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

statistics-lab™ 为您的留学生涯保驾护航 在代写抽样调查sampling theory of survey方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽样调查sampling theory of survey方面经验极为丰富，各种代写抽样调查sampling theory of survey相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

So far, the performance of a strategy $(p, t)$ has been described by its $\operatorname{MSE} M_{p}(t)$, which is a function defined as the parameter space $\Omega$, the set of all vectors $Y$ relevant in a given situation.
Now, $\Omega$ may be such that
$$\sup {Y \in \Omega} M{p}(t)=R_{p}(t) \text {, say, }$$
is finite for some strategies $(p, t)$ of a class $\Delta$ fixed in advance, especially by budget restrictions. Then it may be of interest to look for a strategy minimizing $R_{p}(t)$, with respect to the pair $(p, t)$.

Let $\Delta$ be the class of all available strategies and $R_{p}(t)$ be finite for at least some elements of $\Delta$. Then
$$r^{}=\inf {(p, t) \in \Delta} R{p}(t)=\inf {(p, t) \in \Delta} \sup {Y \in \Omega} M_{p}(t)<\infty$$ and $r^{}$ is called minimax value with respect to $\Omega$ and $\Delta$; a strategy $\left(p^{}, t^{}\right) \in \Delta$ is called a minimax strategy if
$$R_{p^{}}\left(t^{}\right)=r^{*} .$$
For given size measures $x$ and $z$ with
$$\begin{array}{cc} 0<X_{i} ; & i=1,2, \ldots, N \ 0<Z_{i} \leq Z / 2 ; & i=1,2, \ldots, N \end{array}$$

where $Z=\sum_{1}^{N} Z_{i}$ let us define the parameter space
$$\Omega_{x z}=\left{Y \in \mathbb{R}^{N}: \sum \frac{X_{i}}{X}\left(\frac{Y_{i}}{Z_{i}}-\frac{Y}{Z}\right)^{2} \leq 1\right} .$$
Of special importance is the class of strategies $\Delta_{n}={(p, t): p$ of fixed effective size $n, t$ homogeneously linear}.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Minimax Strategies of Sample Size

We first consider the special case $\Delta_{1}$, consisting of all pairs $(p, t)$ such that
\begin{aligned} p(s) &>0 \text { implies }|s|=1 \ t(s, Y) &=t(i, Y)=Y_{i} / q_{i}, q_{i} \neq 0 . \end{aligned}
Writing $p_{i}=p(i)$ each strategy in $\Delta_{1}$ may be identified with a pair $(p, q) ; p, q \in \mathbb{R}^{N}$, and its MSE is
$$\sum p_{i}\left[\frac{Y_{i}}{q_{i}}-Y\right]^{2} \text {. }$$
Now, following STENGER (1986), we show that
$$\sup {Y \in \Omega{x z}} \sum p_{i}\left[\frac{Y_{i}}{q_{i}}-Y\right]^{2}$$
is minimum for
$$\begin{gathered} p_{i}=\frac{X_{i}}{X}=p_{i}^{}, \text { say, } \ q_{i}=\frac{Z_{i}}{Z}=q_{i}^{}, \text { say, } \end{gathered}$$
$(i=1,2, \ldots, N)$ such that $\left(p^{}, q^{}\right)$ is a minimax strategy. $Y \in \Omega_{x z}$ implies $Y+\lambda Z \in \Omega_{x z}$ for every real $\lambda$ and the MSE of $a \operatorname{strategy}(p, q)$ evaluated for $Y+\lambda Z$ is
$$\sum p_{i}\left[\frac{Y_{i}+\lambda Z_{i}}{q_{i}}-Y-\lambda Z\right]^{2} .$$
This quadratic function of $\lambda$ is bounded if and only if $\frac{Z_{i}}{q_{i}}-Z=0$

