Posted on Categories:Survey sampling, 抽样调查, 统计代写, 统计代考

# 统计代写|抽样调查代考Survey sampling代写|Unbiased Estimation of $Y$

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## 统计代写|抽样调查代考Survey sampling代写|Unbiased Estimation of $Y$

Let $E_1, V_1$ denote expectation variance operators for the sampling design in the first stage and $E_L, V_L$ those in the later stages. Let $R_i$ be independent variables satisfying
(a) $E_L\left(R_i\right)=Y_i$,
(b) $V_L\left(R_i\right)=V_i$ or
(c) $V_L\left(R_i\right)=V_{s i}$
and let there exist (b) $)^{\prime}$ random variables $v_i$ such that $E_L\left(v_i\right)=$ $V_i$ or (c)’ random variables $v_{s i}$ such that $E_L\left(v_{s i}\right)=V_{s i}$.

Let $E=E_1 E_L=E_L E_1$ be the overall expectation and $V=$ $E_1 V_L+V_1 E_L=E_L V_1+V_L E_1$ the overall variance operators. CHAUDHURI, ADHIKARI and DIHIDAR (2000a, 2000b) have illustrated how these commutativity assumptions may be valid in the context of survey sampling.
Let
\begin{aligned} t_b & =\sum b_{s i} I_{s i} Y_i, \ M_1\left(t_b\right) & =E_1\left(t_b-Y\right)^2=\sum \sum d_{i j} y_i y_j, \ d_{i j} & =E_1\left(b_{s i} I_{s i}-1\right)\left(b_{s j} I_{s j}-1\right), \end{aligned}

$d_{s i j}$ be constants free of $Y$ such that
$$E_1\left(d_{s i j} I_{s i j}\right)=d_{i j} \forall_{i, j} \text { in } U .$$
Let $w_i$ ‘s be certain non-zero constants. Then, one gets
\begin{aligned} M_1\left(t_b\right)= & -\sum \sum_{i<j} d_{i j} w_i w_j\left(\frac{Y_i}{w_i}-\frac{Y_j}{w_j}\right)^2 \ & +\sum \beta_i \frac{Y_i^2}{w_i} \text { when } \beta_i=\sum_{j=1}^N d_{i j} w_j . \end{aligned}

## 统计代写|抽样调查代考Survey sampling代写|PPSWR Sampling of First-Stage Units

First, from DES RAJ (1968) we note the following. Suppose a PPSWR sample of fsus is chosen in $n$ draws from $U$ using normed size measures $P_i\left(0<P_i<i, \Sigma P_i=1\right)$. Writing $y_r\left(p_r\right)$ for the $Y_i\left(p_i\right)$ value for the unit chosen on the $r$ th draw, $(r=$ $1, \ldots, n)$ the HANSEN-HURWITZ estimator
$$t_{H H}=\frac{1}{n} \sum_{n=1}^n \frac{y_r}{p_r}$$

might be used to estimate $Y$ because $E_p\left(t_{H H}\right)=Y$ if $Y_i$ could be ascertained. But since $Y_i$ ‘s are not ascertainable, suppose that each time an fsu $i$ appears in one of the $n$ independent draws by PPSWR method, an independent subsample of elements is selected in subsequent stages in such a manner that estimators $\hat{y}r$ for $y_r$ are available such that $E_L\left(\hat{y}_r\right)=y_r$ and $V_L\left(\hat{y}_r\right)=\sigma_r^2$ with uncorrelated $y_1, y_2, \ldots, y_n$. Then, DAS RAJ’s (1968) proposed estimator for $Y$ is $$e_H=\frac{1}{n} \sum{r=1}^n \frac{\hat{y}r}{p_r}$$ for which the variance is \begin{aligned} V\left(e_H\right) & =V_p\left(t{H H}\right)+E_p\left[\frac{1}{n^2} \sum_{r=1}^n \frac{\sigma_r^2}{p_r^2}\right] \ & =\frac{1}{n} \sum P_i\left(\frac{Y_i}{P_i}-Y\right)^2+\frac{1}{n} \sum_1^N \frac{\sigma_i^2}{P_i} \ & =V_H, \text { say. } \end{aligned}
It follows that
\begin{aligned} & v_H=\frac{1}{2 n^2(n-1)} \sum_{\substack{r=1 r^{\prime}=1 \ r \neq r^{\prime}}}^n\left(\frac{\hat{y}{r^{\prime}}}{p{r^{\prime}}}-\frac{\hat{y}r}{p_r}\right)^2 \ & r \neq r^{\prime} \ & \end{aligned} is an unbiased estimator for $V_H$ because \begin{aligned} E_l\left(v_H\right) & =\frac{1}{2 n^2(n-1)} \sum{r \neq r^{\prime}}\left[\frac{y_r^2}{p_r^2}+\frac{y_{r^{\prime}}^2}{p_{r^{\prime}}^2}+\frac{\sigma_r^2}{p_r^2}+\frac{\sigma_{r^{\prime}}^2}{p_{r^{\prime}}^2}-2 \frac{y_r}{p_r} \frac{y_{r^{\prime}}}{p_{r^{\prime}}}\right] \ E v_H & =E_p E_L\left(v_H\right)=\frac{1}{n}\left(\sum \frac{Y_i^2}{P_i}-Y^2\right)+\frac{1}{n} \sum \frac{\sigma_i^2}{P_i} \ & =\frac{1}{n} \sum P_i\left(\frac{Y_i}{P_i}-Y\right)^2+\frac{1}{n} \sum \frac{\sigma_i^2}{P_i}=V\left(e_H\right) . \end{aligned}

# 抽样调查代写

## 统计代写|抽样调查代考Survey sampling代写|Unbiased Estimation of $Y$

(a) $E_L\left(R_i\right)=Y_i$;
(b) $V_L\left(R_i\right)=V_i$或
(c) $V_L\left(R_i\right)=V_{s i}$

\begin{aligned} t_b & =\sum b_{s i} I_{s i} Y_i, \ M_1\left(t_b\right) & =E_1\left(t_b-Y\right)^2=\sum \sum d_{i j} y_i y_j, \ d_{i j} & =E_1\left(b_{s i} I_{s i}-1\right)\left(b_{s j} I_{s j}-1\right), \end{aligned}

$d_{s i j}$ 使用不含$Y$的常量
$$E_1\left(d_{s i j} I_{s i j}\right)=d_{i j} \forall_{i, j} \text { in } U .$$

\begin{aligned} M_1\left(t_b\right)= & -\sum \sum_{i<j} d_{i j} w_i w_j\left(\frac{Y_i}{w_i}-\frac{Y_j}{w_j}\right)^2 \ & +\sum \beta_i \frac{Y_i^2}{w_i} \text { when } \beta_i=\sum_{j=1}^N d_{i j} w_j . \end{aligned}

## 统计代写|抽样调查代考Survey sampling代写|PPSWR Sampling of First-Stage Units

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