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统计代写|概率与统计作业代写Probability and Statistics代考|Comparing Sampling Plans Using R

In practice we may use the computer to help us evaluate, in particular when we want to use more complex sampling plans, to study other population parameters, or when we want to investigate other estimators. For instance, we may want to select a sample of high-school children again, similar to the data we have in our dataset high-school.csv, to investigate if television watching behavior has changed since the year 2000. We could take the same sampling approach as before, but we could also study alternative sampling procedures that may be more complex and could not be studied previously, since a historical dataset was not available then. Improving the sampling plan, i.e., reducing MSE, can help us lower the number of samples and costs. Based on the historical data we may investigate if stratification on grades helps us reduce the standard error, or maybe we can reduce the number of schools with an increase in the number of students within schools. These types of evaluations may be mathematically difficult, in particular if several other stratifications and cluster samples are already involved, but it may not be so difficult to study these plans with the computer (now that some data are available). In other cases we may even generate our own data or extent the historical data to help evaluate sampling plans and estimators.

Here we will provide a generic approach with $\mathrm{R}$ to be able to investigate sampling plans and estimators. The $\mathrm{R}$ codes are relatively straightforward, but they can be extended and made more complex to address specific situations. The goal is to understand the structure, and not the possible sophistication of sampling and programming. Recall that this type of computer approach requires historical data or some knowledge of the population to be able to mimic data from the population. The general structure is the following:

  1. generate population data,
  2. execute a sampling plan and generate sample data,
  3. compute a statistic using the sample data, and
  4. execute these steps a large number of times and compute bias, MSE and SE using the results.

Under the assumption that we have appropriately created the population data, this procedure will give us (approximately) the values for bias, MSE, and SE when we repeat the procedure many times. To illustrate this it may be easiest to think about a simple random sample. Each time we draw a sample from the population with the computer we draw in principle from the set $S_1, S_2, \ldots \ldots, S_K$ with their probabilities $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$. Thus if we repeat this procedure many times, we will see all sample sets $S_1, S_2, \ldots . ., S_K$ appear in the proper proportions $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$.

统计代写|概率与统计作业代写Probability and Statistics代考|Estimation of the Population Mean

For estimators of the population mean $\mu=\sum_{i=1}^N x_i / N$ in the form of weighted averages, the bias, MSE, and SE can be formulated mathematically when the sampling plan is simple random sampling, systematic sampling, stratified sampling, and cluster sampling. When we obtain the values $x_1, x_2, \ldots, x_n$ from sample $S_k$, we may “average” them in different ways. We may feel that some observations are more important or reliable than other observations and we may want to use this in averaging. This can be done using a weighted average, where the weights would help quantify how much more one observation is valued over other observations. A weighted average for the data observed from sample $S_k$ is now defined as $\bar{x}{w, k}=\sum{i \in S_k} w_{i k} x_i$, with $\sum_{i \in S_k} w_{i k}=1$. Note that the weights need to add up to one and that the weights may in principle depend on the sample set $S_k$, although we will restrict ourselves to weights that are independent of $S_k$, i.e., $w_{i k}=w_i$. If we choose weight $w_1=2 / n$ and weight $w_2=1 /(2 n)$, the first observation is four times as important as the second observation. If every observation has the same weight, we obtain the arithmetic average $\bar{x}k=\sum{i \in S_k} x_i / n$. In practice, weights can depend on other variables like sex and age, in particular in stratified sampling. In this section we will describe the bias, MSE, and SE for weighted averages under the four sampling plans. The SE and MSE depend on population variances, which we will define in the following subsections. Since we have the calculation rule $\mathrm{MSE}=$ bias $^2+\mathrm{SE}^2$ we will mainly focus on bias and MSE. A summary of the theory is provided in Table $2.2$ (Cochran 2007).

Recall that for simple random sampling we are drawing a sample of size $n$ from a population of size $N$ where the number of possible samples $K$ is given by $N ! /[n !(N-n) !]$ and each sample $S_k$ is selected with probability $1 / K$ (see Sect. 2.4.1). In this sampling plan, it can be demonstrated that the arithmetic average is the only unbiased estimator for the population mean in the class of weighted averages (see Cochran 2007). Thus we will focus on the arithmetic average $\bar{x}_k$ for simple random sampling.

