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

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## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimating Functions and Equations

Suppose $Y=\left(Y_{1}, \ldots, Y_{N}\right)^{\prime}$ is a random vector and $X=\left(X_{1}, \ldots\right.$, $\left.X_{N}\right)^{\prime}$ is a vector of known numbers $X_{i}(>0), i=1, \ldots, N$. Let the $Y_{i}$ ‘s be independent and normally distributed with means and variances, respectively
$$\theta X_{i} \text { and } \sigma_{i}^{2}, i=1, \ldots, N \text {. }$$
If all the $Y_{i}$ ‘s $i=1, \ldots, N$ are available for observation, then from the joint probability density function (pdf) of $Y$
$$p(Y, \theta)=\prod_{i=1}^{N} \frac{1}{\sigma_{i} \sqrt{2 \pi}} e^{-\frac{1}{2 \sigma_{i}^{2}}\left(Y_{i}-\theta X_{i}\right)^{2}}$$
one gets the well-known maximum likelihood estimator (MLE) $\theta_{0}$, based on $Y$, for $\theta$, given by the solution of the likelihood equation
$$\frac{\partial}{\partial \theta} \log p(Y, \theta)=0$$
as
$$\theta_{0}=\left[\sum_{1}^{N} Y_{i} X_{i} / \sigma_{i}^{2}\right] /\left[\sum_{1}^{N} X_{i}^{2} / \sigma_{i}^{2}\right]$$

On the other hand, let the normality assumption above be dropped, everything else remaining unchanged, that is, consider the linear model
$$Y_{i}=\theta X_{i}+\varepsilon_{i}$$
with $\varepsilon_{i}$ ‘s distributed independently and
$$E_{m}\left(\varepsilon_{i}\right)=0, V_{m}\left(\varepsilon_{i}\right)=\sigma_{i}^{2}, i=1, \ldots, N .$$
Then, if $\left(Y_{i}, X_{i}\right), i=1, \ldots, N$ are observed, one may derive the same $\theta_{0}$ above as the least squares estimator (LSE) or as the best linear unbiased estimator (BLUE) for $\theta$.

Such a $\theta_{0}$, based on the entire finite population vector $Y=\left(Y_{1}, \ldots, Y_{N}\right)^{\prime}$, is really a parameter of this population itself and will be regarded as a census estimator.

If $X_{i}=1, \sigma_{i}=\sigma$ for all $i$ above, then $\theta_{0}$ reduces to $Y / N=\bar{Y}$.
We shall next briefly consider the theory of estimating functions and estimating equations as a generalization that unifies (see GHOSH, 1989) both of these two principal methods of point estimation and, in the next section, illustrate how the theory may be extended to yield estimators in the usual sense of the term based on a sample of $Y_{i}$ values rather than on the entire $Y$ itself.

We start with the supposition that $Y$ is a random vector with a probability distribution belonging to a class $C$ of distributions each identified with a real-valued parameter $\theta$. Let
$$g=g(Y, \theta)$$
be a function involving both $Y$ and $\theta$ such that
(a) $\frac{\partial g}{\partial \theta}(Y, \theta)$ exists for every $Y$
(b) $E_{m} g(Y, \theta)=0$, called the unbiasedness condition
(c) $E_{m} \frac{\partial g}{\partial \theta}(Y, \theta) \neq 0$
(d) the equation $g(Y, \theta)=0$ admits a unique solution $\theta_{0}=$ $\theta_{0}(Y)$
Such a function $g=g(Y, \theta)$ is called an unbiased estimating function and the equation
$$g(Y, \theta)=0$$
is called an unbiased estimating equation.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Applications to Survey Sampling

A further line of approach is now required because $\theta_{0}$ itself needs to be estimated from survey data
$$d=\left(i, Y_{i} \mid i \in s\right)$$
available only for the $Y_{i}$ ‘s with $i \in s, s$ a sample supposed to be selected with probability $p(s)$ according to a design $p$ for which we assume
$$\pi_{i}=\sum_{s \ni i} p(s)>0 \text { for all } i=1,2, \ldots, N \text {. }$$
With the setup of the preceding section, let the $Y_{i}$ ‘s be independent and consider unbiased estimating functions $\phi_{i}\left(Y_{i}, \theta\right) ; i=$ $1,2, \ldots, N$. Let
$$\theta_{0}=\theta_{0}(Y)$$
be the solution of $g(Y, \theta)=0$ where
$$g(Y, \theta)=\sum_{1}^{N} \phi_{i}\left(Y_{i}, \theta\right)$$
and consider estimating this $\theta_{0}$ using survey data $d=\left(i, Y_{i} \mid i \in\right.$ $s$ ). For this it seems natural to start with an unbiased sampling function
$$h=h(s, Y, \theta)$$
which is free of $Y_{j}$ for $j \notin s$ and satisfies
(a) $\frac{\partial h}{\partial \theta}(s, Y, \theta)$ exists for all $Y$
(b) $E_{m} \frac{\partial h}{\partial \theta}(s, Y, \theta) \neq 0$
(c) $E_{p} h(s, Y, \theta)=g(Y, \theta)$ for all $Y$, the unbiasedness condition.

