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

2022年7月19日

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

We will call the collection of the estimators $t=\sum_{i \in s} b_{\mathrm{s}} y_{i}$, whose coefficients $b_{s i}$ ‘s satisfy the unbiasedness condition (2.3.7) as the class of linear homogeneous unbiased estimator $C_{l l 1}$. The class of the linear unbiased estimators $C_{l}$ comprises estimators (2.3.1) and is subject to (2.3.6). The class of all possible unbiased estimators, which includes linear, linear homogeneous, and nonlinear estimators, will be denoted by $C_{u}$ and clearly, $C_{u} \supset C_{l} \supset C_{l l}$.

In Section 2.2, it is shown that for a given sampling design $p$ we can construct numerous linear unbiased estimators for a finite population total $Y$. Therefore, we need to select the best estimator in the sense of having uniformly minimum variance. Godambe (1955) proved that in the class of linear homogeneous unbiased estimators $C_{l l}$, the UMVUE does not exist, i.e., none of the estimators can be termed as the best. Hanurav (1966) modified Godambe’s result and showed the existence of the UMVUE for a unicluster sampling design (defined in the section next). Basu (1971) generalized the result further and proved the nonexistence of the UMVUE in the class of all unbiased estimators $C_{u}$. Godambe (1955) showed that the HTE is the UMVUE in the reduced subspace $R_{0}$ of the parameter space $R^{N}$, where $R_{0}=\bigcup_{i=1}^{N} \mathbf{y}^{(i)}$ and $\mathbf{y}^{(i)}=$ vector $\mathbf{y}$, whose ith coordinate $y_{i}$ is nonzero and the remaining coordinates are zeros.

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

Let $T(s, \mathbf{y})$ be an unbiased estimator for an arbitrary parametric function $\theta=\theta(\mathbf{y})$. The value of $T(s, \mathbf{y})$ depends on the values of $y_{i}$ ‘s belonging to the sample $s$ but is independent of $y_{i}$ ‘s, which do not belong to $s$. The value of $\theta=\theta(\mathbf{y})$ depends on all the values of $y_{i}, i=1, \ldots N$. Let $C_{\theta}$ be the class of all unbiased estimators of $\theta$. Basu (1971) proved the nonexistence of a UMVUE of $\theta(\mathbf{y})$ in the class $C_{\theta}$ of all unbiased estimators. The theorem is described as follows:
Theorem 2.5.3
For a noncensus design, there does not exist the UMVUE of $\theta=\theta(\mathbf{y})$ in the class of all unbiased estimators $C_{\theta}$.
Proof
If possible, let $T_{0}(s, \mathbf{y})\left(\in C_{\theta}\right)$ be the UMVUE of the population parameter $\theta=\theta(\mathbf{y})$. Since the design $p$ is noncensus and the value of $T_{0}(s, \mathbf{y})$ depends on $y_{i}$ ‘s for $i \in s$ but not on the values of $\gamma_{i}$ ‘s for $i \notin s$, we can find a known vector $\mathbf{y}^{(a)}=\left(a_{1}, \ldots, a_{i}, \ldots, a_{N}\right)$ for which $T_{0}\left(s, \mathbf{y}^{(a)}\right) \neq \theta\left(\mathbf{y}^{(a)}\right)$ with $p(s)>0$. Consider the following estimator
$$T^{}(s, \mathbf{y})=T_{0}(s, \mathbf{y})-T_{0}\left(s, \mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)$$ $T^{}(s, \mathbf{y})$ is unbiased for $\theta(\mathbf{y})$ because
$$E_{p}\left[T^{}(s, \mathbf{y})\right]=\theta(\mathbf{y})-\theta\left(\mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)=\theta(\mathbf{y}) .$$ Since $T_{0}(s, \mathbf{y})$ is assumed to be the UMUVE for $\theta(\mathbf{y})$, we must have $$V_{p}\left[T_{0}(s, \mathbf{y})\right] \leq V_{p}\left[T^{}(s, \mathbf{y})\right] \quad \forall \mathbf{y} \in R^{N}$$
Now for $\mathbf{y}=\mathbf{y}^{(a)}, \quad V_{p}\left[T^{}(s, \mathbf{y})\right]=V_{p}\left[T^{}\left(s, \mathbf{y}^{(a)}\right)\right]=V_{p}\left[\theta\left(\mathbf{y}^{(a)}\right)\right]=0$ while $V_{p}\left[T_{0}\left(s, \mathbf{y}^{(a)}\right)\right]>0$ since we assumed $T_{0}\left(s, \mathbf{y}^{(a)}\right) \neq \theta\left(\mathbf{y}^{(a)}\right)$ with $p(s)>0$. Hence the inequality (2.5.10) is violated at $\mathbf{y}=\mathbf{y}^{(a)}$ and the nonexistence of the UMVUE for $\theta(\mathbf{y})$ is proved.

# 抽样调查代考

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

$$T(s, \mathbf{y})=T_{0}(s, \mathbf{y})-T_{0}\left(s, \mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)$$
$T(s, \mathbf{y})$ 是公正的 $\theta(\mathbf{y})$ 因为
$$E_{p}[T(s, \mathbf{y})]=\theta(\mathbf{y})-\theta\left(\mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)=\theta(\mathbf{y}) .$$

$$V_{p}\left[T_{0}(s, \mathbf{y})\right] \leq V_{p}[T(s, \mathbf{y})] \quad \forall \mathbf{y} \in R^{N}$$

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