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

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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|A Unified Theory

Suppose it is considered important to gather ideas about, for example, (1) the total quantity of food grains stocked in all the godowns managed by a state government, (2) the total number of patients admitted in all the hospitals of a country classified by varieties of their complaints, (3) the amount of income tax evaded on an average by the income earners of a city. Now, to inspect all godowns, examine all admission documents of all hospitals of a country, and make inquiries about all income earners of a city will be too expensive and time consuming. So it seems natural to select a few godowns, hospitals, and income earners, to get all relevant data for them and to be able to draw conclusions on those quantities that could be ascertained exactly only by a survey of all godowns, hospitals, and income earners. We feel it is useful to formulate mathematically as follows the essentials of the issues at hand common to the above and similar circumstances.

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

Let $N$ be a known number of units, e.g., godowns, hospitals, or income earners, each assignable identifying labels $1,2, \ldots, N$ and bearing values, respectively, $Y_{1}, Y_{2}, \ldots, Y_{N}$ of a realvalued variable $y$, which are initially unknown to an investigator who intends to estimate the total
where $f_{s i}$ denotes the frequency of $i$ in $s$ such that
$$\sum_{i=1}^{N} f_{s i}=n(s) .$$
$N \bar{y}$ is called the expansion estimator for $Y$.

More generally, an estimator $t$ of the form
$$t(s, Y)=b_{s}+\sum_{i=1}^{N} b_{s i} Y_{i}$$
with $b_{s i}=0$ for $i \notin s$ is called linear (L). Here $b_{s}$ and $b_{s i}$ are free of $Y$. Keeping $b_{s}=0$ we obtain a homogeneous linear (HL) estimator.

We must emphasize that here $t(s, Y)$ is linear (or homogeneous linear) in $Y_{i}, i \in s$. It may be a nonlinear function of two random variables, e.g., when $b_{s}=0$ and $b_{s i}=X / \Sigma_{1}^{N} f_{s i} X_{i}$ so that
$$t(s, Y)=\frac{\sum_{1}^{N} f_{s i} Y_{i}}{\sum_{1}^{N} f_{s i} X_{i}} X .$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|DESIGN-BASED INFERENCE

Let $\Sigma_{1}$ be the sum over samples for which $|t(s, Y)-Y| \geq k>0$ and let $\Sigma_{2}$ be the sum over samples for which $|t(s, Y)-Y|<k$ for a fixed $Y$. Then from
\begin{aligned} M_{p}(t) &=\Sigma_{1} p(s)(t-Y)^{2}+\Sigma_{2} p(s)(t-Y)^{2} \ & \geq k^{2} \operatorname{Prob}[|t(s, Y)-Y| \geq k] \end{aligned}
one derives the Chebyshev inequality:
$$\operatorname{Prob}[|t(s, Y)-Y| \geq k] \leq \frac{M_{p}(t)}{k^{2}} .$$
Hence
$\operatorname{Prob}[t-k \leq Y \leq t+k] \geq 1-\frac{M_{p}(t)}{k^{2}}=1-\frac{1}{k^{2}}\left[V_{p}(t)+B_{p}^{2}(t)\right]$ where $B_{p}(t)=E_{p}(t)-Y$ is the bias of $t$. Writing $\sigma_{p}(t)=$ $\sqrt{V_{p}(t)}$ for the standard error of $t$ and taking $k=3 \sigma_{p}(t)$, it follows that, whatever $Y$ may be, the random interval $t \pm 3 \sigma_{p}(t)$

covers the unknown $Y$ with a probability not less than
$$\frac{8}{9}-\frac{1}{9} \frac{B_{p}^{2}(t)}{V_{p}(t)}$$
So, to keep this probability high and the length of this covering interval small it is desirable that both $\left|B_{p}(t)\right|$ and $\sigma_{p}(t)$ be small, leading to a small $M_{p}(t)$ as well.

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

∑一世=1ñFs一世=n(s).
ñ是¯被称为扩展估计器是.

$$t(s, Y ) =b_{s}+\sum_{i=1}^{N} b_{si} Y_{i}$$

t(s, Y )=\frac{\sum_{1}^{N} f_{si} Y_{i}}{\sum_{1}^{N} f_{si} X_{i}} X 。

89−19乙p2(吨)在p(吨)

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