统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Continuous Time-Parametric Inference

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我们提供的多元统计分析Multivariate Statistical Analysis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 高等概率论
  • 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|多元统计分析代写Multivariate Statistical Analysis代考|Continuous Time-Parametric Inference

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Proceed with Caution

The first jibe thrown at the Bayesian by the Frequentist is that his analysis is not “objective.” One retort is that of course it isn’t; the clue is in the phrase subjective probabilities. The argument is that there is no such thing as objectivity or that the data can speak for themselves. Inference from data is always channelled through an interpreter, different interpreters start with different background information, and this is bound to influence the inference. The Frequentist will then say that scientific statements should not depend on the opinions of the speaker, particularly the opinions of one of those dodgy Bayesians. He will add, more seriously, that the subjective approach is a vehicle for individual decision making, but it is not appropriate for objective scientific reporting.

A more practical criticism of the Bayesian approach is the difficulty of creating a prior distribution. When the data are extensive, we know that the posterior is mainly determined by the likelihood, the prior having little impact. However, particularly in multiparameter cases, an apparently harmless prior can hide unsuspected and undesirable features. Further, even if you want your input to be negligible compared with that of the data, there is no such thing as an uninformative prior, though time and again you will see this claimed in published work. The classic example is taking a uniform prior for a probability, say $\pi$, to express no preference for any one value over any other in the range $(0,1)$. Unfortunately, the unintended consequence is that this choice expresses preference for smaller values of $\pi^{2}$ over larger ones on $(0,1)$.

Let us now turn our fire on the Frequentist approach. A hypothesis test produces a p-value: if this is very small, doubt is thrown on the hypothesis. By doubt we must mean that, in the light of the data $D$, we view the hypothesis $H$ as being dubious, unlikely, and improbable. But the $\mathrm{p}$-value arises from $p(D \mid H)$, not $p(H \mid D)$, and for ” $H$ improbable” we need the second one. So, the p-value does not do the job that we might like it to do (but see DeGroot, 1973).

Confidence intervals are open to similar criticism. They do not give probabilities: the carefully calculated interval $(0.19,0.31)$, for example, either spans $\theta$ or does not, which could have been said without getting out of bed. The argument that this interval has been randomly selected from a population of such intervals, $95 \%$ of which do $\operatorname{span} \theta$, sounds like a cunning attempt to persuade the listener that this one spans $\theta$ with probability $0.95$. But the latter statement is invalid to the Frequentist because it confers a probability on the parameter. If you want probabilities out (posterior), you have to put probabilities in (prior).

So, what do I think, you ask? Or maybe you don’t, but I will tell you anyway. Well, I cannot really say, even though this fence has rather sharp spikes. On this, I am not able to be dogmatic-I can see both sides of the argument. The Bayesian approach has some attractive features: it is logically consistent; nuisance parameters are not a nuisance, you just integrate them out; you do not have to rely on sometimes-dubious asymptotic approximations, as you do with the Frequentist approach; the computational problems, which inhibited

the application of the Bayesian approach in years gone by, are now mainly solved. On the other hand, the assignment of appropriate priors is tricky, and the Frequentist stand against subjectivism in scientific inference is eminently reasonable.

In putting this book together, as an outgrowth of Classical Competing Risks, much new material needed to be introduced. Some of the former things had to go, and one of them was the use of McMC to produce Bayesian posteriors in some of the applications. Nevertheless, all the methodology is still based on likelihood functions, which lies from the Bayesian approach but a short step or a long stretch, depending on your point of view. Also, currently the major $R$ package, survival, is mainly Frequentist, and I did feel the need to base things around it and other freely available $\mathrm{R}$ programs.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Random Samples

We have a sample $\left(t_{1}, \ldots, t_{n}\right)$ of observed lifetimes. Strictly speaking, no two observed values should be equal when they arise from a continuous distribution. In practice, though, rounding will often produce such ties.

The likelihood contributions are $f\left(t_{i} ; \theta\right)$ for an observed failure time $t_{i}$ and $F\left(t_{i} ; \theta\right)$ for one right-censored at $t_{i}$. The latter give information from events that have not yet occurred. It is sometimes not appreciated that such non-events, like unobserved failures, can provide useful information. Sheerluck Holmes was well aware of this: he gained a vital clue from the “curious incident” that the dog did not bark in Silver Blaze (Doyle, 1950). The overall likelihood function is
L=\prod_{\text {abs }} f\left(t_{i} ; \theta\right) \times \prod_{\text {cens }} \bar{F}\left(t_{i} ; \theta\right),
where $\prod_{\text {aus }}$ and $\prod_{\text {cens }}$ are the products over the observed and right-censored times, respectively. The appearance of $\bar{F}\left(t_{i} ; \theta\right)$ in the expression for $L$ assumes that the censoring tells us nothing further about the failure time than that it is beyond $t_{i}$. It is not always the case that censoring is non-informative; for example, in some circumstances censoring is associated with imminent failure.

