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

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

ROYALL and CUMBERLAND (1981b, 1985) therefore made empirical studies in an effort to make a right choice of an estimator for $V_m\left(t_r-\bar{Y}\right)$ because a model cannot be correctly postulated in practice. DENG and Wu (1987) also pursued with an empirical investigation to rightly choose from these several variance estimators. But they also examined the design biases and design MSEs of all the above-noted estimators $v$, each taken by them as an estimator for $V$, considering SRSWOR only. The theoretical study concerning them is design based, and because of the complicated nature of the estimators their analysis is asymptotic. From their theoretical results $v_D$ seems to be the most promising variance estimator from the designbased considerations and $v_L$ and $v_{l r}$ are both poor.

In the empirical studies undertaken by ROYALL and CumberLand (1981b, 1985) and DENG and Wu (1987) 1000 simple random samples of size $n=32$ each are simulated from several populations including one of size $N=393$. For each of these 1000 SRSWORs values of $t_r, \bar{x}, v_0, v_1, v_2, v_{\hat{g}}, v_L, v_H, v_D$, and $v_J$ are calculated. The estimator $v_{l r}$ is found too poor to be admitted as a viable competitor and is discarded by the authors mentioned. For each sample again for each of these variance estimators $v$, as above, the SZEs and confidence intervals are also calculated
$$\tau=\left(t_r-\bar{Y}\right) / \sqrt{v} \text { and } t_r \pm \tau_{\alpha / 2} \sqrt{v}$$
with $\tau_{\alpha / 2}$ as the $100 \alpha / 2 \%$ point in the upper tail of the STUDENT’s $t$ distribution with $d f=n-2=30$ in this case.
First, from the study of the entire sample the unconditional behavior is reviewed using the overall averages to denote respectively by
\begin{aligned} \bar{M} & =\frac{1}{1000} \Sigma^{\prime}\left(t_r-\bar{Y}\right)^2, \text { the MSE } \ B & =\frac{1}{1000} \Sigma^{\prime} v-\bar{M}, \text { the bias, } \end{aligned}
$\Sigma^{\prime}$ denoting the sum over the 1000 simulated samples. Again, taking $\bar{x}$ as the ancillary statistic conditional (given $\bar{x}$ ) behavior is examined on dividing the 1000 simulated samples into 10 groups, each consisting of 100 samples with the closest values of $\bar{x}$ within each, the groups being separated according to changes in the values of $\bar{x}$. For each group
$$\frac{1}{100} \Sigma^{\prime} \bar{x}, \frac{1}{100} \Sigma^{\prime} v,$$
are separately calculated, $\Sigma^{\prime}$ denoting the sum over the 100 samples in respective groups and the estimated coverage probabilities associated with the confidence intervals. Thus, the unconditional and the conditional behavior of variance estimators related to $t_r$ are investigated, following the same two approaches as with variance estimation related to the ratio estimator $t_R$ discussed in section 7.1. The estimators are compared with respect to MSE, bias, and associated conditional and unconditional coverage probabilities.

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

We presented the formula for the variance of the HTE $\bar{t}=\sum_{i \in s} \frac{Y_i}{\pi_i}$ based on a fixed sample size design available due to YATES and GRUNDY (1953) and SEN (1953), along with an unbiased estimator $v_{Y G}$ thereof. For designs without restriction on sample size the corresponding formulae given by HORVITZ and THOMPSON (1952) themselves were also noted as
\begin{aligned} V_p(t) & =\sum_i \frac{Y_i^2}{\pi_i}+\sum_{i \neq j} Y_i Y_j \frac{\pi_{i j}}{\pi_i \pi_j}-Y^2 \ v_p(\bar{t}) & =\sum_s Y_i^2 \frac{1-\pi_i}{\pi_i^2}+\sum_{i \neq j \in s} \sum_i Y_i Y_j \frac{\pi_{i j}-\pi_i \pi_j}{\pi_i \pi_j \pi_{i j}} . \end{aligned}
It is well known that $v_p(\bar{t})$ has the defect of bearing negative values for samples with high selection probabilities. The estimator $v_{Y G}$ may also turn out negative for designs not subject to the constraints
$$\pi_i \pi_j \geq \pi_{i j} \text { for all } i \neq j$$
as may be seen in BIYANI’s (1980) work. To get rid of this problem of negative variance estimators, JESSEN (1969) proposed the following variance estimator
$$\left.v_J=\bar{W} \sum_{i<j \in s} \sum_{\frac{Y_i}{\pi_i}}-\frac{Y_j}{\pi_j}\right]^2$$
where
$$\bar{W}=\frac{n-\sum \pi_i^2}{N(N-1)},$$
with $n$ as the fixed sample size.
This is uniformly non-negative and is free of $\pi_{i j}$ and very simple in form.

KUMAR, GUPTA and AGARWAL (1985), following JESSEN (1969), suggest the following uniformly non-negative variance estimator for $V_p(t)$, namely,

# 抽样调查代考

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

$$\tau=\left(t_r-\bar{Y}\right) / \sqrt{v} \text { and } t_r \pm \tau_{\alpha / 2} \sqrt{v}$$

\begin{aligned} \bar{M} & =\frac{1}{1000} \Sigma^{\prime}\left(t_r-\bar{Y}\right)^2, \text { the MSE } \ B & =\frac{1}{1000} \Sigma^{\prime} v-\bar{M}, \text { the bias, } \end{aligned}
$\Sigma^{\prime}$表示1000个模拟样本的和。同样，以$\bar{x}$为辅助统计，条件(给定$\bar{x}$)将1000个模拟样本分为10组，每组由100个样本组成，每组中最接近的值为$\bar{x}$，根据$\bar{x}$值的变化将组分开。对于每一组
$$\frac{1}{100} \Sigma^{\prime} \bar{x}, \frac{1}{100} \Sigma^{\prime} v,$$

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

\begin{aligned} V_p(t) & =\sum_i \frac{Y_i^2}{\pi_i}+\sum_{i \neq j} Y_i Y_j \frac{\pi_{i j}}{\pi_i \pi_j}-Y^2 \ v_p(\bar{t}) & =\sum_s Y_i^2 \frac{1-\pi_i}{\pi_i^2}+\sum_{i \neq j \in s} \sum_i Y_i Y_j \frac{\pi_{i j}-\pi_i \pi_j}{\pi_i \pi_j \pi_{i j}} . \end{aligned}

$$\pi_i \pi_j \geq \pi_{i j} \text { for all } i \neq j$$

$$\left.v_J=\bar{W} \sum_{i<j \in s} \sum_{\frac{Y_i}{\pi_i}}-\frac{Y_j}{\pi_j}\right]^2$$

$$\bar{W}=\frac{n-\sum \pi_i^2}{N(N-1)},$$

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