## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimation of Population Proportion

Let $y_i=1$ if the $i$ th unit belongs to the group $A$, and let $y_i=0$ if the $i$ th unit does not belong to the group $A$. In this case $Y=N_A=$ total number of units that possess the attribute $A$ and $\bar{Y}=N_A / N=\pi_A=$ proportion of units in the population belonging to the group $A ; \bar{\gamma}\left(s_o\right)=n_A / n=$ $\widehat{\pi}A=$ proportion of units in the sample $s_o$ belonging to $A$, where $n_A$ is the total number of units in the sample that fall in group $A$. Now noting that (i) $\sigma_y^2=\sum{i=1}^N\left(y_i-\bar{Y}\right)^2 / N=\sum_{i=1}^N y_i / N-\bar{Y}^2=\pi_A\left(1-\pi_A\right)$ since $y_i=0$ or 1; and
(ii) $\widehat{\sigma}y^2=\sum{r=1}^n\left{\gamma_{(r)}-\bar{\gamma}\left(s_o\right)\right}^2 /(n-1)=\frac{n}{n-1} \widehat{\pi}_A\left(1-\widehat{\pi}_A\right)$ we have the following theorem:
Theorem 3.3.2
(i) $\widehat{\pi}_A$ is an unbiased estimator for the population proportion $\pi_A$.
(ii) Variance of $\widehat{\pi}_A$ is $V\left(\widehat{\pi}_A\right)=\frac{\pi_A\left(1-\pi_A\right)}{n}$
(iii) An unbiased estimator of $V\left(\widehat{\pi}_A\right)$ is $\widehat{V}\left[\widehat{\pi}_A\right]=\frac{\widehat{\pi}_A\left(1-\widehat{\pi}_A\right)}{n-1}$
Remark 3.3.1
It is important to note that Theorems $3.3 .1$ and $3.3 .2$ can be obtained from Theorems $3.2 .2$ and $3.2 .8$ when $N$ is sufficiently large compared to $n$ so that the finite population correction term $f_n=n / N$ is ignored.
Example 3.3.1
From the list of 30 students given in the Example 3.2.1, select a sample of size 8 by the SRSWR method. From the selected sample, (i) estimate the mean height and weight of students (male and female combined) and obtain the variances of the estimator used. Estimate the SEs of the estimators. (ii) Estimate the proportion of the male and female students with their SEs.

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

It follows from Section 2.7.3 that the estimator $\bar{\gamma}\left(s_o\right)$ is inadmissible because it is based on ordered data, which may consist of repletion of units, hence $\bar{y}\left(s_o\right)$ is not a function of a sufficient statistic. Let $s=\left(j_1, \ldots, j_v\right)$ denote the unordered sample obtained by taking distinct $\nu$ units $j_1, \ldots, j_\nu$ with $j_1<\cdots<j_v$. The unordered sample $s$ is a sufficient statistic. Hence, we can improve the inefficient estimator $\bar{y}\left(s_o\right)$ by applying the RaoBlackwellization technique. The improved estimator is given by $E\left[\bar{\gamma}\left(s_o\right) \mid s\right]=\bar{\gamma}s=\sum{i \in s} \gamma_i / \nu=$ the sample mean based on the distinct units. The details have been given in the following theorems.
Theorem 3.3.3
Let $\bar{\gamma}s=\sum{i \in s} \gamma_i / \nu=\sum_{k=1}^\nu \gamma_{j_k} / \nu$ be the sample mean based on the distinct units of $s_0$. Then,
(i) $E\left[\bar{\gamma}s\right]=E\left[\bar{\gamma}\left(s_o\right)\right]=\bar{Y}$ (ii) $V\left[\bar{y}_s\right] \leq V\left[\bar{\gamma}\left(s_o\right)\right]$ Proof Let $n_i\left(s_o\right)$ denote the number of times the ith unit appears in $s_o$. Then writing, \begin{aligned} \bar{y}\left(s_o\right) &=\sum{r=1}^n \gamma_{(r)} / n \ &=\sum_{i=1}^N n_i\left(s_o\right) y_i / n \ &=\sum_{k=1}^p n_{j k}\left(s_e\right) \gamma_{j k} / n \end{aligned}
where $n_{j_1}\left(s_o\right), \ldots, n_{j_\gamma}\left(s_o\right)$ denote the number of times the distinct units $j_1, \ldots, j_\nu$ appear in $s_0$.

# 抽样调查代考

## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimation of Population Proportion

ii)

(i) $\widehat{\pi}_A$ 是人口比例的无偏估计量 $\pi_A$.
(ii) 差异 $\widehat{\pi}_A$ 是 $V\left(\widehat{\pi}_A\right)=\frac{\pi_A\left(1-\pi_A\right)}{n}$
(iii) 无偏估计 $V\left(\widehat{\pi}_A\right)$ 是 $\widehat{V}\left[\widehat{\pi}_A\right]=\frac{\hat{\pi}_A\left(1-\widehat{\pi}_A\right)}{n-1}$

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

(一) $E[\bar{\gamma} s]=E\left[\bar{\gamma}\left(s_o\right)\right]=\bar{Y}$ (二) $V\left[\bar{y}s\right] \leq V\left[\bar{\gamma}\left(s_o\right)\right]$ 证明让 $n_i\left(s_o\right)$ 表示第 i 个单位出现的次数 $s_o$. 然后 写作， $$\bar{y}\left(s_o\right)=\sum r=1^n \gamma{(r)} / n \quad=\sum_{i=1}^N n_i\left(s_o\right) y_i / n=\sum_{k=1}^p n_{j k}\left(s_e\right) \gamma_{j k} / n$$

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