### 统计代写|统计推断作业代写statistical inference代考|Theories of Estimation

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• 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|统计推断作业代写statistical inference代考|ELEMENTS OF POINT ESTIMATION

Essentially there are three stages of sophistication with regard to estimation of a parameter:

1. At the lowest level-a simple point estimate;
2. At a higher level-a point estimate along with some indication of the error of that estimate;
3. At the highest level-one conceives of estimating in terms of a “distribution” or probability of some sort of the potential values that can occur.

This entails the specification of some set of values presumably more restrictive than the entire set of values that the parameter can take on or relative plausibilities of those values or an interval or region.

Consider the I.Q. of University of Minnesota freshmen by taking a random sample of them. We could be satisfied with the sample average as reflective of that population. More insight, however, may be gained by considering the variability

of scores by estimating a variance. Finally one might decide that a highly likely interval for the entire average of freshmen would be more informative.

Sometimes a point estimate is about all you can do. Representing distances on a map, for example. At present there is really no way of reliably illustrating a standard error on a map-so a point estimate will suffice.

An “estimate” is a more or less reasonable guess at the true value of a magnitude or parameter or even a potential observation, and we are not necessarily interested in the consequences of estimation. We may only be concerned in what we should believe a true value to be rather than what action or what the consequences are of this belief. At this point we separate estimation theory from decision theory, though in many instances this is not the case.

## 统计代写|统计推断作业代写statistical inference代考|POINT ESTIMATION

1. An estimate might be considered “good” if it is in fact close to the true value on average or in the long run (pre-trial).
2. An estimate might be considered “good” if the data give good reason to believe the estimate will be close to the true value (post trial).
A system of estimation will be called an estimator
3. Choose estimators which on average or very often yield estimates which are close to the true value.
4. Choose an estimator for which the data give good reason to believe it will be close to the true value that is, a well-supported estimate (one that is suitable after the trials are made.)

With regard to the first type of estimators we do not reject one (theoretically) if it gives a poor result (differs greatly from the true value) in a particular case (though you would be foolish not to). We would only reject an estimation procedure if it gives bad results on average or in the long run. The merit of an estimator is judged, in general, by the distribution of estimates it gives rise to-the properties of its sampling distribution. One property sometimes stressed is unbiasedness. If

$T(D)$ is the estimator of $\theta$ then unbiasedness requires
$$E[T(D)]=\theta .$$
For example an unbiased estimator of a population variance $\sigma^{2}$ is
$$(n-1)^{-1} \sum\left(x_{i}-\bar{x}\right)^{2}=s^{2}$$
since $E\left(s^{2}\right)=\sigma^{2}$.
Suppose $Y_{1}, Y_{2}, \ldots$ are i.i.d. Bernoulli random variables $P\left(Y_{i}=1\right)=\theta$ and we sample until the first “one” comes up so that probability that the first one appears after $X=x$ zeroes is
$$P(X=x \mid \theta)=\theta(1-\theta)^{x} \quad x=0,1, \ldots \quad 0<\theta<1 .$$ Seeking an unbiased estimator we have $$\theta=E(T(Y))=\sum_{x=0}^{\infty} t(x) \theta(1-\theta)^{x}=t(0) \theta+t(1) \theta(1-\theta)+\cdots .$$ Equating the terms yields the unique solution $t(0)=1, t(x)=0$ for $x \geq 1$. This is flawed because this unique estimator always lies outside of the range of $\theta$. So unbiasedness alone can be a very poor guide. Prior to unbiasedness we should have consistency (which is an asymptotic type of unbiasedness, but considerably more). Another desideratum that many prefer is invariance of the estimation procedure. But if $E(X)=\theta$, then for $g(X)$ a smooth function of $X, E(g(X)) \neq g(\theta)$ unless $g(\cdot)$ is linear in $X$. Definitions of classical and Fisher consistency follow: Consistency: An estimator $T_{n}$ computed from a sample of size $n$ is said to be a consistent estimator of $\theta$ if for any arbitrary $\epsilon>0$ and $\delta>0$ there is some value, $N$, such that
$$P\left[\left|T_{n}-\theta\right|<\epsilon\right]>1-\delta \quad \text { for all } n>N,$$

## 统计代写|统计推断作业代写statistical inference代考|Fisher’s Definition of Consistency for i.i.d. Random Variables

“A function of the observed frequencies which takes on the exact parametric value when for those frequencies their expectations are substituted.”

