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# 统计代写|抽样调查代考Survey sampling代写|Global Empirical Studies

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## 统计代写|抽样调查代考Survey sampling代写|Global Empirical Studies

Fortunately, considerable empirical studies have been reported by ROYALL and CUMBERLAND (1978b, 1981a, 1981b, 1985) and also by WU and DENG (1983), in light of which the following brief comments seem useful concerning comparative performances of $v_0, v_1, v_2, v_{\hat{g}}, v_{\tilde{g}}, v_{r e g}, v_H, v_D, v_J$, and $v_{\text {gopt }}$ leaving out $v_L$, which is generally disapproved as a viable competitor.
Keeping in mind three key features namely, (1) linear trend, (2) zero intercept, and (3) increasing squared residuals with $x$ in the scatter diagram of $(x, y)$, ROYALL et al. studied appropriate actual populations including one with $N=393$ hospitals with $x$ as the number of beds and $y$ as the number of patients discharged in a particular month. They took $n=32$ for (1) extreme samples, (2) balanced samples with $|\bar{x}-\bar{X}|$ suitably bounded above, (3) SRSWOR samples, (4) best fit samples with a minimal discrepancy among sample- and populationbased cumulative distribution functions. WU and DENG (1983), however, considered only SRSWORs with $n=32$ from the same populations and also from a few others, purposely violating one or the other of the above three characteristics.

Two types of studies have been made. Simulating 1000 SRSWORs of $n=32$ from each population the values of $t_R$ and the above 10 variance estimators $v$, in general, are calculated. The MSE of $t_R$ is taken as
$$M=\frac{1}{1000} \sum^{\prime}\left(\bar{t}R-\bar{Y}\right)^2 .$$ and the bias of $v$ is taken as $$B=\frac{1}{1000} \sum^{\prime} v-M$$ and the root MSE of $v$ is taken as $$R M=\left[\frac{1}{1000} \sum^{\prime}(v-M)^2\right]^{1 / 2} .$$ Each sum $\Sigma^{\prime}$ is over the 1000 simulated samples. Also, for each of the 1000 simulated samples the SZEs $\tau=\left(\bar{t}_R-\bar{Y}\right) / \sqrt{v}$ and the intervals $t_R \pm \tau{\alpha / 2} \sqrt{v}$ are calculated to examine the closeness of $t$ to $\tau$ in terms of mean, standard deviation, skewness, and kurtosis. The df of $t$ is taken as $n-1=31$.
With respect to RM,
(a) $v_{\text {gopt }}$ is found the best, with $v_{\hat{g}}, v_{\tilde{g}}, v_{\text {reg }}$ closely behind.
(b) Among $v_0, v_1, v_2$ the one closest to $v_{\text {gopt }}$ is found the best.
(c) $v_H$ is found to be close to $v_2$ and fairly good, but $v_D$ is found to be poor, and $v_J$ is found to be the worst.

## 统计代写|抽样调查代考Survey sampling代写|Conditional Empirical Studies

From these global studies, where the averages are taken over all of the 1000 simulated samples, it is apparent that different variance estimators may suit different purposes. For example, one with a small MSE may yield a poor coverage probability, while one with a coverage probability close to the nominal value may not be stable, bearing an unacceptably high MSE. To get over this anomaly, these investigators adopt a conditional approach, which seems to be promising.

In a variance estimator alternative to $v_0$ the term $\bar{x}$ occurs as a prominent factor and its closeness to or deviation from $X$ seems to be a crucial factor in determining its performance characteristics. This $\bar{x}$ is an ancillary statistic, that is, the distribution of $\bar{x}$ is free of $\underline{Y}$, and it seems proper to examine how each $v$ performs for a given value of $\bar{x}$ or over several disjoint intervals of values of $\bar{x}$. In other words, for conditional biases, conditional MSEs, and conditional confidence intervals, given $\bar{x}$ may be treated as suitable criteria for judging the relative performances of these variance estimators.

