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统计代写|概率与统计作业代写Probability and Statistics代考|Mean Squared Error

A small bias of an estimator under a sampling plan does not guarantee that individual sample results $\hat{\theta}_k$ are actually close to the population parameter $\theta$; it just states that they are close on average, if we were to sample over and over again. However, we often only collect one sample, and thus the performance of an estimator on average is not our only concern. We are also concerned with the variability of the estimator across different samples. To capture the variability in the sample results $\hat{\theta}1, \hat{\theta}_2, \ldots, \hat{\theta}_K$ with respect to the true value $\theta$, we use the so-called mean squared error (MSE). This is defined as $$ \mathrm{MSE}=\sum{k=1}^K\left(\hat{\theta}_k-\theta\right)^2 \pi_k
$$
The MSE measures the weighted average squared distance of the sample results $\hat{\theta}_1, \hat{\theta}_2, \ldots, \hat{\theta}_K$ from the population parameter $\theta$. The weights are again determined by the sampling probabilities. Often, the smaller the MSE the better the sampling plan. Sometimes the root mean square error (RMSE) is reported, which is simply $\mathrm{RMSE}=\sqrt{\mathrm{MSE}}$.

Another measure that is relevant to sampling plans, and closely related to the bias and MSE, is the standard error (SE). The standard error is defined as
$$
\mathrm{SE}=\sqrt{\sum_{k=1}^K\left(\hat{\theta}_k-\mathbb{E}(T)\right)^2 \pi_k}
$$
It represents the variability of the sampling plan with respect to the expected population parameter $\mathbb{E}(T)$ instead of using the true population parameter $\theta$. Note that the standard error of an estimator is used as a measure to represent our uncertainty regarding an estimate. In many examples the standard error contains population parameters (see Sect. 2.6) that we do not know. To use standard errors in practice we have to estimate the standard error as well, and this is what researchers and professionals typically do. We will explore this in more detail below when we derive analytical expressions for the bias and standard error of the sample mean, sample variance, and sample proportion for simple random sampling, systematic sampling, stratified sampling, and one-stage and two-stage cluster random sampling, respectively.

Figure $2.2$ shows how bias, MSE, and SE relate: if the bias is small, $\mathbb{E}(T)$ is close to the parameter value $\theta$. On the other hand, if the bias is large, $\mathbb{E}(T)$ is not close to $\theta$. If the MSE is small, the variability of the $\hat{\theta}_k$ ‘s around $\theta$ is small, while if the MSE is large, the variability around $\theta$ is large. If the SE is small, the variability of the $\hat{\theta}_k$ ‘s around $\mathbb{E}(T)$ is small. Finally, note that if the sampling plan is unbiased and thus $\mathbb{E}(T)=\theta$, the RMSE and the SE are identical. More generally, it can be demonstrated that
$$
\mathrm{RMSE}=\sqrt{\mathrm{SE}^2+(\mathbb{E}(T)-\theta)^2}
$$
Thus the RMSE is never smaller than the SE.

统计代写|概率与统计作业代写Probability and Statistics代考|Illustration of a Comparison of Sampling Plans

Here we will assume that we have full knowledge about the population, so that we can evaluate the bias, standard error, and mean squared error for different sampling plans. Clearly, in practice we never have this information, otherwise the sampling becomes obsolete. However, we often do have some knowledge of the population in practice, using information from registries or historical data, and this information can be used to evaluate different strategies, often in combination with simulations (see Sect. 2.5.4).

Our population of interest is provided in Table 2.1, which is taken from Table $2.3$ of Levy and Lemeshow (2013). The population consists of six schools in a community with in each school the total number of students and the number of students that were not immunized for measles. In total there are 314 students, of which 30 students are not immunized for measles. The population parameter of interest is $\theta=30 / 314=$ $0.09554$, the proportion of students not being immunized for measles. We assume that schools 1,3, and 4 are located in the north of the community and the schools 2 , 5 , and 6 are located in the south. Two sampling approaches are being considered: a single-stage cluster sample with a simple random sample of two clusters (schools) and a single-stage cluster sample with stratified sampling of two clusters. For the stratified sampling, the strata are north and south.

For the single-stage cluster sampling with simple random sampling, there are $K=6 ! /[2 ! \times 4 !]=15$ possible samples of two schools: $S_1=(1,2) ; S_2=$ $(1,3) ; \ldots ; S_6=(1,6) ; S_7=(2,3) ; \ldots ; S_{10}=(2,6) ; \ldots ; S_{15}=(5,6)$. Each pair of schools has the same probability of being collected, i.e., $\pi_k=1 / K=1 / 15$ for $k=1,2, \ldots, 15$. The expected population parameter for this sampling approach is given by
$$
\begin{aligned}
\mathbb{E}(T)=& \sum_{k=1}^K \hat{\theta}_k \pi_k \
=& {\left[\frac{9}{87}+\frac{7}{149}+\frac{7}{103}+\frac{11}{95}+\frac{12}{116}+\frac{8}{118}+\frac{8}{72}+\frac{12}{64}\right.} \
\left.\quad+\frac{13}{85}+\frac{6}{134}+\frac{10}{126}+\frac{11}{147}+\frac{10}{80}+\frac{11}{101}+\frac{15}{93}\right] \times \frac{1}{15} \
=& 0.10341 .
\end{aligned}
$$
The bias is therefore determined by bias $=0.10341-0.09554=0.00787$. Thus the simple random sample of two schools (single stage cluster sample) is not fully unbiased, but the bias is relatively small.