Let $c_{i}=I\left(t_{i}\right.$ observed $)$, in terms of the indicator function. So, $c_{i}$ is the censoring indicator, $c_{i}=1$ if $t_{i}$ is observed and $c_{i}=0$ if $t_{i}$ is right-censored. The likelihood function can then be written as
L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right)^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right)\right}
Different symbols are used for the censoring indicator by different authors. Some use $\delta_{i}$, but we will mostly reserve Greek letters for parameters here; further, I prefer to spell censoring with a $c$. Perhaps we should use $C_{i}$ instead of $c_{i}$, adhering to the convention that capitals are used for random variables. However, that looks a bit odd, against most usage. Finally, the term censoring indicator should, strictly speaking, be replaced by non-censoring, or observation, indicator; but let’s not be too fussy.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Type-I Censoring

Consider a random sample from an exponential distribution with mean $\xi$. The observations are right-censored at fixed time $a>0$, that is, we only observe $t_{a}=\min (a, t)$ : this is known as Type-I censoring. Thus,
\mathrm{E}\left(t_{a}\right)=\int_{0}^{a} t\left(\xi^{-1} \mathrm{e}^{-t / \xi}\right) d t+a \mathrm{P}(t>a)=\xi\left(1-\mathrm{e}^{-a / \xi}\right) .
Suppose that the data comprise $t_{1}, \ldots, t_{r}$ (observed values, all $\leq a$ ) and $t_{r+1}, \ldots, t_{n}$ (right-censored, all $\left.=a\right)$. Then,
\mathrm{E}(r)=n \mathrm{P}(t \leq a)=n\left(1-\mathrm{e}^{-a / \xi}\right)
The log-likelihood function is given by
I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+}
where $t_{+}=\sum_{i=1}^{r} t_{i}+(n-r) a$ is the Total Time on Test, a term from reliability engineering. The score function is $l^{\prime}(\xi)=-r \xi^{-1}+t_{+} \xi^{-2}$, and the information function is $-l^{\prime \prime}(\xi)=-r \xi^{-2}+2 t_{+} \xi^{-3}$. The mle, obtained as the solution of $l^{\prime}(\xi)=0$, is $\xi=t_{+} / r$, and its variance is approximated by $-l^{\prime \prime}(\xi)^{-1}=\xi^{2} / r$ (Appendix B).

Consider a random sample from an exponential distribution with mean $\xi$. However, this time we observe only the $r$ smallest $t_{i}$ s, where $r$ is a predetermined number: this is known as Type-II censoring. Let $t_{(1)}, \ldots, t_{(n)}$ be the sample order statistics (the $t_{i} s$ rearranged in ascending order). To calculate the likelihood function we use (a) the density $\xi^{-1} \mathrm{e}^{-\mathrm{f} / \xi}$ for $t_{(1)}, \ldots, t_{(r)}$ (since their values are observed) and (b) the survivor function $e^{-t / \xi}$ evaluated at $t=t_{(r)}$ for $t_{(r+1), \ldots,} t_{(n)}$ (since we know only that their values exceed $t_{(r)}$ ). The log-likelihood function is now
l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi}\right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}
which looks much the same as for Type-I censoring though now $r$ is nonrandom and $t_{+}=\sum_{i=1}^{r} t_{(i)}+(n-r) t_{(r)}$. The score function is $l^{\prime}(\xi)=-r \xi^{-1}+$ $t_{+} \xi^{-2}$ and the information function is $-l^{\prime \prime}(\xi)=-r \xi^{-2}+2 t_{+} \xi^{-3}$. The mle is $\hat{\xi}=t_{+} / r$, and its variance is approximated by $-l^{\prime \prime}(\hat{\xi})^{-1}=\hat{\xi}^{2} / r$.

