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## 统计代写|抽样调查作业代写sampling theory of survey代考|Further Model-Based Optimality Results

Avoiding details, we may briefly mention a few recently available optimality results of interest under certain superpopulation models related to the models considered so far.
Postulating independence of $Y_{i}$ ‘s subject to
(a) $E_{m}\left(Y_{i}\right)=\alpha_{i}+\beta X_{i}$
with $X_{i}(>0), \underline{\alpha}=\left(\alpha_{1}, \ldots, \alpha_{N}\right)^{\prime}, \beta$ known and
(b) $V_{m}\left(Y_{i}\right)=\sigma^{2} f_{i}^{2}$
$\sigma(>0)$ unknown, $f_{i}(>0)$ known, $i=1, \ldots, N$
GoDAMBE (1982) showed that a strategy $\left(p_{n}^{}, e^{}\right)$ is optimal among all strategies $\left(p_{n}, e\right)$ with $E_{p_{n}}(e)=Y$ in the sense that
$$E_{m} V_{p_{n}}(e) \geq \sigma^{2}\left[\left(\sum f_{i}\right)^{2} / n-\sum f_{i}^{2}\right]=E_{m} V_{p_{n}^{}}\left(e^{}\right)$$
for all $\underline{Y}$. Here $p_{n}^{}$ is a $p_{n}$ for which $\pi_{i}$ equals $$\pi_{i}^{}=n f_{i} / \sum_{j=1}^{N} f_{j}$$
and
\begin{aligned} e^{} &=\sum_{i \in s}\left(Y_{i}-\alpha_{i}-\beta X_{i}\right) / \pi_{i}^{}+\sum_{1}^{N}\left(\alpha_{i}+\beta X_{i}\right) \ &=t(\underline{\alpha}, \beta), \text { say } \end{aligned}
which is the generalized difference estimator (GDE) in this case.

TAM (1984) revised the above model, relaxing independence and postulating the covariance structure specified by
$$C_{m}\left(Y_{i}, Y_{j}\right)=\rho \sigma^{2} f_{i} f_{j}$$
with $\rho(0 \leq \rho \leq 1)$ unknown, but considered only LUEs $e=a_{s}+\sum_{i \in s} b_{s i} Y_{i}=e_{L}$, say.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimating Functions and Equations

Suppose $\underline{Y}=\left(Y_{1}, \ldots, Y_{N}\right)^{\prime}$ is a random vector and $\underline{X}=\left(X_{1}, \ldots,\right.$, $\left.X_{N}\right)^{\prime}$ is a vector of known numbers $X_{i}(>0), i=1, \ldots, N$. Let the $Y_{i}$ ‘s be independent and normally distributed with means and variances, respectively
$\theta X_{i}$ and $\sigma_{i}^{2}, i=1, \ldots, N .$
If all the $Y_{i}$ ‘s $i=1, \ldots, N$ are available for observation, then from the joint probability density function (pdf) of $\underline{Y}$
$$p(\underline{Y}, \theta)=\prod_{i=1}^{N} \frac{1}{\sigma_{i} \sqrt{2 \pi}} e^{-\frac{1}{2 \sigma_{i}^{2}}\left(Y_{i}-\theta X_{i}\right)^{2}}$$
one gets the well-known maximum likelihood estimator (MLE) $\theta_{0}$, based on $\underline{Y}$, for $\theta$, given by the solution of the likelihood equation
$$\frac{\partial}{\partial \theta} \log p(\underline{Y}, \theta)=0$$
as
$$\theta_{0}=\left[\sum_{1}^{N} Y_{i} X_{i} / \sigma_{i}^{2}\right] /\left[\sum_{1}^{N} X_{i}^{2} / \sigma_{i}^{2}\right]$$

On the other hand, let the normality assumption above be dropped, everything else remaining unchanged, that is, consider the linear model
$$Y_{i}=\theta X_{i}+\varepsilon_{i}$$
with $\varepsilon_{i}$ ‘s distributed independently and
$$E_{m}\left(\varepsilon_{i}\right)=0, V_{m}\left(\varepsilon_{i}\right)=\sigma_{i}^{2}, i=1, \ldots, N .$$
Then, if $\left(Y_{i}, X_{i}\right), i=1, \ldots, N$ are observed, one may derive the same $\theta_{0}$ above as the least squares estimator (LSE) or as the best linear unbiased estimator (BLUE) for $\theta$.

Such a $\theta_{0}$, based on the entire finite population vector $\underline{Y}=\left(Y_{1}, \ldots, Y_{N}\right)^{\prime}$, is really a parameter of this population itself and will be regarded as a census estimator.

# 抽样调查代考

## 统计代写|抽样调查作业代写sampling theory of survey代考|Further Model-Based Optimality Results

(a) $E_{m}\left(Y_{i}\right)=\alpha_{i}+\beta X_{i}$

(b) $V_{m}\left(Y_{i}\right)=\sigma^{2} f_{i}^{2}$
$\sigma(>0)$ 末知， $f_{i}(>0)$ 已知， $i=1, \ldots, N$
GoDAMBE (1982) 表明，一种策略 $\left(p_{n}, e\right)$ 在所有策略中是最优的 $\left(p_{n}, e\right)$ 和 $E_{p_{n}}(e)=Y$ 在某种意义上说
$$E_{m} V_{p_{n}}(e) \geq \sigma^{2}\left[\left(\sum f_{i}\right)^{2} / n-\sum f_{i}^{2}\right]=E_{m} V_{p_{n}}(e)$$

$$\pi_{i}=n f_{i} / \sum_{j=1}^{N} f_{j}$$
$$e=\sum_{i \in s}\left(Y_{i}-\alpha_{i}-\beta X_{i}\right) / \pi_{i}+\sum_{1}^{N}\left(\alpha_{i}+\beta X_{i}\right) \quad=t(\underline{\alpha}, \beta), \text { say }$$

TAM (1984) 修改了上述模型，放宽了独立性并假设了由下式指定的协方差结构
$$C_{m}\left(Y_{i}, Y_{j}\right)=\rho \sigma^{2} f_{i} f_{j}$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimating Functions and Equations

$$p(\underline{Y}, \theta)=\prod_{i=1}^{N} \frac{1}{\sigma_{i} \sqrt{2 \pi}} e^{-\frac{1}{22_{i}^{2}}\left(Y_{i}-\theta X_{i}\right)^{2}}$$

$$\frac{\partial}{\partial \theta} \log p(\underline{Y}, \theta)=0$$
$$\theta_{0}=\left[\sum_{1}^{N} Y_{i} X_{i} / \sigma_{i}^{2}\right] /\left[\sum_{1}^{N} X_{i}^{2} / \sigma_{i}^{2}\right]$$

$$Y_{i}=\theta X_{i}+\varepsilon_{i}$$

$$E_{m}\left(\varepsilon_{i}\right)=0, V_{m}\left(\varepsilon_{i}\right)=\sigma_{i}^{2}, i=1, \ldots, N .$$

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