### 统计代写|抽样理论作业代写sampling theory 代考|STAT506

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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|抽样理论作业代写sampling theory 代考|Notation and Estimation

We are interested in a finite population $U_N$ consisting of $N$ statistical units, which are supposed to be easily identified by a label. Therefore, it is common practice to make no distinction between a unit and its label, and we simply write the population as
$$U_N={1, \ldots, N}$$
We are interested in some quantitative variable of interest $y$, taking the value $y_k$ on unit $k$.

We suppose that the population of interest $U_N$ is embedded into a nested sequence $\left{U_N\right}$ of finite populations with increasing sizes, and all limiting processes will be taken as $N \rightarrow \infty$. This is essentially the asymptotic framework of [41], which is often used to study the asymptotic properties of a sampling design and of the assorted estimators. Also, this is a natural framework if we are interested in a population which is growing over time, for example, if we wish to select a sample in a data stream.

A without-replacement sampling design $p_N(\cdot)$ is a probability distribution on the subsets in $U_N$, namely $$\forall s \subset U_N \quad p_N(s) \geq 0 \text { and } \sum_{s \subset U_N} p_N(s)=1$$
It enables selecting the random sample $S_N$ of units used for estimation, in the sense that $\operatorname{Pr}\left(S_N=s\right)=p_N(s)$. Once the sampling design is defined, we need to choose a sampling algorithm, which is an effective procedure for the selection of the sample. For a given sampling design, there is usually a variety of sampling algorithms possible [58], see Sect. $1.3$ for an illustration on simple random sampling.

The quantity $\pi_{k \mid N} \equiv \operatorname{Pr}\left(k \in S_N\right)$ for unit $k$ to be selected is called the first-order inclusion probability. The $\pi_{k \mid N}$ ‘s are involved in the computation of estimators, and their sum
$$\sum_{k \in U_N} \pi_{k \mid N} \equiv n$$
gives the average sample size. The probability $\pi_{k l \mid N} \equiv \operatorname{Pr}\left(k, l \in S_N\right)$ for units $k$ and $l$ to be jointly selected in $S_N$ is called the second-order inclusion probability. The $\pi_{k l \mid N}$ ‘s are involved in the computation of the variance of estimators. For a given set of first-order inclusion probabilities $\pi_{k \mid N}, k \in U_N$, the second-order inclusion probabilities depend on the design used for the selection of the sample.

## 统计代写|抽样理论作业代写sampling theory 代考|Horvitz–Thompson Estimator

The Horvitz and Thompson [39] estimator (HT) of the total $t_{y N}=\sum_{k \in U_N} y_k$ is
$$\hat{t}{y \pi}=\sum{k \in S_N} \frac{y_k}{\pi_{k \mid N}}=\sum_{k \in U_N} I_{k N} \frac{y_k}{\pi_{k \mid N}}$$
with $I_{k N}$ the sample membership indicator of unit $k$. We note
$$I_N=\left(I_{1 N}, \ldots, I_{k N}, \ldots, I_{N N}\right)$$
the vector of sample membership indicators. If all the $\pi_{k \mid N}$ ‘s are positive, which is assumed in the rest of the paper, there is no selection bias. In such case, the HT-estimator is design-unbiased for $t_{y N}$, i.e., unbiased under the randomization associated with the sampling design. It is remarkable that this property comes completely model-free. It holds for any variable of interest, without requiring any model assumptions.

There is no severe loss of generality in focusing on the total $t_{y N}$, since many other parameters of interest can be written as smooth functions of totals. Such parameters are therefore easily estimated in a plug-in principle once an estimator of a total is available, see [20]. For example, the population mean is $\mu_{y N}=N^{-1} \sum_{k \in U_N} y_k$, and is estimated by $$\hat{\mu}{y \pi}=\frac{\hat{t}{y \pi}}{\hat{N}\pi}$$ where $\hat{N}\pi=\sum_{k \in S_N} \frac{1}{\pi_{k \mid N}}$ is the HT-estimator of the population size $N$. Similarly, the population distribution function for some real number $t$ is $F_{y N}(t)=$ $N^{-1} \sum_{k \in U_N} 1\left(y_k \leq t\right)$, with $1(\cdot)$ the indicator function. The plug-in estimator of $F_{y N}(t)$ is
$$\hat{F}{y \pi}=\frac{1}{\hat{N}\pi} \sum_{k \in S_N} \frac{1\left(y_k \leq t\right)}{\pi_{k \mid N}}$$

# 抽样调查代考

## 统计代写|抽样理论作业代写sampling theory 代考|Notation and Estimation

$$U_N=1, \ldots, N$$

$$\forall s \subset U_N \quad p_N(s) \geq 0 \text { and } \sum_{s \subset U_N} p_N(s)=1$$

$$\sum_{k \in U_N} \pi_{k \mid N} \equiv n$$

## 统计代写|抽样理论作业代写sampling theory 代考|Horvitz–Thompson Estimator

$$\hat{\mu} y \pi=\frac{\hat{t} y \pi}{\hat{N} \pi}$$

$$\hat{F} y \pi=\frac{1}{\hat{N} \pi} \sum_{k \in S_N} \frac{1\left(y_k \leq t\right)}{\pi_{k \mid N}}$$

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