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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统计推断是利用数据分析来推断概率基础分布的属性的过程。 推断性统计分析推断人口的属性,例如通过测试假设和得出估计值。

statistics-lab™ 为您的留学生涯保驾护航 在代写统计推断statistics interference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。我们的专家在代写统计推断statistics interference方面经验极为丰富,各种代写统计推断statistics interference相关的作业也就用不着说。

我们提供的属性统计推断statistics interference及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等楖率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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| Some concepts and simple applications

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.

Simple Random Sampling: Definition and Examples
统计代写|统计推断作业代写statistics interference代考| Some concepts and simple applications


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

首先被认为是θ给定的是. 大多数情况下,我们使用它的对数一世(θ;是), 通常缩写为一世(θ). 有时这被视为随机向量的函数是而不是是. 日志形式很方便,特别是因为F通常是组件项的乘积。有时我们会直接使用似然函数本身。对于几乎所有目的,将可能性正式乘以任意函数是,或者等价地向对数似然添加任意这样的函数,将保持分析中依赖于可能性的直接计算的部分不变。



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

统计量这个术语经常(相当奇怪地)用来表示观察到的随机变量的任何函数是或其观察到的对应物。一个统计小号=小号(是)如果条件分布是给定小号=s独立于θ对全部s,θ. 等效地
适合功能H和米. 等价形成了所谓的内曼分解定理。离散情况下的证明最明确地遵循定义任何新变量在,一个函数是, 这样是在(1,1)与(小号,在),即,这样(小号,在)决定是. 概率的单个原子不会因变换而改变。那是,
最后一项独立于θ根据定义。在连续情况下,雅可比行列式存在较小的修改,不涉及θ, 转换时需要是到(小号,在). 见说明2.2.

我们使用最小形式小号; 即,总是可以将额外的组件添加到任何给定的小号并保留充足的财产。这种添加是不希望的,并且被最小化的要求排除在外。最小的形式总是存在并且本质上是独一无二的。



  • 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).

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

例 2.1。指数分布 (ctd)。示例的可能性1.6是
并且,假设n待修复,仅通过以下方式涉及数据Σ是到或等效地通过是¯=Σ是到/n. 因此,通过因式分解定理,总和(或均值)就足够了。请注意,如果样本量也是随机的,那么足够的统计量将是(n,Σ是到); 见例子2.4进一步讨论。

在这个例子中,密度小号=Σ是到是ρ(ρs)n−1和−ρs/(n−1)!,伽马分布。它遵循ρ小号有固定的分布。还可以得出,联合条件密度是到给定小号=s在单纯形上是一致的0≤是到≤s;Σ是到=s. 这可以用来测试模型的充分性。
例 2.2。线性模型 (ctd)。例 1.4 线性模型的最小足够统计量由最小二乘估计和残差平方和组成。使用最小二乘估计的这种强有力的理由取决于将对数似然写成形式
凭借定义最小二乘估计的方程。最后一个恒等式具有直接的几何解释。由两者之差定义的向量的平方范数是及其期望值和b被分解为由两者之间的差异定义的组件是和估计的平均值和b^和通过定义的正交分量b^−b. 请参见图 1.1。



对于参数中的非线性回归模型,例如指数回归 (1.5),此论点失败。在没有错误的情况下n×1然后,观测向量位于曲面上,虽然最小二乘估计仍然由正交投影给出,但它们满足非线性方程,并且作为充分性论证基础的对数似然分解仅作为通过处理曲面局部平坦。

统计代写|统计推断作业代写statistics interference代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。统计代写|python代写代考


在概率论概念中,随机过程随机变量的集合。 若一随机系统的样本点是随机函数,则称此函数为样本函数,这一随机系统全部样本函数的集合是一个随机过程。 实际应用中,样本函数的一般定义在时间域或者空间域。 随机过程的实例如股票和汇率的波动、语音信号、视频信号、体温的变化,随机运动如布朗运动、随机徘徊等等。


贝叶斯统计概念及数据分析表示使用概率陈述回答有关未知参数的研究问题以及统计范式。后验分布包括关于参数的先验分布,和基于观测数据提供关于参数的信息似然模型。根据选择的先验分布和似然模型,后验分布可以解析或近似,例如,马尔科夫链蒙特卡罗 (MCMC) 方法之一。贝叶斯统计概念及数据分析使用后验分布来形成模型参数的各种摘要,包括点估计,如后验平均值、中位数、百分位数和称为可信区间的区间估计。此外,所有关于模型参数的统计检验都可以表示为基于估计后验分布的概率报表。





随着AI的大潮到来,Machine Learning逐渐成为一个新的学习热点。同时与传统CS相比,Machine Learning在其他领域也有着广泛的应用,因此这门学科成为不仅折磨CS专业同学的“小恶魔”,也是折磨生物、化学、统计等其他学科留学生的“大魔王”。学习Machine learning的一大绊脚石在于使用语言众多,跨学科范围广,所以学习起来尤其困难。但是不管你在学习Machine Learning时遇到任何难题,StudyGate专业导师团队都能为你轻松解决。


基础数据: $N$ 个样本, $P$ 个变量数的单样本,组成的横列的数据表
变量定性: 分类和顺序;变量定量:数值
数学公式的角度分为: 因变量与自变量


随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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。



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