which is equivalent to $q_{i}=q_{i}^{}$. So $R_{p}(t)<\infty$ for $(p, q)=(p, t) \in$ $\Delta_{1}$ if and only if $q=q^{}$. Now, for
$$A(p)=\sup {Y \in \Omega{\mathrm{xz}}} \sum p_{i}\left[\frac{Y_{i}}{q_{i}^{}}-Y\right]^{2}$$ we have $$A\left(p^{}\right)=\sup {Y \in \Omega{x z}} \sum p_{i}^{}\left[\frac{Y_{i}}{q_{i}^{}}-Y\right]^{2}=Z^{2} .$$
For $p \neq p^{}$ there exists $j$ with $p_{j}=p_{j}^{}+\varepsilon, \varepsilon>0$.
It is easily seen that
$$p_{j}^{}-2 p_{j}^{} q_{j}^{}+q_{j}^{ 2}>0 .$$
So we may define
\begin{aligned} Y_{i}^{(j)} &=q_{j}^{} / \sqrt{p_{j}^{}-2 p_{j}^{} q_{j}^{}+q_{j}^{* 2}} \text { for } i=j \ &=0 \text { for } i \neq j . \end{aligned}
The total $Y^{(j)}$ of $Y^{(j)}$ is equal to $Y_{j}^{(j)}$ and
\begin{aligned} \sum p_{i}\left[\frac{Y_{i}^{(j)}}{q_{i}^{}}-Y^{(j)}\right]^{2} &=Z^{2} \frac{p_{j}-2 p_{j} q_{j}^{}+q_{j}^{* 2}}{p_{j}^{}-2 p_{j}^{} q_{j}^{}+q_{j}^{ 2}} \ &=Z^{2}\left[1+\frac{\varepsilon\left(1-2 q_{j}^{}\right)}{p_{j}^{}-2 p_{j}^{} q_{j}^{}+q_{j}^{* 2}}\right] \ & \geq Z^{2} \end{aligned}
because $Z_{j} \leq Z / 2$ implies $1-2 q_{j}^{} \geq 0$. Obviously, $Y^{(j)} \in \Omega_{x z}$ and $$A(p) \geq Z^{2}=A\left(p^{}\right)$$
for all $p$.

## 统计代写|抽样调查作业代写sampling theory of survey代考|Minimax Strategies of Sample Size

In the special case $X_{i}=Z_{i}=1$ we have the parameter space
$$\Omega_{11}=\left{Y \in \mathbb{R}^{N}: \frac{1}{N} \sum\left(Y_{i}-\bar{Y}\right)^{2} \leq 1\right}$$
and, according to the above result, the minimax strategy within $\Delta_{1}$ consists of choosing every unit with a probability $1 / N$ and employing the estimator $N Y_{i}$ for $Y$ if the unit $i$ is selected.
A much stronger result has been proved by AGGARWAL (1959) and BICKEL and LEHMANN (1981). They consider $\Omega_{11}$ and the class $\Delta_{n}^{+}$of all strategies $\left(p_{n}, t\right), p_{n}$ a design of fixed effective size $n$ and $t$ arbitrary, and show that the expansion estimator $N \bar{y}$ based on SRSWOR of size $n$ is minimax.

Unfortunately, it seems impossible to find analogously general results for other choices of $X$ and $Z$; however, in chapter 6 we report some results valid at least for large samples.
In the present section we give two results for $n \geq 1$ postulating additional conditions on $n$ in relation to $N$ and $X_{1}$, $X_{2}, \ldots, X_{N}$.
Assume for $i=1,2, \ldots, N$
$$Z_{i}=1$$
and
$$\frac{X_{i}}{X}>\frac{n-1}{n} \frac{1}{N-2} .$$
According to the last condition, the variance of the values $X_{1}, X_{2}, \ldots, X_{N}$ must be small. This condition implies that
$$P_{i}=n \frac{N-2}{N-2 n} \frac{X_{i}}{X}-\frac{n-1}{N-2 n}$$
$(i=1,2, \ldots, N)$ are positive with sum 1 . Denote by pLMS the LAHIRI-MIDZUNO-SEN design based on the probabilities $P_{1}$, $P_{2}, \ldots, P_{N}$, that is, in the first draw unit $i$ is selected with probability $P_{i} ; i=1,2, \ldots, N$ and subsequently $n-1$ distinct units are selected by SRSWOR from the $N-1$ units left after the first draw. STENGER and GABLER (1996) have shown:

