如果你也在 怎样代写抽样调查Survey sampling 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。抽样调查Survey sampling是数学工程这一广泛新兴领域中的一个自然组成部分。例如，我们可以断言，数学工程之于今天的数学系，就像数学物理之于一个世纪以前的数学系一样;毫不夸张地说，数学在诸如语音和图像处理、信息理论和生物医学工程等工程学科中的基本影响。
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|Model-Based Regression Analysis
In the above, we really considered a two-step randomization: the finite population is supposedly a realization from an infinite hypothetical superpopulation with reference to which a regression relationship is postulated connecting a dependent variable and a set of independent regressor variables. Then, from the given or realized finite population a sample is randomly drawn because the population is too large to be completely investigated. The sample is then utilized to make inference with reference to the two-step randomization. But now let us consider a purely model-based approach that takes account of the structure of the finite population at hand by postulating an appropriate model.
Suppose for a sample of $c$ clusters from a given finite population, observations are taken on a dependent variable $y$ and a set of independent regressor variables $x_1, \ldots, x_r$ for independently drawn samples of second stage units (SSUs) of sizes $m_i$ from the respective sampled clusters labeled $i=1, \ldots, c$ so that $\sum_1^c m_i=n$, the total sample size. Let $Y_n$ be an $n \times 1$ vector of observations on $y$, successive rows in it giving values on the $m_i$ observations in the order $i=1, \ldots, c$ and the observations $X_j$ ‘s, $j=1, \ldots, r$ be also similarly arranged in succession. Now it is only to be surmised that the observations within the same cluster should be substantially well and positively correlated compared to those across the clusters. So, after postulating a regression relation of $Y_n$ on $X_n$, which is an $n \times r$ matrix, the successive rows in it arranging the values for the clusters taken in order $i=1, \ldots, c$, which states that
$$
E_m\left(Y_n\right)=X_n \beta
$$
where $\beta$ is an $r \times 1$ vector of unknown regression parameters, one should carefully postulate about the distribution of the error vector
$$
\epsilon_n=Y_n-X_n \beta .
$$
统计代写|抽样调查作业代写sampling theory of survey代考|Design Variables
Next we consider an important situation where, besides the regressor variables, there exist another set of variables that are utilized in determining the selection probabilities, called the design variables. For example, one may plan to examine how expenses on certain items of consumption, the dependent variable $y$, vary with the annual income, the single regressor variable $x$. Then, if accounts of the taxes paid by the relevant individuals in the last financial year, values of a variable $z$, are available, this information can be utilized in stratifying the population accordingly. Then $z$ is a design variable obviously well-correlated with $x$ and $y$.
Following the works of NATHAN and Holt (1980), HoLT, Smith and Winter (1980), and PfeffermanN and Holmes (1985) let us consider the simple case of a single dependent (endogeneous) variable $y$, a single regressor (exogeneous, independent) variable $x$, and a single design variable $z$. Assume the regression model
$$
y=\alpha+\beta x+\epsilon
$$
with $E_m(\epsilon \mid x)=0, V_m(\epsilon \mid x)=\sigma^2(\sigma>0)$. Suppose a random sample $s$ of size $n$ is taken following a design $p$ using the values $Z_1, Z_2, \ldots, Z_N$ of $z$ and define
$$
v_z=\frac{1}{N} \sum_1^N Z_i, \sigma_z^2=\frac{1}{N-1} \sum_1^N\left(Z_i-v_z\right)^2 .
$$
Also, let $\bar{y}, \bar{x}, \bar{z}$ denote sample means of $y, x, z, s_y^2, s_x^2, s_z^2$ the sample variances and $s_{y x}, s_{y z}, s_{x z}$ the sample covariances. The problem is to infer about $\beta$, the regression coefficient of $y$ on $x$ under the model-based approach.
