### 统计代写|抽样调查作业代写sampling theory of survey代考| MODEL-DEPENDENT ESTIMATION

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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|Linear Models and BLU Predictors

Let a superpopulation be modeled as follows:
$$Y_{i}=\beta X_{i}+\varepsilon_{i}, i=1, \ldots, N$$
where $X_{i}$ ‘s are the known positive values of a nonstochastic real variable $x ; \varepsilon_{i}$ ‘s are random variables with
$$E_{m}\left(\varepsilon_{i}\right)=0, V_{m}\left(\varepsilon_{i}\right)=\sigma_{i}^{2}, C_{m}\left(\varepsilon_{i}, \varepsilon_{j}\right)=\rho_{i j} \sigma_{i} \sigma_{j},$$
writing $E_{m}, V_{m}, C_{m}$ as operators for expectation, variance and covariance with respect to the modeled distribution.

To estimate $Y=\Sigma_{s} Y_{i}+\Sigma_{r} Y_{i}$, where $\Sigma_{r} Y_{i}$ is the value of a random variable, is actually to predict this value, add that predicted value to the observed quantity $\Sigma_{s} Y_{i}$, and hence obtain a predicted value of $Y$, which also is a random variable in the present formulation of the problem.
Since
$$\sum_{r} Y_{i}=\beta \sum_{r} X_{i}+\sum_{r} \varepsilon_{i}$$
with $E_{m} \Sigma_{r} \varepsilon_{i}=0$, a predictor for $\Sigma_{r} Y_{i}$ may be $\hat{\beta} \Sigma_{r} X_{i}$. Here $\hat{\beta}$ is a function of $d$ (and $X$ ) and for simplicity we will take it as linear in $Y$,
$$\hat{\beta}=\sum_{s} B_{i} Y_{i} \text {, say. }$$
The resulting predictor for $Y$
$$t=\sum_{s} Y_{i}+\hat{\beta} \sum_{r} X_{i}$$

will then be model-unbiased ( $m$-unbiased) if
\begin{aligned} 0 &=E_{m}(t-Y) \ &=E_{m}\left(\sum_{s} Y_{i}+\beta \sum_{r} X_{i}-\sum_{s} Y_{i}-\sum_{r} Y_{i}\right) \ &=E_{m}\left(\hat{\beta} \sum_{r} X_{i}-\beta \sum_{r} X_{i}-\sum_{r} \varepsilon_{i}\right) \ &=\left[E_{m}(\hat{\beta})-\beta\right] \sum_{r} X_{i} \end{aligned}
that is, if
\begin{aligned} \beta &=E_{m} \hat{\beta} \ &=E_{m} \sum_{i \in s} B_{i}\left(\beta X_{i}+\varepsilon_{i}\right) \ &=\beta \sum_{i \in s} B_{i} X_{i} \end{aligned} which is equivalent to $$\sum_{i \in s} B_{i} X_{i}=1 .$$
Note that the predictor for $Y$ then takes the form
\begin{aligned} t &=\sum_{i \in s}\left(1+B_{i} \sum_{r} X_{j}\right) Y_{i} \ &=\sum_{i \in s} a_{s i} Y_{i}, \text { say } \end{aligned}
and
\begin{aligned} \sum a_{s i} X_{i} &=\sum_{i \in s} X_{i}\left(1+B_{i} \sum_{r} X_{j}\right) \ &=\sum_{s} X_{i}+\sum_{s} X_{i} B_{i} \cdot \sum_{r} X_{j} \ &=X \end{aligned}
This is the equation known from representativity and calibration.

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

We introduce some notations for easy reference to several models.

