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

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

Let $p$ be a design. Consider a size measure $x$ and assume that, approximately,
$$Y_{i} \propto X_{i} .$$
Then it seems natural to look for an estimator
$$t=\sum_{i=1}^{N} b_{s i} Y_{i}$$
with $b_{s i}=0$ for $i \notin s$, such that
$$\sum_{i=1}^{N} b_{s i} X_{i}=X$$
for all $s$ with $p(s)>0$. With reference to HÁJEK (1959), a strategy with this property is called representative with respect to $X=\left(X_{1}, X_{2}, \ldots, X_{N}\right)^{\prime}$.

For the mean square error (MSE) of a strategy $(p, t)$ we have
\begin{aligned} M_{p}(t) &=E_{p}(t-Y)^{2} \ &=E_{p}\left(\sum Y_{i}\left(b_{s i}-1\right)\right)^{2} \ &=\sum_{i} \sum_{j} Y_{i} Y_{j} d_{i j} \end{aligned}
where
$$d_{i j}=E_{p}\left(b_{s i}-1\right)\left(b_{s j}-1\right) .$$
A strategy ( $p, t)$ is representative if and only if there exists a vector $X=\left(X_{1}, X_{2}, \ldots, X_{N}\right)^{\prime}$ such that $M_{p}(t)=0$ for $Y_{i} \propto X_{i}$ implying
$$\sum_{i} \sum_{j} X_{i} X_{j} d_{i j}=0 .$$
It may be advisable to use strategies that are representative with respect to several auxiliary variables $x_{1}, x_{2}, \ldots, x_{K}$. Let
$$x{i}=\left(X{i 1}, X_{i 2}, \ldots, X_{i K}\right)^{\prime}$$

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

The ratio estimator
$$t_{1}=X \frac{\sum_{i \in s} Y_{i}}{\sum_{i \in s} X_{i}}$$
is of special importance because of its traditional use in practice. Here, $\left(p, t_{1}\right)$ is obviously representative with respect to a size measure $x$, more precisely to $\left(X_{1}, \ldots, X_{N}\right)$, whatever the sampling design $p$.

Note, however, that $t_{1}$ is usually combined with SRSWOR or SRSWR. The sampling scheme of LAHIRI-MIDZUNO-SEN (LAHIRI, 1951; MIDZUNO, 1952; SEN, 1953) (LMS) yields a design of interest to be employed in conjunction with $t_{1}$ by rendering it design unbiased.
The Hansen-Hurwitz (HH, 1943) estimator (HHE)
$$t_{2}=\frac{1}{n} \sum_{i=1}^{N} f_{s i} \frac{Y_{i}}{P_{i}},$$

with $f_{s i}$ as the frequency of $i$ in $s, i \in \mathcal{U}$, combined with any design $p$, gives rise to a strategy representative with respect to $\left(P_{1}, \ldots, P_{N}\right)^{\prime}$. For the sake of design unbiasedness, $t_{2}$ is usually based on probability proportional to size (PPS) with replacement (PPSWR) sampling, that is, a scheme that consists of $n$ independent draws, each draw selecting unit $i$ with probability $P_{i}$.

Another representative strategy is due to RAO, HARTLEY and COCHRAN (RHC, 1962). We first describe the sampling scheme as follows: On choosing a sample size $n$, the population $\mathcal{U}$ is split at random into $n$ mutually exclusive groups of sizes suitably chosen $N_{i}\left(i=1, \ldots, n ; \sum_{1}^{n} N_{i}=N\right)$ coextensive with $\mathcal{U}$, the units bearing values $P_{i}$, the normed sizes $\left(0<P_{i}<1, \sum P_{i}=1\right)$. From each of the $n$ groups so formed independently one unit is selected with a probability proportional to its size given the units falling in the respective groups. Writing $P_{i j}$ for the $j$ th unit in the $i$ th group,
$$Q_{i}=\sum_{i=1}^{N_{i}} P_{i j},$$
the selection probability of $j$ is $P_{i j} / Q_{i}$. For simplicity, suppressing $j$ to mean by $P_{i}$ the $P$ value for the unit chosen from the $i$ th group, the Rao-Hartley-Cochran estimator (RHCE)
$$t_{3}=\sum_{i=1}^{n} Y_{i} \frac{Q_{i}}{P_{i}},$$
writing $Y_{i}$ for the $y$ value of the unit chosen from the $i$ th group $(i=1,2, \ldots, n)$. This strategy is representative with respect to $P=\left(P_{1}, \ldots, P_{N}\right)^{\prime}$ because $\Sigma_{1}^{n} Q_{i}=1$.

