统计代写|统计推断作业代写statistics interference代考| Some concepts and simple applications

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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statistics interference代考|Likelihood

The likelihood for the vector of observations $y$ is defined as
$$\operatorname{lik}(\theta ; y)=f_{Y}(y ; \theta),$$
considered in the first place as a function of $\theta$ for given $y$. Mostly we work with its logarithm $l(\theta ; y)$, often abbreviated to $l(\theta)$. Sometimes this is treated as a function of the random vector $Y$ rather than of $y$. The log form is convenient, in particular because $f$ will often be a product of component terms. Occasionally we work directly with the likelihood function itself. For nearly all purposes multiplying the likelihood formally by an arbitrary function of $y$, or equivalently adding an arbitrary such function to the log likelihood, would leave unchanged that part of the analysis hinging on direct calculations with the likelihood.

Any calculation of a posterior density, whatever the prior distribution, uses the data only via the likelihood. Beyond that, there is some intuitive appeal in the idea that differences in $I(\theta)$ measure the relative effectiveness of different parameter values $\theta$ in explaining the data. This is sometimes elevated into a principle called the law of the likelihood.

A key issue concerns the additional arguments needed to extract useful information from the likelihood, especially in relatively complicated problems possibly with many nuisance parameters. Likelihood will play a central role in almost all the arguments that follow.

统计代写|统计推断作业代写statistics interference代考|Sufficiency

The term statistic is often used (rather oddly) to mean any function of the observed random variable $Y$ or its observed counterpart. A statistic $S=S(Y)$ is called sufficient under the model if the conditional distribution of $Y$ given $S=s$ is independent of $\theta$ for all $s, \theta$. Equivalently
$$l(\theta ; y)=\log h(s, \theta)+\log m(y),$$
for suitable functions $h$ and $m$. The equivalence forms what is called the Neyman factorization theorem. The proof in the discrete case follows most explicitly by defining any new variable $W$, a function of $Y$, such that $Y$ is in $(1,1)$ correspondence with $(S, W)$, i.e., such that $(S, W)$ determines $Y$. The individual atoms of probability are unchanged by transformation. That is,
$$f_{Y}(y ; \theta)=f_{S, W}(s, w ; \theta)=f_{S}(s ; \theta) f_{W \mid S}(w ; s),$$
where the last term is independent of $\theta$ by definition. In the continuous case there is the minor modification that a Jacobian, not involving $\theta$, is needed when transforming from $Y$ to $(S, W)$. See Note $2.2$.

We use the minimal form of $S$; i.e., extra components could always be added to any given $S$ and the sufficiency property retained. Such addition is undesirable and is excluded by the requirement of minimality. The minimal form always exists and is essentially unique.

Any Bayesian inference uses the data only via the minimal sufficient statistic. This is because the calculation of the posterior distribution involves multiplying the likelihood by the prior and normalizing. Any factor of the likelihood that is a function of $y$ alone will disappear after normalization.

In a broader context the importance of sufficiency can be considered to arise as follows. Suppose that instead of observing $Y=y$ we were equivalently to be given the data in two stages:

• first we observe $S=s$, an observation from the density $f_{S}(s ; \theta)$;
• then we are given the remaining data, in effect an observation from the density $f_{Y \mid S}(y ; s)$.
Now, so long as the model holds, the second stage is an observation on a fixed and known distribution which could as well have been obtained from a random number generator. Therefore $S=s$ contains all the information about $\theta$ given the model, whereas the conditional distribution of $Y$ given $S=s$ allows assessment of the model.

统计代写|统计推断作业代写statistics interference代考|Simple examples

Example 2.1. Exponential distribution (ctd). The likelihood for Example $1.6$ is
$$\rho^{n} \exp \left(-\rho \Sigma y_{k}\right),$$
so that the log likelihood is
$$n \log \rho-\rho \Sigma y_{k},$$
and, assuming $n$ to be fixed, involves the data only via $\Sigma y_{k}$ or equivalently via $\bar{y}=\Sigma y_{k} / n$. By the factorization theorem the sum (or mean) is therefore sufficient. Note that had the sample size also been random the sufficient statistic would have been $\left(n, \Sigma y_{k}\right)$; see Example $2.4$ for further discussion.

In this example the density of $S=\Sigma Y_{k}$ is $\rho(\rho s)^{n-1} e^{-\rho s} /(n-1)$ !, a gamma distribution. It follows that $\rho S$ has a fixed distribution. It follows also that the joint conditional density of the $Y_{k}$ given $S=s$ is uniform over the simplex $0 \leq y_{k} \leq s ; \Sigma y_{k}=s$. This can be used to test the adequacy of the model.
Example 2.2. Linear model (ctd). A minimal sufficient statistic for the linear model, Example 1.4, consists of the least squares estimates and the residual sum of squares. This strong justification of the use of least squares estimates depends on writing the log likelihood in the form
$$-n \log \sigma-(y-z \beta)^{T}(y-z \beta) /\left(2 \sigma^{2}\right)$$
and then noting that
$$(y-z \beta)^{T}(y-z \beta)=(y-z \hat{\beta})^{T}(y-z \hat{\beta})+(\hat{\beta}-\beta)^{T}\left(z^{T} z\right)(\hat{\beta}-\beta),$$
in virtue of the equations defining the least squares estimates. This last identity has a direct geometrical interpretation. The squared norm of the vector defined by the difference between $Y$ and its expected value $z \beta$ is decomposed into a component defined by the difference between $Y$ and the estimated mean $z \hat{\beta}$ and an orthogonal component defined via $\hat{\beta}-\beta$. See Figure 1.1.

It follows that the log likelihood involves the data only via the least squares estimates and the residual sum of squares. Moreover, if the variance $\sigma^{2}$ were

known, the residual sum of squares would be a constant term in the log likelihood and hence the sufficient statistic would be reduced to $\hat{\beta}$ alone.

This argument fails for a regression model nonlinear in the parameters, such as the exponential regression (1.5). In the absence of error the $n \times 1$ vector of observations then lies on a curved surface and while the least squares estimates are still given by orthogonal projection they satisfy nonlinear equations and the decomposition of the log likelihood which is the basis of the argument for sufficiency holds only as an approximation obtained by treating the curved surface as locally flat.

统计代写|统计推断作业代写statistics interference代考|Sufficiency

F是(是;θ)=F小号,在(s,在;θ)=F小号(s;θ)F在∣小号(在;s),

• 首先我们观察小号=s, 从密度观察F小号(s;θ);
• 然后我们得到剩余的数据，实际上是对密度的观察F是∣小号(是;s).
现在，只要模型成立，第二阶段就是对固定且已知分布的观察，该分布也可以从随机数生成器中获得。所以小号=s包含有关的所有信息θ给定模型，而条件分布是给定小号=s允许评估模型。

统计代写|统计推断作业代写statistics interference代考|Simple examples

ρn经验⁡(−ρΣ是到),

n日志⁡ρ−ρΣ是到,

−n日志⁡σ−(是−和b)吨(是−和b)/(2σ2)

(是−和b)吨(是−和b)=(是−和b^)吨(是−和b^)+(b^−b)吨(和吨和)(b^−b),

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。