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## 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 $\gamma_i$ ‘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 11 bbiased 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 $y_i$ ‘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.

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

We have seen in Section $2.5$ that in almost all practical situations, the UMVUE for a finite population total does not exist. The criterion of admissibility is used to guard against the selection of a bad estimator.
An estimator $T$ is said to be admissible in the class $C$ of estimators for a given sampling design $p$ if there does not exist any other estimator in the class $C$ better than $T$. In other words, there does not exist an alternative estimator $T^(\neq T) \in C$, for which following inequalities hold. (i) $V_p\left(T^\right) \leq V_p(T) \quad \forall T^(\neq T) \in C$ and $\mathbf{y} \in R^N$ and (ii) $V_p\left(T^\right)<V_p(T)$ for at least one $\mathbf{y} \in R^N$
Theorem 2.6.1
In the class of linear homogeneous unbiased estimators $C_{l l}$, the HTE $\widehat{Y}{h t}$ based on a sampling design $p$ with $\pi_i>0 \forall i=1, \ldots, N$ is admissible for estimating the population total $Y$. Proof The proof is immediate from Theorem 2.5.2. Since $\widehat{Y}{h t}$ is the UMVUE when $\mathbf{y} \in R_0$, we cannot find an estimator $\forall T^*\left(\neq \widehat{Y}{h t}\right) \in C{l t}$ for which (2.6.1) holds.

The Theorem $2.6 .1$ of admissibility of the HTE $\widehat{Y}{h t}$ in the class $C{l h}$ was proved by Godambe (1960). Godambe and Joshi (1965) proved the admissibility of $\widehat{Y}{h t}$ in the class of all unbiased estimators $C_u$, and it is given in Theorem 2.6.2. Theorem $2.6 .2$ For a given sampling design $p$ with $\pi_i>0 \forall i=1, \ldots, N$, the HTE $\widehat{Y}{h t}$ is admissible in the class $C_u$ of all unbiased estimator for estimating the total $Y$.

# 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

$$\left.T^{(} s, \mathbf{y}\right)=T_0(s, \mathbf{y})-T_0\left(s, \mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)$$
$\left.T^{(} s, \mathbf{y}\right)$ 是公正的 $\theta(\mathbf{y})$ 因为
$$\left.E_p\left[T^{(} s, \mathbf{y}\right)\right]=\theta(\mathbf{y})-\theta\left(\mathbf{y}^{(a)}\right)+\theta\left(\mathbf{y}^{(a)}\right)=\theta(\mathbf{y}) .$$

$$V_p\left[T_0(s, \mathbf{y})\right] \leq V_p\left[T^{(s, \mathbf{y})}\right] \quad \forall \mathbf{y} \in R^N$$

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