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抽样调查Survey sampling是主流统计的边缘。这里的特殊之处在于，我们有一个具有某些特征的有形物体集合，我们打算通过抓住其中一些物体并试图对那些未被触及的物体进行推断来窥探它们。这种推论传统上是基于一种概率论，这种概率论被用来探索观察到的事物与未观察到的事物之间的可能联系。这种概率不被认为是在统计学中，涵盖其他领域，以表征我们感兴趣的变量的单个值之间的相互关系。但这是由调查抽样调查人员通过任意指定的一种技术从具有预先分配概率的对象群体中选择样本而创建的。

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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

因此，ROYALL和CUMBERLAND (1981b, 1985)为了正确选择$V_m\left(t_r-\bar{Y}\right)$的估计量进行了实证研究，因为模型在实践中无法正确假设。DENG和Wu(1987)也进行了实证调查，以正确地从这几个方差估计量中进行选择。但是他们也检查了所有上述估计器$v$的设计偏差和设计mse，每个都被他们作为$V$的估计器，只考虑SRSWOR。关于它们的理论研究是基于设计的，并且由于估计量的复杂性，它们的分析是渐近的。从他们的理论结果来看，$v_D$似乎是基于设计的考虑中最有希望的方差估计器，而$v_L$和$v_{l r}$都很差。

在ROYALL和CumberLand (1981b, 1985)以及DENG和Wu(1987)进行的实证研究中，从几个种群中模拟了1000个简单随机样本，每个样本的大小为$n=32$，其中一个种群的大小为$N=393$。对于这1000个SRSWORs中的每一个，计算$t_r, \bar{x}, v_0, v_1, v_2, v_{\hat{g}}, v_L, v_H, v_D$和$v_J$的值。估计器$v_{l r}$被发现太差，不能被承认为可行的竞争者，并被上述作者丢弃。对于每个样本，对于每个方差估计器$v$，如上所述，还计算了SZEs和置信区间
$$
\tau=\left(t_r-\bar{Y}\right) / \sqrt{v} \text { and } t_r \pm \tau_{\alpha / 2} \sqrt{v}
$$
将$\tau_{\alpha / 2}$作为STUDENT的$t$分布的上尾的$100 \alpha / 2 \%$点，在本例中为$d f=n-2=30$。
首先，从整个样本的研究，无条件的行为是用总体平均值分别表示
$$
\begin{aligned}
\bar{M} & =\frac{1}{1000} \Sigma^{\prime}\left(t_r-\bar{Y}\right)^2, \text { the MSE } \
B & =\frac{1}{1000} \Sigma^{\prime} v-\bar{M}, \text { the bias, }
\end{aligned}
$$
$\Sigma^{\prime}$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
$$
\frac{1}{100} \Sigma^{\prime} \bar{x}, \frac{1}{100} \Sigma^{\prime} v,
$$
分别计算，$\Sigma^{\prime}$表示各自组中100个样本的总和以及与置信区间相关的估计覆盖概率。因此，研究与$t_r$相关的方差估计量的无条件和条件行为，遵循与7.1节中讨论的与比率估计量$t_R$相关的方差估计相同的两种方法。对估计量进行MSE、偏差和相关的条件和无条件覆盖概率的比较。

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

我们给出了HTE方差的公式$\bar{t}=\sum_{i \in s} \frac{Y_i}{\pi_i}$，该公式基于YATES和GRUNDY(1953)和SEN(1953)提供的固定样本量设计，以及其中的无偏估计量$v_{Y G}$。对于没有样本量限制的设计，HORVITZ和THOMPSON(1952)自己给出的相应公式也注明为
$$
\begin{aligned}
V_p(t) & =\sum_i \frac{Y_i^2}{\pi_i}+\sum_{i \neq j} Y_i Y_j \frac{\pi_{i j}}{\pi_i \pi_j}-Y^2 \
v_p(\bar{t}) & =\sum_s Y_i^2 \frac{1-\pi_i}{\pi_i^2}+\sum_{i \neq j \in s} \sum_i Y_i Y_j \frac{\pi_{i j}-\pi_i \pi_j}{\pi_i \pi_j \pi_{i j}} .
\end{aligned}
$$
众所周知，$v_p(\bar{t})$对于高选择概率的样本具有承载负值的缺陷。对于不受约束的设计，估计器$v_{Y G}$也可能是负的
$$
\pi_i \pi_j \geq \pi_{i j} \text { for all } i \neq j
$$
从BIYANI(1980)的作品中可以看出。为了解决负方差估计量的问题，JESSEN(1969)提出了如下的方差估计量
$$
\left.v_J=\bar{W} \sum_{i<j \in s} \sum_{\frac{Y_i}{\pi_i}}-\frac{Y_j}{\pi_j}\right]^2
$$
在哪里
$$
\bar{W}=\frac{n-\sum \pi_i^2}{N(N-1)},
$$
以$n$为固定样本量。
它是非负的，不含$\pi_{i j}$，形式很简单。

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