统计代写|抽样调查作业代写sampling theory of survey代考| Choosing Good Sampling Strategies

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|抽样调查作业代写sampling theory of survey代考|Nonexistence Results

Let a design $p$ be given and consider a $p$-unbiased estimator $t$, that is, $B_{p}(t)=E_{p}(t-Y)=0$ uniformly in $Y$. The performance of such an estimator is assessed by $V_{p}(t)=E_{p}(t-Y)^{2}$ and we would like to minimize $V_{p}(t)$ uniformly in $Y$. Assume $t^{}$ is such a uniformly minimum variance (UMV) unbiased estimator (UMVUE), that is, for every unbiased $t$ (other than $\left.t^{}\right)$ one has $V_{p}\left(t^{}\right) \leq V_{p}(t)$ for every $Y$ and $V_{p}\left(t^{}\right)<V_{p}(t)$ at least for one $Y$.

Let $\Omega$ be the range (usually known) of $Y$; for example, $\Omega=\left{Y: a_{i}<Y_{i}<b_{i}, i=1, \ldots, N\right}$ with $a_{i}, b_{i}(i=1, \ldots, N)$ as known real numbers. If $a_{i}=-\infty$ and $b_{i}=+\infty$, then $\Omega$ coincides with the $N$-dimensional Euclidean space $\mathbb{R}^{N}$; otherwise $\Omega$ is a subset of $\mathbb{R}^{N}$. Let us choose a point $A=\left(A_{1}, \ldots, A_{i}, \ldots\right.$, $\left.A_{N}\right)^{\prime}$ in $\Omega$ and consider as an estimator for $Y$
\begin{aligned} t_{A} &=t_{A}(s, Y) \ &=t^{}(s, Y)-t^{}(s, A)+A \end{aligned}

where $A=\Sigma A_{i}$. Then,
$$E_{p}\left(t_{A}\right)=E_{p} t^{}(s, Y)-E_{p} t^{}(s, A)+A=Y-A+A=Y$$
that is, $t_{A}$ is unbiased for $Y$. Now the value of
$$V_{p}\left(t_{A}\right)=E_{p}\left[t^{}(s, Y)-t^{}(s, A)+A-Y\right]^{2}$$
equals zero at the point $Y=A$. Since $t^{}$ is supposed to be the UMVUE, $V_{p}\left(t^{}\right)$ must also be zero when $Y=A$. Now $A$ is arbitrary. So, in order to qualify as the UMVUE for $Y$, the $t^{}$ must have its variance identically equal to zero. This is possible only if one has a census, that is, every unit of $U$ is in $s$ rendering $t^{}$ coincident with $Y$. So, for no design except a census design, for which the entire population is surveyed, there may exist a UMV estimator among all UE’s for $Y$. The same is true if, instead of $Y$, one takes $\bar{Y}$ as the estimand. This important nonexistence result is due to GODAMBE and JOSHI (1965) while the proof presented above was given by BASU (1971).

Let us now seek a UMV estimator for $Y$ within the restricted class of HLU estimators of the form
$$t=t_{b}=t(s, Y)=\sum_{i \in s} b_{s i} Y_{i} .$$
Because of the unbiasedness of the estimator we need, uniformly in $Y, Y$ equal to
$$E\left(t_{b}\right)=\sum_{s} p(s)\left[\sum_{i \in s} b_{s i} Y_{i}\right]=\sum_{i=1}^{N} Y_{i}\left[\sum_{s \ni i} b_{s i} p(s)\right]$$

统计代写|抽样调查作业代写sampling theory of survey代考|Rao-Blackwellization

An estimator $t=t(s, Y)$ may depend on the order in which the units appear in $s$ and may depend on the multiplicities of the appearances of the units in $s$.

EXAMPLE 3.1 Let $P_{i}\left(0<P_{i}<1, \Sigma_{1}^{N} P_{i}=1\right)$ be known numbers associated with the units $i$ of $U$. Suppose on the first draw a unit $i$ is chosen from $U$ with probability $P_{i}$ and on the second draw a unit $j(\neq i)$ is chosen with probability $\frac{P_{j}}{1-P_{i}}$.

