### 统计代写|抽样调查作业代写sampling theory of survey代考|STAT7124

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

A unified theory is developed by noting that it is enough to establish results concerning $(p, t)$ without heeding how one may actually succeed in choosing samples with preassigned probabilities. A method of choosing a sample draw by draw, assigning selection probabilities with each draw, is called a sampling scheme. Following HANURAV (1966), we show below that starting with an arbitrary design we may construct a sampling scheme.

Suppose for each possible sample $s$ from $U$ the selection probability $p(s)$ is fixed. Let
$$\begin{array}{lll} \beta_{i 1}=p\left(i_{1}\right), & \beta_{i_{1}, i_{2}}=p\left(i_{1}, i_{2}\right), \ldots, & \beta_{i_{1}, \ldots, i_{n}}=p\left(i_{1}, \ldots, i_{n}\right) \ \alpha_{i 1}=\Sigma_{1} p(s), & \alpha_{i_{1}, i_{2}}=\Sigma_{2} p(s), \ldots, & \alpha_{i_{1}, \ldots, i_{n}}=\Sigma_{n} p(s) \end{array}$$
where $\Sigma_{1}$ is the sum over all samples $s$ with $i_{1}$ as the first entry; $\Sigma_{2}$ is the sum over all samples with $i_{1}, i_{2}$, respectively, as the first and second entries in $s, \ldots$, and $\Sigma_{n}$ is the sum over all samples of which the first, second, $\ldots, n$th entries are, respectively, $i_{1}, i_{2}, \ldots, i_{n}$.

Then, let us consider the scheme of selection such that on the first draw from $U, i_{1}$ is chosen with probability $\alpha_{i 1}$, a second draw from $U$ is made with probability
$$\left(1-\frac{\beta_{i 1}}{\alpha_{i 1}}\right) \text {. }$$
On the second draw from $U$ the unit $i_{2}$ is chosen with probability
$$\begin{gathered} \alpha_{i_{1}, i_{2}} \ \alpha_{i 1}-\beta_{i 1} \end{gathered}$$
A third draw is made from $U$ with probability
$$\left(1-\frac{\beta_{i_{1}, i_{2}}}{\alpha_{i_{1}, i_{2}}}\right)$$

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

Now, consider an arbitrary design $p$ of fixed size $n$ and a linear estimator $t$; suppose a subset $S_{0}$ of all samples is less desirable from practical considerations like geographical location, inaccessibility, or, more generally, costliness. Then, it is advantageous to replace design $p$ by a modified one, for example, $q$, which attaches minimal values $q(s)$ to the samples $s$ in $S_{0}$ keeping
$$\begin{gathered} E_{p}(t)=E_{q}(t) \ E_{p}(t-Y)^{2}-E_{q}(t-Y)^{2} \end{gathered}$$
and even maintaining other desirable properties of $p$, if any. A resulting $q$ is called a controlled design and a corresponding scheme of selection is called a controlled sampling scheme. Quite a sizeable literature has grown around this problem of finding appropriate controlled designs. The methods of implementing such a scheme utilize theories of incomplete block designs and predominantly involve ingeneous devices of reducing the size of support of possible samples demanding trials and errors. But RAO and NIGAM (1990) have recently presented a simple solution by posing it as a linear programming problem and applying the well-known simplex algorithm to demonstrate their ability to work out suitable controlled schemes.
Taking $t$ as the HOR VIT7-THOMPSON estimator $\bar{t}=\sum_{i \in S}$ $Y_{i} / \pi_{i}$, they minimize the objective function $F=\sum_{s \in S_{0}} q(s)$ subject to the linear constraints
\begin{aligned} \sum_{s \ni i, j} q(s) &=\sum_{s \ni i, j} p(s)=\pi_{i j} \ q(s) & \geq 0 \text { for all } s \end{aligned}
where $\pi_{i j}{ }^{\prime} s$ are known quantities in terms of the original uncontrolled design $p$.

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

$$\beta_{i 1}=p\left(i_{1}\right), \quad \beta_{i_{1}, i_{2}}=p\left(i_{1}, i_{2}\right), \ldots, \quad \beta_{i_{1}, \ldots, i_{n}}=p\left(i_{1}, \ldots, i_{n}\right) \alpha_{i 1}=\Sigma_{1} p(s), \quad \alpha_{i_{1}, i_{2}}=\Sigma_{2} p(s), \ldots,$$

$$\left(1-\frac{\beta_{i 1}}{\alpha_{i 1}}\right) .$$

$$\alpha_{i_{1}, i_{2}} \alpha_{i 1}-\beta_{i 1}$$

$$\left(1-\frac{\beta_{i_{1}, i_{2}}}{\alpha_{i_{1}, i_{2}}}\right)$$

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

$$E_{p}(t)=E_{q}(t) E_{p}(t-Y)^{2}-E_{q}(t-Y)^{2}$$

$$\sum_{s \ni i, j} q(s)=\sum_{s \ni i, j} p(s)=\pi_{i j} q(s) \quad \geq 0 \text { for all } s$$

## 广义线性模型代考

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