### 统计代写|抽样调查作业代写sampling theory of survey代考|STAT 7124

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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|Probability Proportional to Size Without Replacement Sampling

In probability proportional to size WOR (PPSWOR) sampling scheme, probability of selection of $i_{1}$ at the first draw is $p_{i_{1}}(1)=p_{i_{1}}$. Probability of selecting $i_{2}$ at the second draw is $p_{i_{2}}(2)=\frac{p_{i_{2}}}{1-p_{i_{1}}}$ if the unit $i_{1}\left(i_{2} \neq i_{1}\right)$ is selected at the first draw and $p_{i_{2}}(2)=0$ when the unit $i_{2}$ is selected at the first draw, i.e., $i_{2}=i_{1}$. In general, the probability of selection of $i_{k}$ at the $k$ th draw is $p_{i_{k}}(k)=p_{1-p_{i_{1}}-p_{i_{2}}-\cdots-p_{i_{k-1}}}$, if the units $i_{1}, i_{2}, \ldots, i_{k-1}$ are selected in any of the first $k-1$ draws and $p_{i_{k}}(k)=0$ if the unit $i_{k}$ is selected in any of the first $k-1$ draws for $k=2, \ldots, n ; i=1, \ldots, N$. So, for a PPSWOR sampling scheme, the probability of selecting $i_{1}$ at the first draw, $i_{2}$ at the second draw, and $i_{n}$ at the $n$th draw is
\begin{aligned} p\left(i_{1}, \ldots, i_{n}\right)=& p_{i_{1}} \frac{p_{i_{2}}}{1-p_{i_{1}}} \cdots \frac{p_{i_{k}}}{1-p_{i_{1}}-\cdots-p_{i_{k-1}}} \cdots \frac{p_{i_{n}}}{1-p_{i_{1}}-\cdots-p_{i_{n-1}}} \text { for } \ 1 \leq i_{1} \neq i_{2} \neq \cdots \neq i_{n} \leq N \end{aligned}
It should be noted that PPSWOR reduces to SRSWOR sampling scheme if $p_{i}=1 / N$ for $i=1, \ldots, N$.

## 统计代写|抽样调查作业代写sampling theory of survey代考|HANURAV’S ALGORITHM

Hanurav (1966) established a correspondence between a sampling design and a sampling scheme. He proved that any sampling scheme results in a sampling design. Similarly, for a given sampling design, one can construct at least one sampling scheme, which can implement the sampling design. In fact, Hanurav proposed the most general sampling scheme, known as Hanurav’s algorithm, using which one can derive various types of sampling schemes or sampling designs. Henceforth, we will not differentiate between the terms “sampling design” and “sampling scheme”.

Let $n_{0}$ denote the maximum sample size that might be required from a sampling scheme. Then, Hanurav’s (1966) algorithm is defined as follows:
$$\mathscr{A}=\mathscr{A}\left{q_{1}(i) ; q_{2}(s) ; q_{3}(s, i)\right}$$
where
(i) $0 \leq q_{1}(i) \leq 1, \quad \sum_{i=1}^{N} q_{1}(i)=1$ for $i=1, \ldots, N$
(ii) $0 \leq q_{2}(s) \leq 1$ for any sample $s \in \mathscr{S}$, where $\mathscr{\mathcal { S }}$ be the set of all possible samples.
(iii) $q_{3}(s, i)$ is defined when $q_{2}(s)>0$ and subject to $0 \leq q_{3}(s, i) \leq 1$,
$$\sum_{i=1}^{N} q_{3}(s, i)=1 \text { for } i=1, \ldots, N$$
Samples are selected using the following steps:
Step 1: At the first draw a unit $i_{1}$ is selected with probability $q_{1}\left(i_{1}\right)$; $i_{1}=1, \ldots, N$

Step 2: In this step, we decide whether the sampling procedure will be terminated or continued. Let $s_{(1)}=i_{1}$ be the unit selected in the first draw. A Bernoulli trial is performed with success probability $q_{2}\left(s_{(1)}\right)$. If the trial results in a failure, the sampling procedure is terminated and the selected sample is $s_{(1)}=i_{1}$. On the other hand, if the trial results in a success, we go to step 3 .

## 统计代写|抽样调查作业代写sampling theory of survey代考|Probability Proportional to Size Without Replacement Sampling

$$p\left(i_{1}, \ldots, i_{n}\right)=p_{i_{1}} \frac{p_{i_{2}}}{1-p_{i_{1}}} \cdots \frac{p_{i_{k}}}{1-p_{i_{1}}-\cdots-p_{i_{k-1}}} \cdots \frac{p_{i_{n}}}{1-p_{i_{1}}-\cdots-p_{i_{n-1}}} \text { for } 1 \leq i_{1} \neq i_{2} \neq \cdots$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|HANURAV’S ALGORITHM

Hanurav (1966) 建立了抽样设计和抽样方案之间的对应关系。他证明了任何抽样方案都会导致抽样设计。类似 地，对于给定的抽样设计，可以构建至少一种抽样方案，该方案可以实现抽样设计。事实上，Hanurav 提出了最 通用的抽样方案，称为 Hanurav 算法，利用该算法可以推导出各种类型的抽样方案或抽样设计。此后，我们将 不再区分”抽样设计”和”抽样方案”这两个术语。

$\backslash$ mathscr ${\mathrm{A}}=\backslash$ mathscr ${\mathrm{A}} \backslash \operatorname{left}\left{\mathrm{q}{-}{1}(\mathrm{i}) ; \mathrm{q}{-}{2}(\mathrm{s}) ; \mathrm{q}{-}{3}(\mathrm{s}, \mathrm{i}) \backslash\right.$ right $}$ 其中 (i) $0 \leq q{1}(i) \leq 1, \quad \sum_{i=1}^{N} q_{1}(i)=1$ 为了 $i=1, \ldots, N$
(二) $0 \leq q_{2}(s) \leq 1$ 对于任何样品 $s \in \mathscr{S}$ ，在哪里 $\mathcal{S}$ 是所有可能样本的集合。
$\Leftrightarrow q_{3}(s, i)$ 定义为 $q_{2}(s)>0$ 并受 $0 \leq q_{3}(s, i) \leq 1$ ，
$$\sum_{i=1}^{N} q_{3}(s, i)=1 \text { for } i=1, \ldots, N$$

## 广义线性模型代考

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## MATLAB代写

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