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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 $t=\sum_{i \in s} b_{s i} y_i$, whose coefficients $b_{s i}$ ‘s satisfy the unbiasedness condition (2.3.7) as the class of linear homogeneous unbiased estimator $C_{l l i}$. 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 $C_{11}$. 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 $i$ th 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 Linear Homogeneous Unbiased Estimators

Theorem 2.5.1
In the class of linear homogeneous unbiased estimators $C_{l l}$, the UMVUE of a finite population total $Y$ based on a noncensus sampling design, $p$ with $\pi_i>0 \quad \forall i=1, \ldots, N$ does not exist if the sampling design $p$ is nonunicluster. However, the UMVUE does exist if $p$ is a unicluster design.
Proof
The class $C_{l h}$ consists of the estimators $t(s, \mathbf{y})=\sum_{i \in s} b_{s i} y_i$, where the constants $b_{s i}$ ‘s satisfy the unbiasedness condition (2.3.7) viz.
$$\sum_{s \supset i} b_{s i} p(s)=1 \text { for every } i=1, \cdots, N$$

Here we want to find the constants, $b_{s i}$ ‘s, that minimize
$$V_p(t)=\sum_{s \in \mathcal{J}}\left(\sum_{i \in s} b_{s i} y_i\right)^2 p(s)-Y^2$$
subject to the unbiasedness condition (2.3.7).
For minimization, we consider
$$\Psi=\sum_{s \in \mathcal{J}}\left(\sum_{i \in s} b_{s i} y_i\right)^2 p(s)-Y^2-\sum_{i=1}^N \lambda_i\left{\sum_{s \supset i} b_{s i} p(s)-1\right}$$
with $\lambda_i$ ‘s as undetermined Lagrange multipliers.
Differentiating $\Psi$ with respect to $b_{s i}$ and equating to zero, we get
$$\frac{\partial \Psi}{\partial b_{s i}}=2 \gamma_i\left(\sum_{i \in s} b_{s i} y_i\right) p(s)-\lambda_i p(s)=0$$
Eq. (2.5.2) should be valid for $V \mathbf{y} \in R^N$. In particular for $\mathbf{y}=\mathbf{y}^{(i)}=\left(0, \ldots, 0, \gamma_i, 0, \ldots, 0\right)$ with $\gamma_i \neq 0$, Eq. (2.5.2) yields the optimum value of $b_{s i}$ as
$$b_{s i}=\lambda_i /\left(2 y_i^2\right)=b_{0 i} \text { (say) }$$
The unbiasedness condition (2.3.7) and (2.5.3) yield
$$b_{s i}=b_{0 i}=1 / \pi_i$$
Thus the UMVUE should necessarily be the HTE viz. $\sum_{i \in s} \gamma_i / \pi_i$ and should satisfy Eq. (2.5.2), i.e.,
$$\sum_{i \in s} y_i / \pi_i=\frac{\lambda_i}{2 y_i} \text { for } \forall i \in s, \gamma_i \neq 0$$

# 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 Linear Homogeneous Unbiased Estimators

$\sum_{s \supset i} b_{s i} p(s)=1$ for every $i=1, \cdots, N$

$$V_p(t)=\sum_{s \in \mathcal{J}}\left(\sum_{i \in s} b_{s i} y_i\right)^2 p(s)-Y^2$$

$\backslash$ left 的分隔符缺失或无法识别

$$\frac{\partial \Psi}{\partial b_{s i}}=2 \gamma_i\left(\sum_{i \in s} b_{s i} y_i\right) p(s)-\lambda_i p(s)=0$$

$$b_{s i}=\lambda_i /\left(2 y_i^2\right)=b_{0 i} \text { (say) }$$

$$b_{s i}=b_{0 i}=1 / \pi_i$$

$$\sum_{i \in s} y_i / \pi_i=\frac{\lambda_i}{2 y_i} \text { for } \forall i \in s, \gamma_i \neq 0$$

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