Arbitrary random variables $Y_{1}, Y_{2}, \ldots, Y_{N}$ may be written as
$$Y_{i}=\mu_{i}+\varepsilon_{i}$$
where $\varepsilon_{1}, \varepsilon_{2}, \ldots, \varepsilon_{N}$ are random variables with
$$E_{m}\left(\varepsilon_{i}\right)=0, V_{m}\left(\varepsilon_{i}\right)=\sigma_{i}^{2}, C_{m}\left(\varepsilon_{i}, \varepsilon_{j}\right)=\rho_{i j} \sigma_{i} \sigma_{j}$$
for $i, j=1,2, \ldots, N$ and $i \neq j$.
A superpopulation model of special importance is defined by the restrictions
\begin{aligned} \mu_{i} &=\beta X_{i} \ \sigma_{i}^{2} &=\sigma^{2} X_{i}^{\gamma} \end{aligned}
with known positive values $X_{i}$ of a nonstochastic variable $x$. This model is denoted by
$$\begin{array}{ll} \mathcal{M}{0 \gamma} & \text { if } \rho{i j}=\rho \text { for all } i \neq j \ \mathcal{M}{1 \gamma} & \text { if } \rho{i j}=0 \text { for all } i \neq j \ \mathcal{M}{2 \gamma} & \text { if } \varepsilon{1}, \varepsilon_{2}, \ldots, \varepsilon_{N} \text { are independent } \end{array}$$
(cf. section 3.2.4). If the assumption $\mu_{i}=\beta X_{i}$ is replaced by
$$\mu_{i}=\alpha+\beta X_{i}$$
we write $\mathcal{M}{j \gamma}^{\prime}$ instead of $\mathcal{M}{j \gamma}$ for $j=0,1,2$.
In the previous section we have shown that the ratio predictor $t_{R}$ is BLU under $\mathcal{M}{11}$ and has the MSE $$M{0}=\frac{N^{2}}{n}(1-f) \frac{\bar{X} \bar{x}{r}}{\bar{x}} \sigma^{2} .$$ It follows from the last formula that if the $n$ units with the largest $X{i}$ ‘s are chosen as to constitute the sample on which to base the BLUP $t_{R}$, then the value of $M_{0}$ will be minimal. So, an optimal sampling design is a purposive one that prescribes to select with probability one a sample of $n$ units with the largest $X_{i}$ values.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Balancing and Robustness for M11

In practice, we never will be sure as to which particular model is appropriate in a given situation. Let us suppose that the model $\mathcal{M}_{11}$ is considered adequate and one contemplates

adopting the optimal strategy $\left(p_{n o}, t_{R}\right)$ for which
$$V_{m}\left(t_{R}-Y\right)=M_{0}=\frac{N^{2}(1-f)}{n} \frac{\bar{X} \bar{x}{r}}{\bar{x}} \sigma^{2}$$ as noted in section 4.1.1. We intend to examine what happens to the performance of this strategy if the correct model is $\mathcal{M}{11}^{\prime}$.
Under $\mathcal{M}{11}^{\prime}$, $$E{m}\left(t_{R}\right)=N \alpha \frac{\bar{X}}{\bar{x}}+\beta X$$
and thus $t_{R}$ has the bias
$$B_{m}\left(t_{R}\right)=E_{m}\left(t_{R}-Y\right)=N \alpha\left(\frac{\bar{X}}{\bar{x}}-1\right)$$
which vanishes if and only if $\bar{x}$ equals $\bar{X}$. So, if instead of the design $p_{n o}$, which is optimal under $\mathcal{M}{11}$, one adopts a design for which $\bar{x}$ equals $\bar{X}$, then $t{R}$, which is $m$-unbiased under $\mathcal{M}{11}$, continues to be $m$-unbiased under $\mathcal{M}{11}^{\prime}$ as well.

A sample for which $\bar{x}$ equals $\bar{X}$ is called a balanced sample and a design that prescribes choosing a balanced sample with probability one is called a balanced design. Hence, based on a balanced sample, $t_{R}$ is robust in respect of model failure.

It is important to note that $t_{R}$ based on a balanced sample is identical to the expansion predictor $N \bar{y}$.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Linear Models and BLU Predictors

∑r是一世=b∑rX一世+∑re一世

0=和米(吨−是) =和米(∑s是一世+b∑rX一世−∑s是一世−∑r是一世) =和米(b^∑rX一世−b∑rX一世−∑re一世) =[和米(b^)−b]∑rX一世

b=和米b^ =和米∑一世∈s乙一世(bX一世+e一世) =b∑一世∈s乙一世X一世这相当于∑一世∈s乙一世X一世=1.

∑一种s一世X一世=∑一世∈sX一世(1+乙一世∑rXj) =∑sX一世+∑sX一世乙一世⋅∑rXj =X

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

μ一世=bX一世 σ一世2=σ2X一世C

（参见第 3.2.4 节）。如果假设μ一世=bX一世被替换为
μ一世=一种+bX一世

## 广义线性模型代考

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