$ be given. Then, the MSE of the BLU predictor $t_o$ for $Y$ is minimum for a sample $s$ of size $n$ if
$$\frac{1}{n} \sum_s X_i^j=\frac{1}{N} \sum_1^N X_i^j \text { for } j=0,1, \ldots, k .$$
If these equalities hold we have
$$t_o(s, Y)=N \bar{y}$$
A sample satisfying the equalities in Result 4.2 is said to be balanced up to order $k$.

Now, assume the true model $\mathcal{M}{k^{\prime}}$ agrees with a statistician’s working model $\mathcal{M}_k$ in all respects except that $$E_m\left(Y_i\right)=\sum_0^{k^{\prime}} \beta_j X_i^j$$ with $k^{\prime}>k$. The statistician will use $t_o$ instead of $t_o^{\prime}$, the BLU predictor for $Y$ on the base of $\mathcal{M}{k^{\prime}}$. However, if he selects a sample that is balanced up to order $k^{\prime}$
$$t_o^{\prime}(s, Y)=t_o(s, Y)=N \bar{y}$$
and his error does not cause losses.
It is, of course, too ambitious to realize exactly the balancing conditions even if $k^{\prime}$ is of moderate size, for example, $k^{\prime}=4$ or 5 . But if $n$ is large the considerations outlined in Result 4.1 apply again for SRSWOR or SRSWOR independently from within strata after internally homogeneous strata are priorly constructed.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Linear Models in Matrix Notation

Suppose $x_1, x_2, \ldots, x_k$ are real variables, called auxiliary or explanatory variables, each closely related to the variable of interest $y$. Let
$$xi=\left(X{i 1}, X_{i 2}, \ldots, X_{i k}\right)^{\prime}$$
be the vector of explanatory variables for unit $i$ and assume the linear model
$$Y_i=xi^{\prime} \beta+\varepsilon_i$$ for $i=1,2, \ldots, N$. Here $$\beta=\left(\beta_1, \beta_2, \ldots, \beta_k\right)^{\prime}$$ is the vector of (unknown) regression parameters; $\varepsilon_1, \varepsilon_2, \ldots$, $\varepsilon_N$ are random variables satisfying \begin{aligned} E_m \varepsilon_i & =0 \ V_m \varepsilon_i & =v{i i} \ C_m\left(\varepsilon_i, \varepsilon_j\right) & =v_{i j}, i \neq j \end{aligned}
where $E_m, V_m, C_m$ are operators for expectation, variance, and covariance with respect to the model distribution; and the ma$\operatorname{trix} V=\left(v_{i j}\right)$ is assumed to be known up to a constant $\sigma^2$.
To have a more compact notation define
\begin{aligned} Y & =\left(Y_1, Y_2, \ldots, Y_N\right)^{\prime} \ X & =\left(x1, x_2, \ldots, x_N\right)^{\prime}=\left(X{i j}\right) \ \varepsilon & =\left(\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_N\right)^{\prime} \end{aligned}
and write the linear model as
$$Y=X \beta+\varepsilon$$
where
\begin{aligned} E_{m \varepsilon} & =0 \ V_m(\varepsilon) & =V \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|Balancing for Polynomial Models

4.2设$\mathcal{M}_k$。然后，对于大小为$n$ if的样本$s$，对于$Y$的BLU预测器$t_o$的MSE是最小的
$$\frac{1}{n} \sum_s X_i^j=\frac{1}{N} \sum_1^N X_i^j \text { for } j=0,1, \ldots, k .$$

$$t_o(s, Y)=N \bar{y}$$

$$t_o^{\prime}(s, Y)=t_o(s, Y)=N \bar{y}$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Linear Models in Matrix Notation

$$xi=\left(X{i 1}, X_{i 2}, \ldots, X_{i k}\right)^{\prime}$$

$$Y_i=xi^{\prime} \beta+\varepsilon_i$$代表$i=1,2, \ldots, N$。其中$$\beta=\left(\beta_1, \beta_2, \ldots, \beta_k\right)^{\prime}$$为(未知)回归参数的向量;$\varepsilon_1, \varepsilon_2, \ldots$, $\varepsilon_N$是满足\begin{aligned} E_m \varepsilon_i & =0 \ V_m \varepsilon_i & =v{i i} \ C_m\left(\varepsilon_i, \varepsilon_j\right) & =v_{i j}, i \neq j \end{aligned}的随机变量

\begin{aligned} Y & =\left(Y_1, Y_2, \ldots, Y_N\right)^{\prime} \ X & =\left(x1, x2, \ldots, x_N\right)^{\prime}=\left(X{i j}\right) \ \varepsilon & =\left(\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_N\right)^{\prime} \end{aligned} 把线性模型写成 $$Y=X \beta+\varepsilon$$ 在哪里 \begin{aligned} E{m \varepsilon} & =0 \ V_m(\varepsilon) & =V \end{aligned}

## 广义线性模型代考

statistics-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。

## 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。

## 统计代写|抽样调查作业代写sampling theory of survey代考|Equicorrelation Model

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey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。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

