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# 统计代写|抽样调查代考SURVEY SAMPLING代考|SOC6160 Kriging or Spatial Prediction

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## 统计代写|抽样调查代考SURVEY SAMPLING代考|Kriging or Spatial Prediction

To this topic the present author has yet no documented contribution. This section is only an abridged quote from Thompson’s (1992) text inserted in order to get my readers acquainted with a topic closely related to the prediction approach discussed in this section.
‘Kriging’ relates to random variables, taking real values of characteristics of living creatures that are ecological or of fossiled features of those that are dead and gone and buried under the earth, like iron ores or rocks that may turn into fuels that are located in various sites.

Suppose $y_{1}, \cdots, y_{n}$ are $n$ observable random variables related respectively to $n$ locations $t_{1}, \cdots, t_{n}$ in a specified region. Let our interest be to predict the value of $y_{0}$ related to some particular location $t_{0}$ in the region other than these. Suppose $E\left(y_{i}\right)=\mu_{i}, i=1, \cdots n$ and $y_{i}$ s are observed and we intend to predict the value of $E\left(y_{0}\right)$ by a quantity
$$\hat{y}{0}=\sum{i=1}^{n} a_{i} y_{i} \quad \text { such that }$$
$E\left(\hat{y}{0}\right)=y{0}$ and the quantities $a_{i}, i=1, \cdots, n$ are to be determined subject to
$$E\left(\hat{y}{0}-y{0}\right)^{2}$$
minimized with respect to $a_{i}, i=1, \cdots, n$.
This is linear spatial prediction or kriging because it is in respect to locations and not temporal and the $y_{i}$ s are variables and not constants.
In this context the covariance functions
$$E\left(y_{t+h}-E\left(y_{t+h}\right)\right)(t-E(t))=C_{h}$$
and the variograms
$$\operatorname{Var}\left(y_{t+h}-y_{t}\right)=2 r(h)$$
are important concepts for study.
Thompson (1992) and the references cited by him are important subjects for studying for interested readers.

## 统计代写|抽样调查代考SURVEY SAMPLING代考|Estimating Equations and Estimating Functions

In our 2005 monograph by Chaudhuri and Stenger(2005) we discussed rather elaborately the above topic. But Mukhopadhyay’s (2004) text contains much more. Confining to what is relevant to survey sampling alone let us briefly extend our coverage in Chaudhuri and Stenger (2005).

Mukhopadhyay (2004) in his landmark text book has covered comprehensively almost all about this subject though being unaware of this at the time Chaudhuri and Stenger (2005) presented at least a readable gist though they have not contributed any substance beyond that as yet. So, here we continue to remain brief.

Continuing with our super-population model-based coverage we suppose $\underline{\mathrm{Y}}=\left(y_{1}, \cdots, y_{i}, \cdots, y_{N}\right)$ is a finite dimensional random vector of independent random variables $y_{i}, i=1, \cdots, N$, with distributions involving an unknown common real-valued parameter $\theta$ which is needed to be suitably estimated. In addition, we suppose $\underline{X}=\left(x_{1}, \cdots x_{i}, \cdots, x_{N}\right)$ is a vector of known real numbers $x_{i}, i=1, \cdots, N$. As usual we shall write $Y=\sum_{1}^{N} y_{i}$ and $X=\sum_{1}^{N} x_{i}$.
If $y_{i}$ ‘s are independently normally distributed with the joint pdf as
$$p(\underline{\mathrm{Y}} \mid \theta)=\frac{1}{(\sqrt{2 \pi})^{N}} \bar{e}^{\frac{1}{2}} \sum_{1}^{N} \frac{\left(y_{i}-\theta x_{i}\right)^{2}}{\sigma_{i}^{2}}$$
then on solving the log likelihood equation
$$\frac{\partial}{\partial \theta} \log p(\underline{\mathrm{Y}}(\theta))=0$$
with respect to $\theta$ one derives for $\theta$ the maximum likelihood estimator
$$\hat{\theta}=\frac{\sum_{1}^{N} y_{i} x_{i} / \sigma_{i}^{2}}{\sum_{1}^{N} x_{i}^{2} / \sigma_{i}^{2}}$$
This ‘census estimator’ is available if $y_{i}$ ‘s are observed for every $i=$ $1, \cdots, N$.
Without postulating normality but supposing
$$E_{m}\left(Y_{i}\right)=\theta x_{i} \quad \text { and } \quad V_{m}\left(y_{i}\right)=\sigma_{i}^{2}$$
on solving with respect to $\theta$ the equation
$$\frac{d}{d \theta} \sum_{1}^{N}\left(y_{i}-\theta x_{i}\right)^{2}-\sigma_{i}^{2}=0$$
one may derive the ‘Least Squares Estimator’ (LSE) or the ‘Best Linear Unbiased Estimator'(BLUE) for $\theta$ as
$$\hat{\theta}=\frac{\sum_{1}^{N} y_{i} x_{i} / \sigma_{i}^{2}}{\sum_{1}^{N} x_{i}^{2} / \sigma_{i}^{2}}$$

## 统计代写|抽样调查代考SURVEY SAMPLING代考|Kriging or Spatial Prediction

“克里金法”与随机变量有关，取生物的生态特征或已死、消失并埋在地下的化石特征的真 实值，例如可能变成燃料的铁矿石或岩石，位于各种网站。

$$\hat{y} 0=\sum i=1^{n} a_{i} y_{i} \quad \text { such that }$$
$E(\hat{y} 0)=y 0$ 和数量 $a_{i}, i=1, \cdots, n$ 将根据以下情况确定
$$E(\hat{y} 0-y 0)^{2}$$

$$E\left(y_{t+h}-E\left(y_{t+h}\right)\right)(t-E(t))=C_{h}$$

$$\operatorname{Var}\left(y_{t+h}-y_{t}\right)=2 r(h)$$

Thompson (1992) 和他引用的参考文献是感兴趣的读者学习的重要课题。

## 统计代写|抽样调查代考SURVEY SAMPLING代考|Estimating Equations and Estimating Functions

Mukhopadhyay (2004) 在他具有里程碑意义的教科书中全面涵盖了几乎所有关于这个主 题的内容，㞔管在 Chaudhuri 和 Stenger (2005) 提出了至少一个可读的要点时并没有意 识到这一点，尽管他们还没有贡南任何内容。所以，在这里我们继续保持简短。

$$p(\underline{\mathrm{Y}} \mid \theta)=\frac{1}{(\sqrt{2 \pi})^{N}} \bar{e}^{-\frac{1}{2}} \sum_{1}^{N} \frac{\left(y_{i}-\theta x_{i}\right)^{2}}{\sigma_{i}^{2}}$$

$$\frac{\partial}{\partial \theta} \log p(\mathrm{Y}(\theta))=0$$

$$\hat{\theta}=\frac{\sum_{1}^{N} y_{i} x_{i} / \sigma_{i}^{2}}{\sum_{1}^{N} x_{i}^{2} / \sigma_{i}^{2}}$$

$$E_{m}\left(Y_{i}\right)=\theta x_{i} \quad \text { and } \quad V_{m}\left(y_{i}\right)=\sigma_{i}^{2}$$

$$\frac{d}{d \theta} \sum_{1}^{N}\left(y_{i}-\theta x_{i}\right)^{2}-\sigma_{i}^{2}=0$$

$$\hat{\theta}=\frac{\sum_{1}^{N} y_{i} x_{i} / \sigma_{i}^{2}}{\sum_{1}^{N} x_{i}^{2} / \sigma_{i}^{2}}$$

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