With this end in view, in their empirical studies ROYALL and CUMBERLAND (1978b, 1981a, 1981b, 1985) and Wu and DENG (1983) divided the 1000 simulated samples each of size $n=32$ into 20 groups of 50 each in increasing order of $\bar{x}$ values for the samples. Thus, the first 50 smallest $\bar{x}$ values are placed in the first group, the next 50 larger $\bar{x}$ values are taken in the second group, and so on. Then they calculate
(a) the average of $\bar{x}, A_{\bar{x}}=\frac{1}{50} \Sigma^{\prime} \bar{x}$ for respective groups
(b) the conditional MSE of $t_R$ within respective groups as $M_{\bar{x}}=\frac{1}{50} \Sigma^{\prime \prime}\left(\bar{t}R-\bar{Y}\right)^2$ (c) averages $v{\bar{x}}=\frac{1}{50} \Sigma^{\prime} v$ of each of the $v$ ‘s within respective groups where $\Sigma^{\prime}$ denotes summation over 50 samples within respective groups.

# 抽样调查代写

## 统计代写|抽样调查代考Survey sampling代写|Global Empirical Studies

ROYALL等人考虑到三个关键特征，即(1)线性趋势，(2)零截距，(3)$(x, y)$散点图中$x$的残差平方增加，研究了适当的实际人群，包括$N=393$医院，其中$x$为床位数，$y$为特定月份的出院患者数。他们取$n=32$作为(1)极端样本，(2)平衡样本，$|\bar{x}-\bar{X}|$适当地在上面有界，(3)SRSWOR样本，(4)样本和基于总体的累积分布函数之间差异最小的最佳拟合样本。然而，WU和DENG(1983)只考虑了来自同一种群的$n=32$的SRSWORs，也考虑了来自少数其他种群的SRSWORs，故意违反了上述三个特征中的一个或另一个。

$$M=\frac{1}{1000} \sum^{\prime}\left(\bar{t}R-\bar{Y}\right)^2 .$$ 和偏见 $v$ 被认为是 $$B=\frac{1}{1000} \sum^{\prime} v-M$$ 的根MSE $v$ 被认为是 $$R M=\left[\frac{1}{1000} \sum^{\prime}(v-M)^2\right]^{1 / 2} .$$ 每一笔 $\Sigma^{\prime}$ 超过1000个模拟样本。此外，对于1000个模拟样本中的每个样本，SZEs $\tau=\left(\bar{t}R-\bar{Y}\right) / \sqrt{v}$ 还有间隔 $t_R \pm \tau{\alpha / 2} \sqrt{v}$ 都是用来检查的 $t$ 到 $\tau$ 在均值，标准差，偏度和峰度方面。的df $t$ 被认为是 $n-1=31$． 对于RM， (a) $v{\text {gopt }}$ 是找到了最好的，用了 $v_{\hat{g}}, v_{\tilde{g}}, v_{\text {reg }}$ 紧跟在后面。
(b)其中 $v_0, v_1, v_2$ 最接近的一个 $v_{\text {gopt }}$ 是找到了最好的。
(c) $v_H$ 被发现接近吗 $v_2$ 而且相当不错，但是 $v_D$ 被发现是贫穷的，而 $v_J$ 是最糟糕的。

## 统计代写|抽样调查代考Survey sampling代写|Conditional Empirical Studies

(a)各组别的平均值$\bar{x}, A_{\bar{x}}=\frac{1}{50} \Sigma^{\prime} \bar{x}$
(b)各组内$t_R$的条件均方差为$M_{\bar{x}}=\frac{1}{50} \Sigma^{\prime \prime}\left(\bar{t}R-\bar{Y}\right)^2$ (c)各组内各$v$的平均$v{\bar{x}}=\frac{1}{50} \Sigma^{\prime} v$，其中$\Sigma^{\prime}$为各组内50个样本的总和。

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