For a discrete random variable with $P\left(X_{j}=x_{j} \mid \theta\right)=p_{j}(\theta)$ let $T_{n}$ be a function of the observed frequencies $n_{j}$ whose expectations are $E\left(n_{j}\right)=n p_{j}(\theta)$. Then the linear function of the frequencies $T_{n}=\frac{1}{n} \sum_{j} c_{j} n_{j}$ will assume the value
$$\tau(\theta)=\Sigma c_{j} p_{j}(\theta)$$
when $n p_{j}(\theta)$ is substituted for $n_{j}$ and thus $n^{-1} T_{n}$ is a consistent estimator of $\tau(\theta)$.
Another way of looking at this is:
Let $F_{n}(x)=\frac{1}{n} \times #$ of observations $\leq x$
$$=\frac{i}{n} \text { for } x_{(i-1)}<x \leq x_{(i)}$$
where $x_{(j)}$ is the $j$ th smallest observation. If $T_{n}=g\left(F_{n}(x)\right)$ and $g(F(x \mid \theta))=\tau(\theta)$ then $T_{n}$ is Fisher consistent for $\tau(\theta)$. Note if
$$T_{n}=\int x d F_{n}(x)=\bar{x}{n}$$ and if $$g(F)=\int x d F(x)=\mu$$ then $\bar{x}{n}$ is Fisher consistent for $\mu$.
On the other hand, if $T_{n}=\bar{x}{n}+\frac{1}{n}$, then this is not Fisher consistent but is consistent in the ordinary sense. Fisher Consistency is only defined for i.i.d. $X{1}, \ldots, X_{n}$.
However, as noted by Barnard (1974), “Fisher consistency can only with difficulty be invoked to justify specific procedures with finite samples” and also “fails because not all reasonable estimates are functions of relative frequencies.” He also presents an estimating procedure that does meet his requirements that the estimate lies within the parameter space and is invariant based on pivotal functions.

## 统计代写|统计推断作业代写statistical inference代考|ELEMENTS OF POINT ESTIMATION

1. 在最低层——简单的点估计；
2. 在更高的层次上——一个点估计以及该估计错误的一些指示；
3. 在最高级别，第一级设想根据可能发生的某种潜在值的“分布”或概率进行估计。

“估计”是对幅度或参数甚至潜在观察的真实值或多或少合理的猜测，我们不一定对估计的后果感兴趣。我们可能只关心我们应该相信一个真正的价值是什么，而不是这种信念的行动或后果是什么。在这一点上，我们将估计理论与决策理论分开，尽管在许多情况下并非如此。

## 统计代写|统计推断作业代写statistical inference代考|POINT ESTIMATION

1. 如果实际上平均或长期（预审）接近真实值，则估计值可能被认为是“好”的。
2. 如果数据有充分的理由相信估计值将接近真实值（试验后），则估计值可能被认为是“好的”。
估计系统将被称为估计器
3. 选择平均或经常产生接近真实值的估计值的估计器。
4. 选择一个数据有充分理由相信它会接近真实值的估计量，即一个有充分支持的估计值（在进行试验后适合的估计值。）

(n−1)−1∑(X一世−X¯)2=s2

## 统计代写|统计推断作业代写statistical inference代考|Fisher’s Definition of Consistency for i.i.d. Random Variables

“观察到的频率的函数，当这些频率的期望被替换时，它具有精确的参数值。”

τ(θ)=ΣCjpj(θ)

=一世n 为了 X(一世−1)<X≤X(一世)

## 广义线性模型代考

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## 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。