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

\sup { Y \in \Omega} M{p}(t)=R_{p}(t) \text { 比如说 }
$$对于某些策略是有限的(p,吨)一类的Δ提前固定，尤其是受预算限制。那么寻找最小化策略可能会很有趣Rp(吨), 关于对(p,吨). 让Δ是所有可用策略的类，并且Rp(吨)对至少某些元素是有限的Δ. 那么$$
r^{}=\inf {(p, t) \in \Delta} R{p}(t)=\inf {(p, t) \in \Delta} \sup { Y \in \Omega } M_{p}(t)<\infty一种nd$r$一世sC一种ll和d米一世n一世米一种X在一种l在和在一世吨Hr和sp和C吨吨这$Ω$一种nd$Δ$;一种s吨r一种吨和G是$(p,吨)∈Δ$一世sC一种ll和d一种米一世n一世米一种Xs吨r一种吨和G是一世F
R_{p^{}}\left(t^{}\right)=r^{*} 。
F这rG一世在和ns一世和和米和一种s在r和s$X$一种nd$和$在一世吨H
0<X一世;一世=1,2,…,ñ 0<从一世≤从/2;一世=1,2,…,ñ
$$在哪里从=∑1ñ从一世让我们定义参数空间$$
\Omega_{xz}=\left{ Y \in \mathbb{R}^{N}: \sum \frac{X_{i}}{X}\left(\frac{Y_ {i}}{Z_{i}}-\frac{Y}{Z}\right)^{2} \leq 1\right} 。
$$特别重要的是策略类 \Delta_{n}={(p, t): p这FF一世X和d和FF和C吨一世在和s一世和和n, t 齐次线性}。 ## 统计代写|抽样调查作业代写sampling theory of survey代考|Minimax Strategies of Sample Size 我们首先考虑特殊情况Δ1, 由所有对组成(p,吨)这样$$
\begin{aligned}
p(s) &>0 \text { 意味着 }|s|=1 \
t(s, Y ) &=t(i, Y )=Y_{i} / q_{i }, q_{i} \neq 0 。
\end{aligned}
$$写作p一世=p(一世)中的每个策略Δ1可以用一对 ( p , q ) 来标识；p , q \in \mathbb{R}^{N},一种nd一世吨s米小号和一世s∑p一世[是一世q一世−是]2. ñ这在,F这ll这在一世nG小号吨和ñG和R(1986),在和sH这在吨H一种吨 \sup { Y \in \Omega{xz}} \sum p_{i}\left[\frac{Y_{i}}{q_{i}}-Y\right]^{2} 一世s米一世n一世米在米F这r p一世=X一世X=p一世, 说， q一世=从一世从=q一世, 说，$$
(一世=1,2,…,ñ)这样 $\left( p ^{}, q ^{}\right)一世s一种米一世n一世米一种Xs吨r一种吨和G是.Y \in \Omega_{xz}一世米pl一世和sY +\lambda Z \in \Omega_{xz}F这r和在和r是r和一种lλ一种nd吨H和米小号和这F一个\operatorname{策略}( p , q )和在一种l在一种吨和dF这rY + λ Z一世s∑p一世[是一世+λ从一世q一世−是−λ从]2.吨H一世sq在一种dr一种吨一世CF在nC吨一世这n这Fλ一世sb这在nd和d一世F一种nd这nl是一世F\frac{Z_{i}}{q_{i}}-Z=0$

A( p )=\sup { Y \in \Omega{\mathrm{xz}}} \sum p_{i}\left[\frac{Y_{i}}{q_{i}^{}}-Y \右]^{2}在和H一种在和A\left(p^{}\right)=\sup { Y \in \Omega{xz}} \sum p_{i}^{}\left[\frac{Y_{i}}{q_{i}^ {}}-Y\right]^{2}=Z^{2} 。
F这r$p≠p$吨H和r和和X一世s吨s$j$在一世吨H$pj=pj+e,e>0$.一世吨一世s和一种s一世l是s和和n吨H一种吨
p_{j}^{}-2 p_{j}^{} q_{j}^{}+q_{j}^{ 2}>0 。

$$总计是(j) Y ^{(j)}一世s和q在一种l吨这Y_{j}^{(j)}一种nd∑p一世[是一世(j)q一世−是(j)]2=从2pj−2pjqj+qj∗2pj−2pjqj+qj2 =从2[1+e(1−2qj)pj−2pjqj+qj∗2] ≥从2b和C一种在s和Z_{j} \leq Z / 2一世米pl一世和s1-2 q_ {j} ^ {} \ geq 0.这b在一世这在sl是,Y ^{(j)} \in \Omega_{xz}一种nd A( p ) \geq Z^{2}=A\left( p^{} \right)$$

## 统计代写|抽样调查作业代写sampling theory of survey代考|Minimax Strategies of Sample Size

$$\Omega_{11}=\left{ Y \in \mathbb{R}^{N}: \frac{1}{N} \sum\left(Y_{i}-\bar{ Y}\right)^{2} \leq 1\right}$$
，根据上面的结果，内的极小极大策略Δ1包括以概率选择每个单位1/ñ并使用估算器ñ是一世为了是如果单位一世被选中。
AGGARWAL (1959) 和 BICKEL 和 LEHMANN (1981) 证明了一个更强有力的结果。他们认为Ω11和班级Δn+在所有策略中(pn,吨),pn固定有效尺寸的设计n和吨任意的，并表明扩展估计量ñ是¯基于大小的 SRSWORn是极小极大。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。