## 统计代写|概率与统计作业代写Probability and Statistics代考|MATH1342

2022年9月29日

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## 统计代写|概率与统计作业代写Probability and Statistics代考|Evaluating Estimators Given Different Sampling Plans

In Chap. 1, several descriptive statistics were discussed, like the average, standard deviation, median, quartiles, quantiles, etc. They were introduced to summarize the collected data, but in the context of sampling they can be viewed as so-called estimators: quantities that we compute using the data in our sample to say something about the population. For example, can we determine how well the sample average $\bar{x}$, as defined in the previous chapter, estimates the population mean, $\mu$ ? This means that we would like to determine the “closeness” of the sample average to the population mean. Whatever measure we would like to use for closeness, the sampling approach will influence the performance of the estimator. For instance, the sample average may generally be closer to the population mean under simple random sampling than under cluster random sampling. We will use bias, mean square error (MSE), and standard error (SE) as measures of closeness and we will illustrate these measures in this section for any type of statistic that we wish to calculate. To do this we will first provide a general framework for random sampling (Cochran 2007), for which simple random sampling, systematic sampling, stratified sampling, and cluster sampling are all a special case. In the third subsection we will illustrate the bias, mean squared error and standard error and in the fourth subsection we show how $\mathrm{R}$ can be used for evaluations.

A formal or mathematical definition for collecting a random sample of size $n$ from a population of units indicated by $\Omega={1,2,3, \ldots \ldots, N}, N \geq n$, can be described as follows: Let $S_1, S_2, \ldots . ., S_K$ be subsets of the population $\Omega, S_k \subset \Omega, k=1,2, \ldots, K$, such that each subset $S_k$ has $n$ unique units from $\Omega$ and the union of all units from $S_1, S_2, \ldots \ldots, S_K$ forms the whole population $\Omega$, i.e., $\Omega=\cup_{k=1}^K S_k$. Then each subset $S_k$ is attached a probability $\pi_k$ such that $\pi_k>0$, for all $k=1,2, \ldots, K$, and $\pi_1+$ $\pi_2+\cdots+\pi_K=1$. A random sample of size $n$ is obtained by drawing just one number from $1,2,3, \ldots, K$ using the probabilities $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$. Subsets $S_1$, $S_2, \ldots ., S_K$ can be assumed to be unique, $S_k \neq S_l$ when $k \neq l$, since otherwise we can create a unique set by adding the probabilities for the subsets that are equal. This does not mean that there is no overlap in units from different subsets, i.e., we do not require $S_k \cap S_l=\emptyset$. Note that simple random sampling, systematic sampling, stratified sampling, and cluster random sampling all satisfy this definition.

The set of samples $S_1, S_2, \ldots, S_K$ with their probabilities $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$ is referred to as a sampling plan. Note that $K$ can be very large and quite different for different sampling plans. It is also good to realize that the sets $S_1, S_2, \ldots, S_K$ and the probabilities $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$ result in a set of probabilities $p_1, p_2, p_3, \ldots, p_N$ for units $1,2,3, \ldots, N$ in the population $\Omega$, with $p_i>0 .{ }^{22}$

The sampling plan contains all the information necessary to analyze the quality of a sampling procedure. As long as we know $S_1, S_2, \ldots ., S_K$ with their respective probabilities $\pi_1, \pi_2, \pi_3, \ldots, \pi_K$ we can use these in any further analysis. Hence, despite the differences between simple random sampling, systematic sampling, stratified sampling, and cluster sampling, our subsequent theory for judging the quality of a sampling plan can be solely stated in terms of the $S_k$ ‘s and $\pi_k$ ‘s.

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

Consider a population of $N$ units and assume that we are interested in one characteristic or variable of the unit. For instance, the variable could represent height, weight, gender, hours of television watching per week, tensile strength, bacterial contamination, a face rating, etc. Each unit $i$ in the population has a theoretical value $x_i$ that may become available in the sample. Note that we consider numerical values only. The population parameter of interest can be defined by $\theta \equiv \theta(\boldsymbol{x})$, with $\boldsymbol{x}=\left(x_1, x_2, \ldots, x_N\right)$, as some kind of calculation on all theoretical values: for instance, the population mean $\mu=\sum_{i=1}^N x_i / N$ or the population variance $\sigma^2=\sum_{i=1}^N\left(x_i-\mu\right)^2 / N$.

A sample $S_k$ of size $n$ can now be seen as the set of units, i.e., $S_k=\left{i_1, i_2, \ldots, i_n\right}$ with $i_h \in{1,2,3, \ldots, N}$ and all indices unique $\left(i_h \neq i_l\right.$ when $\left.h \neq l\right)$. With every sample $S_k$ we have observed a vector of observations $\boldsymbol{x}k^{\prime}=\left(x{i_1}, x_{i_2}, \ldots, x_{i_n}\right)$, with ‘ indicating the transpose. ${ }^{23}$ Based on the observed data we compute the descriptive statistic $\hat{\theta}k=T\left(\boldsymbol{x}_k\right)$ and use it as an estimate for the population parameter $\theta$, with $T$ a function applied to the observed data (i.e., some calculation procedure). In many cases the function $T$ is identical to the calculation $\theta$ at the population level, but alternative functions may be used depending on the sampling plan. For instance, for estimation of the population mean, we may use average $\bar{x}_k=\sum{h=1}^n x_{i_h} / n=\sum_{i \in S_k} x_i / n$, but we may also use a weighted average $\sum_{i \in S_k} w_i x_i / n$, with the weights adding up to one $\left(\sum_{i \in S_k} w_i=1\right)$; see Sect. 2.6. The function $T$ is referred to as the estimator.
In general, the value $\hat{\theta}_k$ can be considered an estimate of the population parameter $\theta$ when sample $S_k$ would be collected. The estimate $\hat{\theta}_k$ will most likely be different from the population parameter $\theta$, hecause the sample is just a subset of the population. When the sample is representative the sample result should be “quite close” to the population parameter and then the sample result may be considered a good estimate of the population parameter.

# 概率与统计作业代考

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。