# 统计代写|抽样调查作业代写sampling theory of survey代考|PSY279

## 统计代写|抽样调查作业代写sampling theory of survey代考|Basu’s (1971) Circus Example

A circus company while transporting its 30 elephants needed their weights to book their tickets. To save the weighing machine the manager decided to weigh only 1 of the elephants and multiply its weight by 30 . But the company had an employee knowledgeable in statistics who advised against choosing the medium sized elephant Sambo but asked to choose 1 with an arbitrary probability and divide its weight by its selection-probability. The manager assigned selection-probability $\frac{29}{30}$ to Sambo and $\left(\frac{1}{29}\right) \frac{1}{30}$ to every other. Sambo was obviously selected and the Horvitz and Thompson’s estimate for the 30 elephants was calculated as $\frac{30}{29}$ weight of Sambo.

If somehow the elephant Jumbo happened to be selected the estimated total weight would be $30 \times 29 \times$ Jumbo’s weight. Basu (1971) wittily concluded with the remarks “This is how the circus statistician lost his circus job and possibly became a university professor.” We have said enough. This problem is to be tackled and we say more to follow.

The reason behind Basu’s (1971) caustic comments against Horvitz and Thompson’s (1952) estimator is apparently because its admissibility (Godambe (1960) and Godambe and Joshi (1965)), necessary bestness (Prabhu Ajgaonkar (1965)) and hyperadmissibility (Hanurav (1968)) among other properties were announced irrespective of any relationships among $\pi_i$ ‘s and $Y_i^{\prime}$ ‘s. But the ‘estimating function approach’ is not indifferent about them and some optimality requirements of the $\pi_i$ ‘s to be employed in practice have been carefully enunciated as stated in Mukhopadhyay’s (2004) book. But we have not touched them here because Godambe (1955, 1960) and Godambe and Thompson (1977) have discussed almost fully and derived similar results as we have also rather comprehensively discussed already in this text. Yet personally I am not very enthusiastic about the utility of Godambe and Durbin’s (1960) ‘estimating function approach’ to ‘survey sampling’ in spite of Godambe and Thompson’s (1986) advocacy of it though this approach has been mentioned energetically by Mukhopadhyay (2004) to other complementary areas of statistics which we have no reason to dispute. But I feel after Godambe and Thompson’s (1977) promising resolutions about the optimality properties of a wider class of estimators generalizing the Horvitz and Thompson’s estimator which have opened up further research in sampling along the lines of generalized regression estimators and other competitors, I am rather amazed at Godambe and Thompson’s (1986) defense of the ‘estimating function approach’ to survey sampling. I feel we need to wait for more useful discoveries in the context which are beyond my capabilities for the present.

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

Warner (1965) gave us $50^{+}$years back his novel ‘Randomized Response (RR) Technique (RRT)’ to procure trustworthy data on sensitive personal items from sampled people from a community, like his/her proneness to tax evasion, illegal driving habits, gambling involvement etc., namely the features people usually like to hide from others.

His device is to present before a chosen respondent a box of identical cards in proportions $p\left(0<p \neq \frac{1}{2}<1\right)$ bearing $A$ and $(1-p)$ bearing $A^C$, the complementary characteristic. The respondent is to randomly draw a card from the box and return to it after telling the interviewer ‘Yes’ if the card mark matched his/her trait $A$ or $A^C$ and ‘No’ if it did not match. Other chosen persons are also to independently repeat this exercise, not of course divulging the card label to the interviewer. Warner (1965) selected samples by simple random sampling with replacement (SRSWR) and easily provided appropriate estimator for the unknown proportion $\theta$ of people bearing $A$ in the community along with an appropriate variance estimator. Chaudhuri $(2001,2011 \mathrm{a}, 2016)$ propagated the view that an RRT has only to elicit a truthful response to a query on implementing an RR trial from a respondent no matter how chosen and provided (i) every sampled person is given a positive inclusion-probability and (ii) every pair of distinct respondents is given a positive inclusion-probability; then (a) an unbiased estimator for $\theta$ along with an (b) unbiased estimator of the variance thereof is available. Let us show how.

Let $U=(1, \cdots i, \cdots, N)$ denote a known collection of people in a community identified and labeled $i=1, \cdots, N$ with values $y_i$ such that
\begin{aligned} & y_i=1 \text { if } i \text { bears } A \ & =0 \text { if } i \text { bears } A^C \ & \end{aligned}
and our intention is to estimate $Y=\sum_1^N y_i$ and $\theta=\frac{Y}{N}$ on obtaining by Warner’s RR method a response from a sampled person $i$ the response
$I_i=1 \quad$ if for $i$ the card-type matches the feature $A$ or $A^C$ $=0, \quad$ if it mis-matches.

# 抽样调查代考

## 统计代写|抽样调查作业代写sampling theory of survey代考|Basu’s (1971) Circus Example

Basu (1971) 对 Horvitz 和 Thompson (1952) 估计量的刻薄评论背后的原因显然是因为它的可乎纳 性 (Godambe (1960) 以及 Godambe 和 Joshi (1965))、必要最佳性 (Prabhu Ajgaonkar (1965)) 和超 可采纳性 (Hanurav ( 1968)) 在其他属性中被公布，而不管它们之间的任何关系 $\pi_i{ }^{\prime}$ ‘沙 $Y_i^{\prime}$ 的。但是 “估计函数方法”对它们和一些最优性要求并不漠不关心 $\pi_i$ 如 Mukhopadhyay (2004) 的书中所述， 在实践中使用的方法已被仔细阐明。但我们在这里没有触及它们，因为 Godambe $(1955,1960)$ 和 Godambe 和 Thompson (1977) 几乎完全讨论并得出了类似的结果，正如我们在本文中已经相当全 面地讨论的那样。然而，尽管 Godambe 和 Thompson（1986 年) 提倡使用 Godambe 和 Durbin (1960 年) 的”估计函数方法”来进行”调查抽样”，但我个人并不是很热心心，尽管 Mukhopadhyay (2004 年) 积极提到了这种方法) 到涐们没有理由质疑的其他补充統计领域。但我觉得在 Godambe 和 Thompson（1977) 关于更广泛的估计器的最优性属性的有前途的解决方案之后，推 广了 Horvitz 和 Thompson 的估计器，这开辟了沿着广义回归估计器和其他竞争者的路线进一步抽 样研究，我宁愿对 Godambe 和 Thompson (1986) 对调查抽样的“估计函数方法”的辩护感到掠讶。 我觉得我们需要等待更多有用的发现，这些发现超出了我目前的能力范围。

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

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