Consider RAJ’s (1956) estimator (see section 2.4.6) $t_{D}=t(i, j)=\frac{1}{2}\left[\frac{Y_{i}}{P_{i}}+\left(Y_{i}+\frac{Y_{j}}{P_{j}}\left(1-P_{i}\right)\right)\right]=\frac{1}{2}\left(e_{1}+e_{2}\right), \quad s a y .$
Now,
$$E_{p}\left(e_{1}\right)=E_{p}\left[\frac{Y_{i}}{P_{i}}\right]=\sum_{1}^{N} \frac{Y_{i}}{P_{i}} P_{i}=Y$$
and
$$e_{2}=Y_{i}+\frac{Y_{j}}{P_{j}}\left(1-P_{j}\right)$$
has the conditionalexpectation, given that $\left(i, Y_{i}\right)$ is observed on the first draw,
$$E_{C}\left(e_{2}\right)=Y_{i}+\sum_{j \neq i}\left[\frac{Y_{j}}{P_{j}}\left(1-P_{i}\right)\right] \frac{P_{j}}{1-P_{i}}=Y_{i}+\sum_{j \neq i} Y_{j}=Y$$
and hence the unconditional expectation $E_{p}\left(e_{2}\right)=Y$. So $t_{D}$ is unbiased for $Y$, but depends on the order in which the units appear in the sample $s=(i, j)$ that is, in general
$$t_{D}(i, j) \neq t_{D}(j, i) .$$
EXAMPLE 3.2 Let n draws be independently made choosing the unit $i$ on every draw with the probability $P_{i}$ and let $t$ be an estimator for $Y$ given by
$$t=\frac{1}{n} \sum_{r=1}^{n} \frac{y_{r}}{p_{r}}$$
where $y_{r}$ is the value of $y$ for the unit selected on the rth draw $(r=1, \ldots, n)$ and $p_{r}$ the value $P_{i}$ if the rth draw produces the unit $i$. This $t$, usually attributed to HANSEN and HURWITZ (1943), may also be written as
$$t_{H H}=\frac{1}{n} \sum_{i=1}^{N} \frac{Y_{i}}{P_{i}} f_{s i}$$
and, therefore, depends on the multiplicity $f_{s i}$ of $i$ in $s$ (see section 2.2).

Next we consider a requirement of admissibility of an estimator in the absence of UMVUEs for useful designs in a meaningful sense.

An unbiased estimator $t_{1}$ for $Y$ is better than another unbiased estimator $t_{2}$ for $Y$ if $V_{p}\left(t_{1}\right) \leq V_{p}\left(t_{2}\right)$ for every $Y \in$ $\Omega$ and $V_{p}\left(t_{1}\right)<V_{p}\left(t_{2}\right)$ at least for one $Y \in \Omega$. Subsequently, the four cases mentioned in section 3.1.2 are considered for $\Omega$ without explicit reference.

If there does not exist any unbiased estimator for $Y$ better than $t_{1}$, then $t_{1}$ is called an admissible estimator for $Y$ within the UE class. If this definition is restricted throughout within the HLUE class, then we have admissibility within HLUE.
RESULT 3.2 The HTE
$$t=\sum_{i \in s} \frac{Y_{i}}{\pi_{i}}$$
is admissible within the HLUE class.
PROOF: For $t_{b}$ in the HLUE class and for the HTE $\bar{t}$ we have
$$V_{p}\left(t_{b}\right)=\sum_{i} Y_{i}^{2}\left[\sum_{s \ni i} b_{s i}^{2} p(s)\right]+\sum_{i \neq j} \sum_{i} Y_{j}\left[\sum_{s \ni i, j} b_{s i} b_{s j} p(s)\right]-Y^{2}$$
$$V_{p}(\bar{t})=\sum_{i} Y_{i}^{2} / \pi_{i}+\sum_{i \neq j} \sum_{i} Y_{j} \frac{\pi_{i j}}{\pi_{i} \pi_{j}}-Y^{2} .$$
Evaluated at a point $Y{0}^{(i)}=\left(0, \ldots, Y{i} \neq 0, \ldots, 0\right),\left[V_{p}\left(t_{b}\right)-\right.$ $\left.V_{p}(\bar{t})\right]$ equals
$$Y_{i}^{2}\left[\sum_{s \ni i} b_{s i}^{2} p(s)-\frac{1}{\pi_{i}}\right] \geq 0$$
on applying Cauchy’s inequality. This degenerates into an equality if and only if $b_{s i}=b_{i}$, for every $s \ni i$, rendering $t_{b}$ equal to the HTE $\bar{t}$. So, for $t_{b}$ other than $\bar{t}$,
$$\left[V_{p}\left(t_{b}\right)-V_{p}(t)\right]{Y=Y{0}^{(i)}}>0 .$$
This result is due to GoDAMBE (1960a). Following GoDAMBE and JOSHI (1965) we have:
RESULT 3.3 The HTE $\bar{t}$ is admissible in the wider UE class.
PROOF: Let, if possible, $t$ be an unbiased estimator for $Y$ better than the HTE $t$. Then, we may write
$$t=t(s, Y)=\bar{t}(s, Y)+h(s, Y)=\bar{t}+h$$
with $h=h(s, Y)=t-\bar{t}$ as an unbiased estimator of zero. Thus,
$$0=E_{p}(h)=\sum_{s} h(s, Y) p(s) .$$

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