Posted on Categories:Sampling Theory, 抽样理论, 统计代写, 统计代考

# 统计代写|抽样理论代考Sampling Theory代写|SURV626 Random Arrangements of Units

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## 统计代写|抽样理论代考Sampling Theory代写|Random Arrangements of Units

Systematic sampling can be treated as SRSWOR sampling if the arrangement of the units in the population is random. So, treating the selected systematic sample $s$ as an SRSWOR sample of size $n$, the variance of the systematic sample mean $V_{s Y}$ can be estimated as
where $f=n / N$ and $s_\gamma^2=\sum_{i \in s}\left{y_i-\bar{\gamma}(s)\right}^2 /(n-1)$.
The estimator $\widehat{v}_1$ performs well if the arrangement of the units is random, i.e., if no trend or periodicity in the arrangements of the units persists. However, if the arrangement of the units is such that the systematic sampling is more efficient than SRSWOR sampling, then $\widehat{v}_1$ certainly overestimates the variance. On the other hand, $\widehat{v}_1$ underestimates the variance if in reality systematic sampling is less efficient than SRSWOR.

Stratified Sampling With One Unit Per Stratum

Systematic sampling stratifies the population into $k$ strata each of size $n$. Let $n$ be even and equal to $2 q$. Now consider the units $(r, r+k)$ as a stratified sample of size 2 selected by the SRSWOR method from a stratum comprising the first $2 k$ units viz. $1, \ldots, 2 k ;(r+2 k, r+3 k)$ as a stratified sample of size 2 from a stratum comprising the next $2 k$ viz. $2 k+1, \ldots, 4 k$; in general, units ${r+2(j-1) k, r+(2 j-1) k}$ as a stratified sample of size 2 from the $j$ th stratum comprising units $2(j-1) k+1, \ldots, 2 j k ; j=1, \ldots, q$. An approximate estimate of $V_{s Y}$ is obtained as
$$\widehat{v}2=\frac{(1-f)}{n^2} \sum{j=1}^{n / 2} \delta_j^2$$
where $\delta_j=y_{r+(2 j-1) k}-y_{r+2(j-1) k}$.

## 统计代写|抽样理论代考Sampling Theory代写|Presence of Linear Trend

Let us construct the following difference table from the selected systematic sample $s_r$ :

The variance of the systematic sample mean $V_{s Y}$ is estimated by using the following formulae:
$$\widehat{v}3=\frac{(1-f)}{n} \frac{1}{2(n-1)} \sum{j=1}^{n-1}{\Delta(j)}^2$$
and
$$\widehat{v}4=\frac{(1-f)}{n} \frac{1}{6(n-2)} \sum{j=1}^{n-2}\left{\Delta^2(j)\right}^2$$
The estimators $\widehat{v}_3$ and $\widehat{v}_4$ perform well if the ordering of units produces a linear trend. Otherwise, they underestimate variance. Several variance estimators based on higher-order differences are also proposed, e.g., Yates (1949).

Cochran (1946) provided an alternative variance estimator involving $\widehat{\rho}k$ as an estimate of $\rho_k$, intraclass correlation of order $k(=N / n)$. Cochran’s variance estimator is given by where $\widehat{\boldsymbol{\rho}}_k=\left[\frac{1}{(n-1)} \sum{j=0}^{n-2}\left{y_{r+(j+1) k}-\bar{y}(s)\right}\left{y_{r+j k}-\bar{y}(s)\right}\right] / s_y^2$.
The proposed variance estimators mentioned earlier are based on underlying certain superpopulation model. The variance estimators may not perform well if the underlying model fails. Wolter (1984) studied in detail the properties of bias, mean-square error, and confidence interval estimation theoretically and empirically.

# 抽样理论代写

## 统计代写|抽样理论代考代写|单位的随机安排

$$\hat{v} 2=frac{(1-f)}{n^2}。\sum j=1^{n / 2} δj^2$$ 其中$delta_j=y{r+(2 j-1) k}-y_{r+2(j-1) k}$。

## 统计代写|抽样理论代考代写|线性趋势的存在

$$\hat{v} 3=frac{(1-f)}{n}。\夫拉克{1}{2(n-1)} 总和 j=1^{n-1}\Delta(j)^2$$

Missing or unrecognized delimiter for \left }

Cochran(1946)提供了一个替代的方差估计器，涉及$hat{\rho} k$作为$rho_{k text { r }}$阶$k(=N / n)$的类内相关的估计。Cochran的方差估计器是由其中Missing or unrecognized delimiter for \left给出的。

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