统计代写|概率与统计作业代写Probability and Statistics代考|EAS305

概率与统计代考

统计代写|概率与统计作业代写概率与统计代考|均方误差


抽样计划下估计量的小偏差并不能保证个别样本结果$\hat{\theta}_k$实际上接近总体参数$\theta$;它只是表明,如果我们反复抽样,它们的平均值接近。然而,我们通常只收集一个样本,因此估计器的平均性能不是我们唯一关心的。我们还关注不同样本间估计量的可变性。为了捕捉样本结果$\hat{\theta}1, \hat{\theta}_2, \ldots, \hat{\theta}_K$相对于真实值$\theta$的可变性,我们使用所谓的均方误差(MSE)。这被定义为$$ \mathrm{MSE}=\sum{k=1}^K\left(\hat{\theta}_k-\theta\right)^2 \pi_k
$$
。MSE度量样本结果$\hat{\theta}_1, \hat{\theta}_2, \ldots, \hat{\theta}_K$到总体参数$\theta$的加权平均平方距离。权重由抽样概率决定。通常,均方误差越小,抽样方案越好。有时报告的均方根误差(RMSE)是$\mathrm{RMSE}=\sqrt{\mathrm{MSE}}$ .


另一种与抽样方案相关并与偏差和MSE密切相关的度量是标准误差(SE)。标准误差定义为
$$
\mathrm{SE}=\sqrt{\sum_{k=1}^K\left(\hat{\theta}_k-\mathbb{E}(T)\right)^2 \pi_k}
$$
它表示抽样计划相对于期望总体参数$\mathbb{E}(T)$而不是使用真实总体参数$\theta$的可变性。注意,估计量的标准误差是用来表示我们对估计的不确定性的一种度量。在许多例子中,标准误差包含我们不知道的总体参数(见第2.6节)。为了在实践中使用标准误差,我们还必须估计标准误差,这是研究人员和专业人员通常做的事情。当我们分别推导简单随机抽样、系统抽样、分层抽样以及一阶段和两阶段聚类随机抽样的样本均值、样本方差和样本占比的偏差和标准误差的解析表达式时,我们将在下面更详细地探讨这一点

图$2.2$显示了偏差、MSE和SE之间的关系:如果偏差很小,$\mathbb{E}(T)$接近参数值$\theta$。另一方面,如果偏差很大,则$\mathbb{E}(T)$不接近$\theta$。当均方误差较小时,$\hat{\theta}_k$在$\theta$附近的变异性较小,当均方误差较大时,$\theta$附近的变异性较大。如果SE很小,$\hat{\theta}_k$在$\mathbb{E}(T)$附近的变异性也很小。最后,请注意,如果抽样计划是无偏的,因此$\mathbb{E}(T)=\theta$,则RMSE和SE是相同的。更一般地,可以证明
$$
\mathrm{RMSE}=\sqrt{\mathrm{SE}^2+(\mathbb{E}(T)-\theta)^2}
$$
因此RMSE从不小于SE。

统计代写|概率与统计作业代写Probability and Statistics代考|Illustration of a Comparison of Sampling Plans


这里我们假设我们对总体有充分的了解,因此我们可以评估不同抽样方案的偏差、标准误差和均方误差。显然,在实际操作中,我们永远得不到这些信息,否则抽样就过时了。然而,在实践中,我们经常使用来自注册表或历史数据的信息,对总体有一定的了解,这些信息可用于评估不同的策略,通常与模拟相结合(见第2.5.4节)


我们感兴趣的总体在表2.1中提供,该表摘自Levy和Lemeshow(2013)的表$2.3$。人口由一个社区的六所学校组成,每所学校的学生总数和未接种麻疹疫苗的学生人数。共有314名学生,其中30名学生没有接种麻疹疫苗。我们感兴趣的总体参数是$\theta=30 / 314=$$0.09554$,即未接种麻疹疫苗的学生的比例。我们假设学校1、3和4位于社区的北部,学校2、5和6位于社区的南部。目前正在考虑两种抽样方法:一种是单阶段聚类抽样,采用两个聚类(学校)的简单随机抽样;另一种是单阶段聚类抽样,采用两个聚类的分层抽样。对于分层采样,地层分为南北两部分

对于简单随机抽样的单阶段聚类抽样,有两个学派的$K=6 ! /[2 ! \times 4 !]=15$可能样本:$S_1=(1,2) ; S_2=$$(1,3) ; \ldots ; S_6=(1,6) ; S_7=(2,3) ; \ldots ; S_{10}=(2,6) ; \ldots ; S_{15}=(5,6)$。每对学校被收集的概率是相同的,即$k=1,2, \ldots, 15$为$\pi_k=1 / K=1 / 15$。该抽样方法的期望总体参数由
$$
\begin{aligned}
\mathbb{E}(T)=& \sum_{k=1}^K \hat{\theta}_k \pi_k \
=& {\left[\frac{9}{87}+\frac{7}{149}+\frac{7}{103}+\frac{11}{95}+\frac{12}{116}+\frac{8}{118}+\frac{8}{72}+\frac{12}{64}\right.} \
\left.\quad+\frac{13}{85}+\frac{6}{134}+\frac{10}{126}+\frac{11}{147}+\frac{10}{80}+\frac{11}{101}+\frac{15}{93}\right] \times \frac{1}{15} \
=& 0.10341 .
\end{aligned}
$$
给出,因此偏差由偏差$=0.10341-0.09554=0.00787$决定。因此,两个学派的简单随机样本(单阶段聚类样本)并非完全无偏,但偏倚相对较小

统计代写|概率与统计作业代写Probability and Statistics代考

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