统计代写|概率与统计作业代写Probability and Statistics代考|MATH107

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|比较抽样方案使用R

.使用R


在实践中,我们可以使用计算机来帮助我们进行评估,特别是当我们想要使用更复杂的抽样方案来研究其他总体参数,或者当我们想要研究其他估计量时。例如,我们可能想再次选择一个高中生的样本,类似于我们的数据集high-school.csv中的数据,来调查从2000年以来看电视的行为是否发生了变化。我们可以采用与以前相同的抽样方法,但我们也可以研究其他的抽样程序,这些程序可能更复杂,以前无法研究,因为当时没有历史数据集。改进抽样计划,即降低MSE,可以帮助我们降低样本数量和成本。根据历史数据,我们可以研究等级分层是否有助于我们减少标准误差,或者我们可以通过增加学校内学生的数量来减少学校的数量。这些类型的评价可能在数学上是困难的,特别是如果已经涉及到其他几个分层和聚类样本,但用计算机研究这些计划可能不是那么困难(现在有了一些数据)。在其他情况下,我们甚至可能生成自己的数据,或扩展历史数据,以帮助评估采样方案和估计器


在这里,我们将提供一个通用的方法$\mathrm{R}$,以便能够研究抽样计划和估计量。$\mathrm{R}$代码相对简单,但是可以对其进行扩展并使其更复杂,以解决特定的情况。我们的目标是理解结构,而不是了解可能复杂的采样和编程。回想一下,这种类型的计算机方法需要历史数据或一些关于人口的知识,以便能够模拟来自人口的数据。一般结构如下:

  1. 生成种群数据,
  2. 执行抽样计划并生成样本数据,
  3. 使用样本数据计算统计量,
  4. 大量执行这些步骤并使用结果计算偏差、MSE和SE


    假设我们已经适当地创建了总体数据,当我们多次重复该过程时,该过程将(近似地)为我们提供偏差、MSE和SE的值。为了说明这一点,最简单的方法是考虑一个简单的随机样本。每次我们用计算机从总体中抽取样本时,原则上我们从集合$S_1, S_2, \ldots \ldots, S_K$中抽取概率$\pi_1, \pi_2, \pi_3, \ldots, \pi_K$。因此,如果我们多次重复这个过程,我们将看到所有的样本集$S_1, S_2, \ldots . ., S_K$以适当的比例$\pi_1, \pi_2, \pi_3, \ldots, \pi_K$ .

统计代写|概率与统计作业代写Probability and Statistics代考|Estimation of the Population Mean


对于以加权平均值形式表示的总体均值的估计量$\mu=\sum_{i=1}^N x_i / N$,当抽样方案为简单随机抽样、系统抽样、分层抽样和聚类抽样时,其偏差、均方误差和均方误差可以用数学方法表示。当我们从样本$S_k$获得值$x_1, x_2, \ldots, x_n$时,我们可以用不同的方法对它们进行“平均”。我们可能会觉得有些观测结果比其他观测结果更重要或更可靠,我们可能会想要在平均时使用这一点。这可以使用加权平均来完成,权重将有助于量化一个观测值比其他观测值高多少。从样本$S_k$观察到的数据的加权平均值现在定义为$\bar{x}{w, k}=\sum{i \in S_k} w_{i k} x_i$,其中$\sum_{i \in S_k} w_{i k}=1$。请注意,权重加起来需要等于1,而且原则上权重可能取决于样本集$S_k$,尽管我们将限制自己的权重独立于$S_k$,即$w_{i k}=w_i$。如果我们选择权重$w_1=2 / n$和权重$w_2=1 /(2 n)$,第一个观察结果的重要性是第二个观察结果的4倍。如果每个观测值的权重相同,则得到算术平均值$\bar{x}k=\sum{i \in S_k} x_i / n$。在实践中,权重可能取决于性别和年龄等其他变量,特别是在分层抽样中。在本节中,我们将描述四种抽样方案下加权平均数的偏差、均方误差和均方误差。SE和MSE依赖于总体方差,我们将在以下小节中定义。由于我们有计算规则$\mathrm{MSE}=$ bias $^2+\mathrm{SE}^2$,我们将主要关注偏差和MSE。表$2.2$ (Cochran 2007)提供了理论的摘要


回想一下,对于简单随机抽样,我们从规模为$N$的总体中抽取规模为$n$的样本,其中可能样本的数量$K$由$N ! /[n !(N-n) !]$给出,每个样本$S_k$的选择概率为$1 / K$(见第2.4.1节)。在这个抽样计划中,可以证明算术平均是加权平均类中总体均值的唯一无偏估计量(见Cochran 2007)。因此,我们将关注简单随机抽样的算术平均数$\bar{x}_k$

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

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