Let $H$ be a class of such unbiased sampling functions. Following the extension of the approach in section 3.3.1 by GoDAMBE and THOMPSON (1986a), we may call a member
$$h_{0}=h_{0}(s, Y, \theta)$$

of $H$ and the corresponding equation $h_{0}=0$, optimal if
$$\frac{E_{m} E_{p} h^{2}(s, Y, \theta)}{\left[E_{m} E_{p} \frac{\partial h}{\partial \theta}(s, Y, \theta)\right]^{2}}$$
as a function of $h \in H$ is minimal for $h=h_{0}$.

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

$$Y_{i}=\theta+\varepsilon_{i}$$
where the $\varepsilon_{i}$ ‘s are independent with $E_{m} \varepsilon_{i}=0, V_{m} \varepsilon_{i}=\sigma_{i}^{2}$. Then the estimating function
$$\sum_{i}^{N} \phi_{i}\left(Y_{i}, \theta\right)=\sum_{i}^{N} \frac{\left(Y_{i}-\theta\right)}{\sigma_{i}^{2}}$$
is linearly optimal, but does not define the survey population parameter $\bar{Y}$, which is usually of interest. Therefore, we may consider the estimating equation $g_{0}=0$ where
$$g_{0}=\sum \phi_{i}\left(Y_{i}, \theta\right)=\sum\left(Y_{i}-\theta\right)$$
is unbiased and, while not linearly optimal, defines
$$\theta_{0}=\bar{Y}$$
and the optimal sample estimator
$$\hat{\theta}{0}=\frac{\sum{s} Y_{i} / \pi_{i}}{\sum_{s} 1 / \pi_{i}}$$
for $\theta_{0}$. Incidentally, this estimator was proposed earlier by HÁJEK (1971).
In general, the solution $\theta_{0}$ of
$$g=\sum \phi_{i}\left(Y_{i}, \theta\right)=0$$
where $\phi_{i}\left(Y_{i}, \theta\right), i=1,2, \ldots, N$ are unbiased estimating functions is an estimator of the parameter $\theta$ of the superpopulation model, provided all $Y_{1}, Y_{2}, \ldots, Y_{N}$ are known. In any case, it may be of interest in itself, that is, an interesting parameter of the population. The solution $\hat{\theta}{0}$ of the optimal unbiased sampling equation $h{0}=0$ is used as an estimator for the population parameter $\theta_{0}$.

If $g$ is linearly optimal, then the population parameter $\theta_{0}$ is especially well-motivated by the superpopulation model.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Estimating Functions and Equations

p( Y , \theta)=\prod_{i=1}^{N} \frac{1}{\sigma_{i} \sqrt{2 \pi}} e^{-\frac{1}{2 \sigma_{i}^{2}}\left(Y_{i}-\theta X_{i}\right)^{2}}
$$一个得到众所周知的最大似然估计（MLE）θ0, 基于  Y,F这r\θ,G一世在和nb是吨H和s这l在吨一世这n这F吨H和l一世ķ和l一世H这这d和q在一种吨一世这n \frac{\partial}{\partial \theta} \log p( Y , \theta)=0 一种s \theta_{0}=\left[\sum_{1}^{N} Y_{i} X_{i} / \sigma_{i}^{2}\right] /\left[\sum_{1}^{ N} X_{i}^{2} / \sigma_{i}^{2}\right]$$

$$g=g( Y , \theta)$$是一个包含 $Y 的函数一种nd\θs在CH吨H一种吨(一种)\frac{\partial g}{\partial \theta}( Y , \theta)和X一世s吨sF这r和在和r是是(b)E_{m} g( Y , \theta)=0,C一种ll和d吨H和在nb一世一种s和dn和ssC这nd一世吨一世这n(C)E_{m} \frac{\partial g}{\partial \theta}( Y , \theta) \neq 0(d)吨H和和q在一种吨一世这ng( Y , \theta)=0一种d米一世吨s一种在n一世q在和s这l在吨一世这n\theta_{0}=\theta_{0}( Y )小号在CH一种F在nC吨一世这ng=g( Y , \theta)一世sC一种ll和d一种n在nb一世一种s和d和s吨一世米一种吨一世nGF在nC吨一世这n一种nd吨H和和q在一种吨一世这n$
g( Y , \theta)=0
$$称为无偏估计方程。 ## 统计代写|抽样调查作业代写sampling theory of survey代考|Applications to Survey Sampling 现在需要进一步的方法，因为θ0本身需要从调查数据中估计 d=(一世,是一世∣一世∈s) 仅适用于是一世与一世∈s,s应该以概率选择的样本p(s)根据设计p我们假设 圆周率一世=∑s∋一世p(s)>0 对全部 一世=1,2,…,ñ. 有了上一节的设置，让是一世是独立的并考虑无偏估计函数φ一世(是一世,θ);一世= 1,2,…,ñ. 令$$
\theta_{0}=\theta_{0}( Y )

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