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

#### Doug I. Jones

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couryes-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

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

Since $n_{d g}$ and $n_{d g h}$ ‘s are very small, if we may believe that the $g$ groups have been so effectively formed that in respect of the characteristics of interest $y$ there is homogeneity within each separate group across the domains, then the following broadbased estimators for $T_d$ may be useful
\begin{aligned} t_{c s d} & =\sum_g N_{d_g}\left(\Sigma_{s . g} Y_k / \pi_k\right) /\left(\Sigma_{s, g} 1 / \pi_k\right) \ t_{c s c d} & =\sum_g\left(\Sigma_{H_d} N_{d g h}\right)\left(\Sigma_{H_d} \Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{H_d} \Sigma_{s, g h} 1 / \pi_k\right) \ t_{c s s d}^{\prime} & =\sum_g \Sigma_{H_d} N_{d g h}\left(\Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{s, g h} 1 / \pi_k\right) \end{aligned}
called the count-synthetic estimators for unstratified, stratified-combined, and stratified-separate sampling, respectively. The corresponding ratio-synthetic estimators for unstratified and stratified sampling are:
\begin{aligned} t_{R s d} & =\sum_g X_{d g}\left(\Sigma_{s, g} Y_k / \pi_k\right) /\left(\Sigma_{s, g} X_k / \pi_k\right) \ t_{R s c d} & =\sum_g\left(\Sigma_{H_d} X_{d g h}\right) \frac{\left(\Sigma_{H_d} \Sigma_{s, g h} Y_k / \pi_k\right)}{\left(\Sigma_{H_d} \Sigma_{s, g h} X_k / \pi_k\right)} \ t_{R_{s e d}} & =\sum_g \Sigma_{H_d} X_{d g h}\left(\Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{s, g h} X_k / \pi_k\right) . \end{aligned}
For SRSWOR from $U$ and independent SRSWORs from $U . . h$, we have the six simpler synthetic estimators
\begin{aligned} t_1 & =\sum_g N_{d g} \bar{y}{g g} \ t_2 & =\sum_g X{d g} \frac{\bar{y}{g g}}{\bar{x}_g}, \ t_3 & =\sum_g \Sigma{H_d} N_{d g h}\left(\Sigma_{H_d} \frac{N . . h}{n . . h} n_{g h} \bar{y}{g h}\right) /\left(\Sigma{H_d} \frac{N . . h}{n . . h} n_{g h}\right) \ t_4 & =\sum_g \Sigma_{H_d} N_{d g h} \bar{y}{g h} \ t_5 & =\sum_g \Sigma{H_d} X_{d g h} \frac{\left(\Sigma_{H_d} \frac{N . h}{n . h} n_{g h} \bar{y}{g h}\right)}{\left(\Sigma{H_d} \frac{N . h}{n . h} n_{g h} \bar{x}{g h}\right)} \ t_6 & =\sum_g \Sigma{H_d} X_{d g h} \frac{\bar{y}{g h}}{\bar{x}{g h}} . \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|Model-Based Estimation

An alternative procedure of small area estimation involving a technique of borrowing strength is the following. Suppose $T_d, d=1, \ldots, D$ are the true values for large number, $D$, of domains of interest and, employing suitable sampling schemes, estimates $t_d$ for $d \in s_0$ are obtained, where $s_0$ is a subset of $m$ domains. Now, suppose auxiliary characters $x_j, j=1, \ldots, K$ are available with known values $X_{j d}, d=1, \ldots, D$. Then, postulating a linear multiple regression
$$T_d=\beta_0+\beta_1 X_{1 d}+\ldots+\beta_K X_{K d}+\epsilon_d ; d=1, \ldots, m$$
one may write for $d \epsilon s_0$
$$t_d=\beta_0+\beta_1 X_{1 d}+\ldots+\beta_K X_{K d}+e_d+\epsilon_d$$
writing $e_d=t_d-T_d$, the error in estimating $T_d$ by $t_d$. Now applying the principle of least squares utilizing the sampled values, one may get estimates $\widehat{\beta}j$ for $j=0,1, \ldots, K$ based on $\left(t_d, X{j d}\right)$ for $d \epsilon s_0$ and $j=1, \ldots, K$, assuming $m>K+1$. Then, we may take $\sum_0^K \widehat{\beta}j X{j d}=\widehat{T}_d$ as estimates for $T_d$ not only for $d \epsilon s_0$ but also for the remaining domains $d \notin s_0$.