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

Let $(p, t)$ be a strategy with
$$t=\sum_{i=1}^{N} b_{s i} Y_{i}$$
where $b_{s i}$ is free of $Y=\left(Y_{1}, \ldots, Y_{N}\right)^{\prime}$ and $b_{s i}=0$ for $i \notin s$. Then, the mean square error may be written as
\begin{aligned} M_{p}(t) &=E_{p}\left[\sum Y_{i}\left(b_{s i}-1\right)\right]^{2} \ &=\sum_{i=1}^{N} \sum_{j=1}^{N} Y_{i} Y_{j} d_{i j} \end{aligned}
with
$$d_{i j}=E_{p}\left(b_{s i}-1\right)\left(b_{s j}-1\right) .$$
Let $(p, t)$ be representative with respect to a given vector $X=$ $\left(X_{1}, \ldots, X_{N}\right)^{\prime}, X_{i}>0, i \in U$. Then, writing
$$Z_{i}=\frac{Y_{i}}{X_{i}}$$
we get
$$M_{p}(t)=\sum \sum Z_{i} Z_{j}\left(X_{i} X_{j} d_{i j}\right)$$
such that
$$\sum_{i} \sum_{j} X_{i} X_{j} d_{i j}=0 .$$
Define $a_{i j}=X_{i} X_{j} d_{i j}$. Then
$$M_{p}(t)=\sum \sum Z_{i} Z_{j} a_{i j}$$

is a non-negative quadratic form in $Z_{i} ; i=1, \ldots, N$ subject to $\sum_{i} \sum_{j} a_{i j}=0 .$
This implies for every $i=1, \ldots, N$
$$\sum_{j} a_{i j}=0 .$$
From this $M_{p}(t)=\sum \sum Z_{i} Z_{j} a_{i j}$ may be written in the form
\begin{aligned} M_{p}(t) &=-\sum_{i<j}\left(Z_{i}-Z_{j}\right)^{2} a_{i j} \ &=-\sum_{i<j} \sum_{i<j}\left(\frac{Y_{i}}{X_{i}}-\frac{Y_{j}}{X_{j}}\right)^{2} X_{i} X_{j} d_{i j} \end{aligned}
This property of a representative strategy leads to an unbiased quadratic estimator for $M_{p}(t)$, an estimator that is nonnegative, uniformly in $Y$, if such an estimator does exist. This may be shown as follows.
Let
$$m_{p}(t)=\sum_{i=1}^{N} \sum_{j=1}^{N} Y_{i} Y_{j} d_{s i j}$$
be a quadratic unbiased estimator for $M_{p}(t)$ with $d_{s i j}$ free of $Y$ and $d_{s i j}=0$ unless $i \in s$ and $j \in s$. Then
$$\sum_{1}^{N} \sum_{1}^{N} Y_{i} Y_{j} d_{i j}=\sum_{s} p(s)\left[\sum_{1}^{N} \sum_{1}^{N} Y_{i} Y_{j} d_{s i j}\right]$$
or
$$\sum_{1}^{N} \sum_{1}^{N} Z_{i} Z_{j} X_{i} X_{j} d_{i j}=\sum_{s} p(s)\left[\sum_{1}^{N} \sum_{1}^{N} Z_{i} Z_{j} X_{i} X_{j} d_{s i j}\right]$$

∑一世=1ñbs一世X一世=X

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