调查的抽样理论 | Sampling Theory of Surveys 代写 STATS 3003 代考

a Suppose we ignore the fpc of a modcl-based estimator. Find
$$V_{M}\left(\sum_{i \in S} \sum_{j \in S_{i}} b_{i j} Y_{i j}\right) \text {. }$$
b Prove (5.39). HINT: Let
$$c_{i j}= \begin{cases}b_{i j}-1 & \text { if } i \in \mathcal{S} \text { and } j \in \mathcal{S}{i} . \ -1 & \text { otherwise. }\end{cases}$$ Then, $\hat{T}-T=\sum{i=1}^{N} \sum_{j=1}^{M_{i}} c_{i j} Y_{i j}$.

(Requires linear aIgebra and calculus.) Although $\hat{T}{r}$ is unbiased for model Ml, constructing an estimator with smaller variance is possible. Let $$c{k}=\frac{m_{k}}{1+\rho\left(m_{k}-1\right)}$$
and
$$\hat{T}{\text {opt }}=\sum{i \in S} \sum_{j \in S_{i}} \frac{c_{i}}{m_{i}}\left[\rho M_{i}+\frac{K-\rho \sum_{k \in S} c_{k} \cdot M_{k}}{\sum_{k \in S} c_{k}}\right] Y_{i j}$$

STATS 3003 COURSE NOTES ：

Then, for one-stage pps sampling, $t_{i} / \psi_{i}=K \bar{y}{i}$, so \begin{aligned} &\hat{t}{\psi}=\frac{K}{n} \sum_{i=1}^{N} Q_{i} \bar{y}{i} \ &\hat{\hat{y}}{\psi}=\frac{1}{n} \sum_{i=1}^{N} Q_{i} \bar{y}_{i} \end{aligned}