This method has been found by ERICKSEN (1974) to work well in many situations of estimating current population figures in large numbers of U.S. counties and in correcting census undercounts. An obvious step forward is to combine the estimators $t_d$ with $\widehat{T}_d$ for $d=1, \ldots, m$ to derive estimators that should outperform both $t_d$ and $\widehat{T}_d, d=1, \ldots, m$. Postulating that $e_d$ ‘s and $\epsilon_d$ ‘s are mutually independent and separately iid random variates respectively distributed as $N\left(0, \sigma^2\right)$ and $N\left(0, \tau^2\right)$, following GHOSH and MEEDEN (1986) one may derive weighted estimators
$$t_d^*=\frac{\tau^2}{\sigma^2+\tau^2} t_d+\frac{\sigma^2}{\sigma^2+\tau^2} \widehat{T}_d, d=1, \ldots, m$$
provided $\sigma$ and $\tau$ are known. If they are unknown, they are to be replaced by suitable estimators. Thus, here we may use JAMES-STEIN or empirical Bayes estimators of the form
$$\widehat{t}_d=\widehat{W} t_d+(1-\widehat{W}) \widehat{T}_d$$
with $0<\widehat{W}<1$, such that according as $t_d\left(\widehat{T}_d\right)$ is more accurate for $T_d$, the weight $\widehat{W}$ goes closer to $1(0)$. These procedures we have explained and illustrated in section 4.2. PRASAD (1988) is an important reference.

# 抽样调查代考

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

\begin{aligned} t_{c s d} & =\sum_g N_{d_g}\left(\Sigma_{s . g} Y_k / \pi_k\right) /\left(\Sigma_{s, g} 1 / \pi_k\right) \ t_{c s c d} & =\sum_g\left(\Sigma_{H_d} N_{d g h}\right)\left(\Sigma_{H_d} \Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{H_d} \Sigma_{s, g h} 1 / \pi_k\right) \ t_{c s s d}^{\prime} & =\sum_g \Sigma_{H_d} N_{d g h}\left(\Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{s, g h} 1 / \pi_k\right) \end{aligned}

\begin{aligned} t_{R s d} & =\sum_g X_{d g}\left(\Sigma_{s, g} Y_k / \pi_k\right) /\left(\Sigma_{s, g} X_k / \pi_k\right) \ t_{R s c d} & =\sum_g\left(\Sigma_{H_d} X_{d g h}\right) \frac{\left(\Sigma_{H_d} \Sigma_{s, g h} Y_k / \pi_k\right)}{\left(\Sigma_{H_d} \Sigma_{s, g h} X_k / \pi_k\right)} \ t_{R_{s e d}} & =\sum_g \Sigma_{H_d} X_{d g h}\left(\Sigma_{s, g h} Y_k / \pi_k\right) /\left(\Sigma_{s, g h} X_k / \pi_k\right) . \end{aligned}

\begin{aligned} t_1 & =\sum_g N_{d g} \bar{y}{g g} \ t_2 & =\sum_g X{d g} \frac{\bar{y}{g g}}{\bar{x}g}, \ t_3 & =\sum_g \Sigma{H_d} N{d g h}\left(\Sigma_{H_d} \frac{N . . h}{n . . h} n_{g h} \bar{y}{g h}\right) /\left(\Sigma{H_d} \frac{N . . h}{n . . h} n_{g h}\right) \ t_4 & =\sum_g \Sigma_{H_d} N_{d g h} \bar{y}{g h} \ t_5 & =\sum_g \Sigma{H_d} X_{d g h} \frac{\left(\Sigma_{H_d} \frac{N . h}{n . h} n_{g h} \bar{y}{g h}\right)}{\left(\Sigma{H_d} \frac{N . h}{n . h} n_{g h} \bar{x}{g h}\right)} \ t_6 & =\sum_g \Sigma{H_d} X_{d g h} \frac{\bar{y}{g h}}{\bar{x}{g h}} . \end{aligned}

## 统计代写|抽样调查作业代写sampling theory of survey代考|Model-Based Estimation

$$T_d=\beta_0+\beta_1 X_{1 d}+\ldots+\beta_K X_{K d}+\epsilon_d ; d=1, \ldots, m$$

$$t_d=\beta_0+\beta_1 X_{1 d}+\ldots+\beta_K X_{K d}+e_d+\epsilon_d$$

ERICKSEN(1974)发现，这种方法在估计大量美国县的当前人口数据和纠正人口普查不足的许多情况下都很有效。一个明显的进步是将$d=1, \ldots, m$的估计器$t_d$与$\widehat{T}_d$结合起来，以推导出性能应该优于$t_d$和$\widehat{T}_d, d=1, \ldots, m$的估计器。假设$e_d$ ‘s和$\epsilon_d$ ‘s是相互独立的独立随机变量，分别分布为$N\left(0, \sigma^2\right)$和$N\left(0, \tau^2\right)$，根据GHOSH和MEEDEN(1986)，可以得到加权估计量
$$t_d^*=\frac{\tau^2}{\sigma^2+\tau^2} t_d+\frac{\sigma^2}{\sigma^2+\tau^2} \widehat{T}_d, d=1, \ldots, m$$

$$\widehat{t}_d=\widehat{W} t_d+(1-\widehat{W}) \widehat{T}_d$$

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