## 英国补考|抽样调查作业代写sampling theory of survey代考|PSY 279

2022年7月19日

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## 英国补考|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) $V_{p}\left(T^{}\right){h t}$ 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 $\forall T^{*}\left(\neq \widehat{Y}{h t}\right) \in C{l l t}$ for which (2.6.1) holds.

The Theorem 2.6.1 of admissibility of the HTE $\widehat{Y}{h t}$ in the class $C{h h}$ was proved by Godambe (1960). Godambe and Joshi (1965) proved the admissibility of $\widehat{Y}{h t}$ in the class of all unbiased estimators $C{t}$, 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 $Y$.

## 英国补考|抽样调查作业代写sampling theory of survey代考|SUFFICIENCY IN FINITE POPULATION

An estimator $e(s, \mathbf{y})$ is said to be inadmissible in a class of estimators $C$ if there exists an estimator $e^{*}(s, \gamma)(\in C)$ better than $e(s, \gamma)$. Hence it is natural to question how an inadmissible estimator could be improved. The method of improvement of an inadmissible estimator with the aid of sufficient statistics is known as Rao-Blackwellization. The concept of sufficient statistics in survey sampling was introduced by Basu $(1958)$, while the concepts of linear sufficiency, distribution-free sufficient statistics, and Bayesian sufficiency were also introduced by Godambe (1966, 1968). Details have been given by Cassel et al. (1977), Chaudhuri and Stenger (1992), and Thompson and Seber (1996).

Let $s=\left(i_{1}, \ldots, i_{k}, \ldots, i_{n_{s}}\right)$ be an ordered sample of size $n_{s}$ selected from a population $U$ with probability $p(s)$ using a sampling design $p$, where the unit $i_{k}$ is selected at the $k$ th draw. After selection of sample $s$, the responses $\gamma_{i_{1}}, \ldots, \gamma_{i_{n_{s}}}$ were obtained from sampled units $i_{1}, \ldots, i_{n_{s}}$, respectively. The ordered data based on the ordered sample $s$ are denoted by $d=\left{\left(i_{1}, \gamma_{i_{1}}\right), \ldots,\left(i_{k}, \gamma_{i_{n s}}\right)\right}=\left(i_{k}, \gamma_{i_{k}} ; i_{k} \in s\right)$.

Let $\tilde{s}=\left(j_{1}, j_{2}, \ldots, j_{v_{s}}\right)$ with $j_{1}<j_{2}<\ldots<j_{v_{s}}$ be the unordered sample obtained from $s$ by taking distinct units of $s$ and arranging them in ascending order of their labels. Let us denote the operator $r$, which transforms the ordered sample $s$ to the unordered sample $\widetilde{s}$, i.e., $r(s)=\widetilde{s}$. The unordered data are denoted by $\tilde{d}=\left{\left(j_{1}, \gamma_{j_{1}}\right), \ldots,\left(j_{v_{s}}, \gamma_{j_{v_{s}}}\right)\right}=\left(j, \gamma_{j} ; j \in \widetilde{s}\right)$. We define the operator $R$ to get unordered data $\tilde{d}$ from ordered data $d$, i.e., $R(d)=\tilde{d}$
Example $2.7 .1$
Let $U=(1,2,3,4,5), \quad \mathbf{y}=(10,15,15,20,10), \quad$ and $\quad s=(5,2,5)$. Here $y_{1}=10, \quad y_{2}=15, \quad y_{3}=15, \quad y_{4}=20, \quad$ and $\quad \gamma_{5}=10 ; \quad r(s)=\widetilde{s}=(2,5)$, $d={(5,10),(2,15),(5,10)}$ and $R(d)=\widetilde{d}={(2,15),(5,10)}$.

# 抽样调查代考

## 英国补考|抽样调查作业代写sampling theory of survey代考|ADMISSIBLE ESTIMATORS

HTE可接治性定理2.6.1 $\widehat{Y} h t$ 在课堂里 $C h h$ Godambe (1960) 证明了这一点。Godambe 和 Joshi (1965) 证明了 $\widehat{Y} h t$ 在所有无偏估计量中 $C t$ ，它在定理 $2.6 .2$ 中给出。

## 英国补考|抽样调查作业代写sampling theory of survey代考|SUFFICIENCY IN FINITE POPULATION

$i_{1}, \ldots, i_{n_{s}}$ ，分别。基于有序样本的有序数据 $s$ 表示为

$y_{1}=10, \quad y_{2}=15, \quad y_{3}=15, \quad y_{4}=20, \quad$ 和
$\gamma_{5}=10 ; \quad r(s)=\tilde{s}=(2,5), d=(5,10),(2,15),(5,10)$ 和
$R(d)=\tilde{d}=(2,15),(5,10)$

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