Consider the ordinary least squares estimator (OLSE),
$$
b=s_{y x} / s_x^2 .
$$
Its performance depends essentially on the relation between the design variable $z$ and the variables $x, y$ in the regression model. In the simplest case $x, y, z$ might follow a trivariate normal distribution. DEMETS and HALPERIN (1977) have shown that, under this assumption, $b$ is biased. Following ANDERSON’s (1957) missing value approach, they derive an alternative estimator, which is the maximum likelihood estimator (MLE) for $\beta$, namely,
$$
\widehat{\beta}=\left[s_{y x}+\frac{s_{y z} s_{x z}}{s_z^2}\left(\frac{\sigma_z^2}{s_z^2}-1\right)\right] /\left[s_x^2+\frac{s_{x z}^2}{s_z^2}\left(\frac{\sigma_z^2}{s_z^2}-1\right)\right] .
$$
抽样调查代考
统计代写|抽样调查作业代写sampling theory of survey代考|Model-Based Regression Analysis
在上面，我们真正考虑的是两步随机化:有限总体是无限假设的超总体的实现，参考这一假设，回归关系连接了因变量和一组独立回归变量。然后，从给定或实现的有限总体中随机抽取样本，因为总体太大而无法完全调查。然后利用样本参照两步随机化进行推理。但是现在让我们考虑一种纯粹基于模型的方法，它通过假设一个适当的模型来考虑手头有限人口的结构。
假设对于来自给定有限总体的$c$簇的样本，对独立绘制的第二阶段单元(ssu)的样本(大小为$m_i$)的因变量$y$和一组独立回归变量$x_1, \ldots, x_r$进行观察，这些样本分别标记为$i=1, \ldots, c$，因此$\sum_1^c m_i=n$为总样本量。设$Y_n$为$y$上的观测值的$n \times 1$向量，其中的连续行以$i=1, \ldots, c$的顺序给出$m_i$观测值的值，而观测值$X_j$ s, $j=1, \ldots, r$也以类似的顺序排列。现在只能推测，同一集群内的观测结果与跨集群的观测结果相比，应该具有良好的正相关关系。因此，假设$X_n$上的回归关系为$Y_n$，这是一个$n \times r$矩阵，其中的连续行将集群的值按$i=1, \ldots, c$的顺序排列，这表明
$$
E_m\left(Y_n\right)=X_n \beta
$$
其中$\beta$是未知回归参数的$r \times 1$向量，应该仔细假设误差向量的分布
$$
\epsilon_n=Y_n-X_n \beta .
$$
统计代写|抽样调查作业代写sampling theory of survey代考|Design Variables
接下来，我们考虑一个重要的情况，除了回归变量之外，还有另一组变量用于确定选择概率，称为设计变量。例如，一个人可能计划检查某些消费项目的费用(因变量$y$)如何随年收入(单回归变量$x$)而变化。然后，如果有有关个人在上一个财政年度所付税款的帐户，即变量$z$的值，则可以利用这一资料对人口进行相应的分层。那么$z$是一个与$x$和$y$明显相关的设计变量。
继NATHAN和Holt(1980)、Holt、Smith和Winter(1980)以及PfeffermanN和Holmes(1985)的研究之后，让我们考虑一个简单的情况，即一个因变量(内源性)$y$、一个回归变量(外源性、自变量)$x$和一个设计变量$z$。假设回归模型
$$
y=\alpha+\beta x+\epsilon
$$
通过$E_m(\epsilon \mid x)=0, V_m(\epsilon \mid x)=\sigma^2(\sigma>0)$。假设取一个大小为$n$的随机样本$s$，遵循使用$z$和define的值$Z_1, Z_2, \ldots, Z_N$的设计$p$
$$
v_z=\frac{1}{N} \sum_1^N Z_i, \sigma_z^2=\frac{1}{N-1} \sum_1^N\left(Z_i-v_z\right)^2 .
$$
同样，设$\bar{y}, \bar{x}, \bar{z}$表示$y, x, z, s_y^2, s_x^2, s_z^2$的样本方差和$s_{y x}, s_{y z}, s_{x z}$的样本协方差的样本均值。问题是在基于模型的方法下，对$y$在$x$上的回归系数$\beta$进行推断。
考虑普通最小二乘估计(OLSE)，
$$
b=s_{y x} / s_x^2 .
$$
其性能主要取决于回归模型中设计变量$z$与变量$x, y$之间的关系。在最简单的情况下$x, y, z$可能遵循三元正态分布。DEMETS和HALPERIN(1977)已经表明，在这个假设下，$b$是有偏见的。根据ANDERSON(1957)的缺失值方法，他们推导出一个替代估计量，即$\beta$的最大似然估计量(MLE)，即:
$$
\widehat{\beta}=\left[s_{y x}+\frac{s_{y z} s_{x z}}{s_z^2}\left(\frac{\sigma_z^2}{s_z^2}-1\right)\right] /\left[s_x^2+\frac{s_{x z}^2}{s_z^2}\left(\frac{\sigma_z^2}{s_z^2}-1\right)\right] .
$$
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