## 统计代写|抽样调查作业代写sampling theory of survey代考|Equicorrelation Model

Following CSW (1976, 1977), consider the model of equicorrelated $Y_i$ ‘s for which
$$E_m\left(Y_i\right)=\alpha_i+\beta X_i$$
$\alpha_i$ known with mean $\bar{\alpha}, \beta$ unknown, $0<X_i$ known with $\Sigma X_i=$ $N$
\begin{aligned} V_m\left(Y_i\right) & =\sigma^2 X_i^2 \ C_m\left(Y_i, Y_j\right) & =\rho \sigma^2 X_i X_j,-\frac{1}{N-1}<\rho<1 . \end{aligned}
Linear unbiased estimators (LUE) for $\bar{Y}$ are of the form
$$t=t(s, Y)=a_s+\sum_{i \in s} b_{s i} Y_i$$
with $a_s, b_{s i}$ free of $Y$ such that for a fixed design $p$
$$E_p\left(a_s\right)=0, \sum_{s \ni i} b_{s i} p(s)=\frac{1}{N} \text { for } i=1, \ldots, N .$$
To find an optimal strategy ( $p, t)$ let us proceed as follows. First note that writing $c_{s i}=b_{s i} X_i$,
$$1=\frac{X}{N}=\frac{1}{N} \sum_1^N X_i=\sum_1^N \sum_{s \ni i} c_{s i} p(s)=\sum_s p(s)\left[\sum_{i \in s} c_{s i}\right] .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Applications to Survey Sampling

A further line of approach is now required because $\theta_0$ itself needs to be estimated from survey data
$$d=\left(i, Y_i \mid i \in s\right)$$
available only for the $Y_i$ ‘s with $i \in s, s$ a sample supposed to be selected with probability $p(s)$ according to a design $p$ for which we assume
$$\pi_i=\sum_{s \ni i} p(s)>0 \text { for all } i=1,2, \ldots, N .$$
With the setup of the preceding section, let the $Y_i$ ‘s be independent and consider unbiased estimating functions $\phi_i\left(Y_i, \theta\right) ; i=$ $1,2, \ldots, N$. Let
$$\theta_0=\theta_0(Y)$$
be the solution of $g(Y, \theta)=0$ where
$$g(Y, \theta)=\sum_1^N \phi_i\left(Y_i, \theta\right)$$
and consider estimating this $\theta_0$ using survey data $d=\left(i, Y_i \mid i \in\right.$ s). For this it seems natural to start with an unbiased sampling function
$$h=h(s, Y, \theta)$$
which is free of $Y_j$ for $j \notin s$ and satisfies
(a) $\frac{\partial h}{\partial \theta}(s, Y, \theta)$ exists for all $Y$
(b) $E_m \frac{\partial h}{\partial \theta}(s, Y, \theta) \neq 0$
(c) $E_p h(s, Y, \theta)=g(Y, \theta)$ for all $Y$, the unbiasedness condition.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Equicorrelation Model

$$E_m\left(Y_i\right)=\alpha_i+\beta X_i$$
$\alpha_i$已知，平均$\bar{\alpha}, \beta$未知，$0<X_i$已知，\Sigma X_i=$$N$$ \begin{aligned} V_m\left(Y_i\right) & =\sigma^2 X_i^2 \ C_m\left(Y_i, Y_j\right) & =\rho \sigma^2 X_i X_j,-\frac{1}{N-1}<\rho<1 . \end{aligned} $$\bar{Y}的线性无偏估计量(LUE)是这样的形式$$ t=t(s, Y)=a_s+\sum_{i \in s} b_{s i} Y_i $$与a_s, b_{s i}免费Y这样，为一个固定的设计p$$ E_p\left(a_s\right)=0, \sum_{s \ni i} b_{s i} p(s)=\frac{1}{N} \text { for } i=1, \ldots, N . $$为了找到一个最佳策略(p, t))，让我们按照以下步骤进行。首先要注意写c_{s i}=b_{s i} X_i，$$ 1=\frac{X}{N}=\frac{1}{N} \sum_1^N X_i=\sum_1^N \sum_{s \ni i} c_{s i} p(s)=\sum_s p(s)\left[\sum_{i \in s} c_{s i}\right] . $$## 统计代写|抽样调查作业代写sampling theory of survey代考|Applications to Survey Sampling 现在需要进一步的方法，因为\theta_0本身需要根据调查数据进行估计$$ d=\left(i, Y_i \mid i \in s\right) $$只适用于Y_i与i \in s, s的一个样本应该是选择的概率p(s)根据我们假设的设计p$$ \pi_i=\sum_{s \ni i} p(s)>0 \text { for all } i=1,2, \ldots, N . $$通过上一节的设置，让Y_i是独立的，并考虑无偏估计函数\phi_i\left(Y_i, \theta\right) ; i=$$1,2, \ldots, N。让
$$\theta_0=\theta_0(Y)$$

$$g(Y, \theta)=\sum_1^N \phi_i\left(Y_i, \theta\right)$$

$$h=h(s, Y, \theta)$$

(a) $\frac{\partial h}{\partial \theta}(s, Y, \theta)$适用于所有$Y$
(b) $E_m \frac{\partial h}{\partial \theta}(s, Y, \theta) \neq 0$
(c) $E_p h(s, Y, \theta)=g(Y, \theta)$对于所有$Y$，为无偏性条件。

## 广义线性模型代考

statistics-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。

## 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。