# 统计代写|抽样调查作业代写sampling theory of survey代考|STAT392

## 统计代写|抽样调查作业代写sampling theory of survey代考|Small Area Estimation

On this topic J.N.K. Rao (2003), J.N.K. Rao and Isabel Molina (2015), and Mukhopadhyay (1998) have three book-level expansions, and in addition, the present author (Chaudhuri, 2012) has an electronic monograph. In the latter’s three textbooks ${($ Chaudhuri and Stenger (2005), Chaudhuri (2010, $2014)}$ also, certain aspects of this topic have been dealt with.

Let me offer here some of our ideas about this. Suppose a sample has been chosen from a survey population following a suitable sampling design $p$. Let such a sample $s$ chosen with probability $p(s)$ be surveyed chiefly to appropriately estimate the population total and mean by some standard procedures. Let further the population be composed of certain discernible non-overlapping parts called ‘domains’ labelled $U_d$ with $s \cap U_d-s_d$ for $d-1, \cdots, D$ and $\cup_{d=1}^D U_d=U$. Let, in addition, it be needed to suitably estimate the totals and means for $U_d$ ‘s, $d=1, \cdots, D$ utilizing the sample data $\left(s, y_i \mid i \in s\right)$ at hand. Suppose the size of $s$ is $n$ which has been suitably hit upon to provide adequate efficiency for the population total/mean estimate. But the size $n_d$ of $s_d$ is a random variable for $d-1, \cdots, D$. If the estimator for the population total $Y=\sum_{i=1}^N y_i$ is taken as the Horvitz and Thompson’s estimator $t_{H T}=\sum_{i \in s} \frac{y_i}{\pi_i}$ and is supposedly adequate enough in efficiency one has to decide on a serviceable estimator for the $d$ th domain total $Y_d=\sum_{i \in U_d} y_i$ as well.
Let us write
\begin{aligned} I_{d i} & =1 \text { if } i \in U_d \ & =0 \text { if } i \notin U_d, \ Y_d & =\sum_{i=1}^N y_i I_{d i}=\sum_{i \in U_d} y_{d i}, y_{d i}=y_i I_{d i} \end{aligned}
and $N_d$ as the size of $U_d$; also $\bar{Y}d=\frac{Y_d}{N_d}$, the $d$ th domain mean, $d=1 \cdots, D$. Now, $t_d=\sum{i \in s} \frac{y_{d i}}{\pi_i}=\sum_{i \in s_d} \frac{y_i}{\pi_i}$ is the Horvitz and Thompson’s (HT, 1952) estimator for $Y_d$. If however $n_d$ be quite small relative to $n=\sum_{d=1}^D n_d$ even if $t_{H T}$ be good enough for $Y$, the estimator $t_d$ for $Y_d$ may not fare well.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Bootstrap in Finite Population Sampling

For large samples an empirical distribution function converges uniformly with probability one to the underlying true distribution function as proved by Loeve (1977). Efron (1982) draws upon this his strength about the closeness of the simulation-based histograms calculated from SRSWR’s from observed sample-values to the unknowable underlying probability distribution. Such SRSWR’s from observed SRSWR’s from theoretical probability distributions are called ‘bootstrap’ samples. If the bootstrap samples are sufficiently numerous, studying them useful inferences are possible.

In case of sampling finite populations also bootstrap samples may be generated and put to suitable uses though no strong theoretical justifications for them are known to have been established and many results have yet remained heuristic.

Suppose $\theta_j=\frac{1}{N} \sum_{i=1}^N \xi_{j i}, j=1, \cdots, K$ are finite population totals of $K$ real variables $\xi_j(j=1, \cdots K)$ with values $\xi_{j i}$ for the units $i$ of a finite population $U=(1, \cdots, i, \cdots, N)$. Let $\theta=\left(\theta_1, \cdots, \theta_j, \cdots, \theta_K\right)$ and $g(\theta)$ be a nonlinear but a well-bahaved function of $\theta$ and our interest be to derive a suitable point estimator for $g(\theta)$ along with an estimator for its Mean Square Error (MSE) and also to construct a Confidence Interval (CI) for $g(\theta)$ with a suitably high confidence coefficient $100(1-\alpha) \%$ with $\alpha$ in $(0,1)$. As an alternative to (i) linearization technique, (ii) IPNS, or (iii) jack-knife the ‘sub-sampling replication’ procedure called ‘bootstrap’ sampling is often found handy. The non-linear functions $g(\theta)$ we find interesting to illustrate are:

Chaudhuri and Stenger (2005), Chaudhuri (2010, 2014) have given detailed accounts of several bootstrap sampling procedures to handle general problems including, in particular, these three.

In this treatise we intend to concentrate on the discussions as in $\mathrm{Pal}$ (2009) and in Chaudhuri and Saha (2004) avoiding repetition of what was comprehensively detailed in Chaudhuri and Stenger (2005).

Pal (2009) starts with Rao and Wu’s(1988)’re-scaling’ bootstrap technique. In the latter, for each $j=1, \cdots, t, t_j$ is taken as the Horvitz and Thompson (HT, 1952) estimator $t_{H T}=\sum_{i \in s} \frac{y_i}{n_i}$ unbiased for $Y=\sum_1^N y_i$ with the Yates and Grundy’s (YG, 1953) variance form
$$V_{Y G}\left(t_{H T}\right)=\sum_{i0 \forall i \neq j and most importantly, having \nu_{Y G} uniformly nonnegative, guaranteed by the design condition \pi_i \pi_j>\pi_{i j} \forall i \neq j in U. # 抽样调查代考 ## 统计代写|抽样调查作业代写sampling theory of survey代考|Small Area Estimation 关于这个主题，JNK Rao (2003)、JNK Rao 和 Isabel Molina (2015) 以及 Mukhopadhyay (1998) 有 三本书级别的扩展，此外，本作者 (Chaudhuri, 2012) 有一本电子专若。在后者的三本教材中 (\ChaudhuriandStenger (2005), Chaudhuri (2010, \ 2014) 此外，还讨论了该主题的某些方 面。 让我在这里提出我们对此的一些想法。假设样本是按照合适的抽样设计从调查人群中选出的 p. 让 这样的样本 s 以概率选择 p(s) 进行调查主要是为了通过一些标准程序适当地估计人口总数和平均 值。进一步让人口由某些可辨别的非重冝部分组成，称为”域”标记 U_d 和 s \cap U_d-s_d 为了 d-1, \cdots, D 和 \cup_{d=1}^D U_d=U. 此外，还需要适当地估计总计和平均数 U_d 的， d=1, \cdots, D 利用 样本数据 \left(s, y_i \mid i \in s\right) 在眼前。假设大小 s 是 n 这已被适当地利用来为人口总数/平均估计提供足够 的效率。但是大小 n_d 的 s_d 是一个随机变量 d-1, \cdots, D. 如果人口总数的估计量 Y=\sum_{i=1}^N y_i 被 视为 Horvitz 和 Thompson 的估计量 t_{H T}=\sum_{i \in s} \frac{y_i}{\pi_i} 并且据说效率足够高，因此必须为 d 第 th 个 域总数 Y_d=\sum_{i \in U_d} y_i 以及。 让我们写$$
I_{d i}=1 \text { if } i \in U_d \quad=0 \text { if } i \notin U_d, Y_d=\sum_{i=1}^N y_i I_{d i}=\sum_{i \in U_d} y_{d i}, y_{d i}=y_i I_{d i}
$$和 N_d 作为大小 U_d; 还 \bar{Y} d=\frac{Y_d}{N_d} ， 这 d 第域意味着， d=1 \cdots, D. 现在， t_d=\sum i \in s \frac{y_{d i}}{\pi_i}=\sum_{i \in s_d} \frac{y_i}{\pi_i} 是 Horvitz 和 Thompson (HT, 1952) 的估计量 Y_d. 然而，如果 n_d 相 对于 n=\sum_{d=1}^D n_d 即使 t_{H T} 足够好 Y ，估计量 t_d 为了 Y_d 可能不会过得很好。 ## 统计代写|抽样调查作业代写sampling theory of survey代考|Bootstrap in Finite Population Sampling 对于大样本，经验分布函数以概率 1 均匀收敛到基础真实分布函数，正如 Loeve (1977) 所证明的 那样。Efron (1982) 利用了他关于从观测样本值的 SRSWR 计算的基于模拟的直方图与不可知的潜 在概率分布的接近度的优势。这种来自从理论概率分布观察到的 SRSWR 的 SRSWR 被称为”自举” 样本。如果 bootstrap 样本足够多，研究它们就可能得出有用的推论。 在对有限总体进行抽样的情况下，也可以生成自举样本并将其用于适当的用途，尽管目前还没有 为它们建立强有力的理论依据，而且许字结果仍然是启发式的。 认为 \theta_j=\frac{1}{N} \sum_{i=1}^N \xi_{j i}, j=1, \cdots, K 是有限总体总数 K 实变量 \xi_j(j=1, \cdots K) 有价值观 \xi_{j i} 对于 \backslash theta_K\right)andg( \backslash \theta) beanonlinearbutawell – bahavedfunctionof \ \theta andourinterestbetoderiveasuitablepointestimator forg (\backslash \Theta) alongwithanestimator forits MeanSquareError (M S E )andalsotoconstructaConfiden \mathrm{g}(\backslash \theta) withasuitablyhighcon fidencecoefficient 100(1-\backslash \mathrm{alpha}) \backslash \% with \backslash \alpha in (0,1) . Asanalternativeto(i)linearizationtechnique, (ii)IPNS, or(iii)jack – knifethe’subg( theta ) \ 我们发现有趣的是: Chaudhuri 和 Stenger（2005 年) 、Chaudhuri（2010 年、2014 年) 详细说明了几种引导抽样程 序来处理一般问题，特别是这三个问题。 在这篇论文中，我们打算集中讨论Pal(2009) 以及 Chaudhuri 和 Saha (2004) 避免重复 Chaudhuri 和 Stenger (2005) 中全面详述的内容。 \mathrm{Pal} (2009) 从 Rao 和 Wu (1988) 的”重新缩放”自举技术开始。在后者中，对于每个 j=1, \cdots, t, t_j 被视为 Horvitz 和 Thompson (HT, 1952) 估计器 t_{H T}=\sum_{i \in s} \frac{y_i}{n_i} 不偏不倚 Y=\sum_1^N y_i 与 Yates 和 Grundy 的 (Y G, 1953) 方差形式 \ \$$
V_{YG $} \backslash$ left(t_{HT}\right) $=\backslash$ sum_{i0 $\backslash$ forall i $\backslash$ neq jandmostimportantly, having $\backslash$ \nu_ ${Y G}$ in 美元。

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