Type-I Censoring 的图像结果
统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Continuous Time-Parametric Inference


统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Proceed with Caution


对贝叶斯方法的一个更实际的批评是创建先验分布的困难。当数据广泛时,我们知道后验主要由可能性决定,先验影响不大。然而,特别是在多参数情况下,明显无害的先验可以隐藏未预料到的和不受欢迎的特征。此外,即使您希望您的输入与数据的输入相比可以忽略不计,也不存在无信息先验之类的东西,尽管您会一次又一次地在已发表的作品中看到这一点。经典的例子是对概率采用统一的先验,比如说圆周率, 表示不偏爱范围内的任何一个值(0,1). 不幸的是,意想不到的结果是这种选择表达了对较小值的偏好圆周率2在较大的(0,1).

现在让我们把注意力转向频率论方法。假设检验产生一个 p 值:如果它非常小,就会对假设产生怀疑。怀疑我们必须是指,根据数据D, 我们查看假设H是可疑的、不太可能的和不可能的。但是p-价值来自p(D∣H), 不是p(H∣D), 并且对于”H不可能”我们需要第二个。因此,p 值并没有做我们可能希望它做的工作(但参见 DeGroot,1973)。

置信区间也会受到类似的批评。他们没有给出概率:仔细计算的区间(0.19,0.31),例如,任一跨度θ或者不,这可以不用起床就可以说出来。这个区间是从这些区间的总体中随机选择的,95%其中做跨度⁡θ, 听起来像是一种狡猾的尝试来说服听众这个跨越θ有概率0.95. 但是后一种说法对频率论者是无效的,因为它赋予了参数一个概率。如果您想要排除概率(后验),则必须将概率放入(先验)。



作为经典竞争风险的产物,在编写本书时,需要引入许多新材料。以前的一些事情不得不去,其中之一是在一些应用程序中使用 McMC 来产生贝叶斯后验。尽管如此,所有的方法仍然是基于似然函数,它来自贝叶斯方法,但是是一小步或一长,取决于你的观点。另外,目前主要R包,生存,主要是Frequentist,我确实觉得有必要围绕它和其他免费提供的东西R程式。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Random Samples


可能性贡献是F(吨一世;θ)对于观察到的故障时间吨一世和F(吨一世;θ)对于一个右删失吨一世. 后者提供尚未发生的事件的信息。有时人们不理解此类非事件(如未观察到的故障)可以提供有用的信息。Sheerluck Holmes 很清楚这一点:他从“奇怪的事件”中获得了一条重要线索,即狗在 Silver Blaze 中没有吠叫(Doyle,1950 年)。整体似然函数是
大号=∏腹肌 F(吨一世;θ)×∏人口普查 F¯(吨一世;θ),
在哪里∏在……之外 和∏人口普查 分别是观察时间和右删失时间的乘积。的出现F¯(吨一世;θ)在表达式中大号假设审查只告诉我们失败时间超出了吨一世. 审查并不总是不提供信息。例如,在某些情况下,审查与即将失败有关。

L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right )^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F \left(t_{i} ; \theta\right)\right}L=\prod_{i=1}^{n}\left{f\left(t_{i} ; \theta\right)^{c_{i}} F\left(t_{i} ; \theta\right )^{1-c_{i}}\right}=\prod_{i=1}^{n}\left{h\left(t_{i} ; \theta\right)^{c_{i}} F \left(t_{i} ; \theta\right)\right}
不同作者使用不同的符号作为审查指标。一些使用d一世, 但我们这里主要保留希腊字母作为参数;此外,我更喜欢用 a 来拼写审查C. 也许我们应该使用C一世代替C一世,遵守大写用于随机变量的约定。但是,与大多数用法相比,这看起来有点奇怪。最后,严格来说,审查指标一词应由非审查指标或观察指标代替;但我们不要太挑剔。

统计代写|多元统计分析代写Multivariate Statistical Analysis代考|Type-I Censoring

考虑一个来自指数分布的随机样本,均值X. 在固定时间对观测值进行右删失一种>0,也就是说,我们只观察吨一种=分钟(一种,吨): 这被称为 I 型审查。因此,
假设数据包括吨1,…,吨r(观察值,所有≤一种) 和吨r+1,…,吨n(右删失,所有=一种). 然后,
I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right ) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+ }I(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{i} / \xi}\right ) \times \prod_{i=r+1}^{n} \mathrm{e}^{-a / \xi}\right}=-r \log \xi-\xi^{-1} t_{+ }
在哪里吨+=∑一世=1r吨一世+(n−r)一种是总测试时间,可靠性工程中的一个术语。评分函数为l′(X)=−rX−1+吨+X−2, 信息函数为−l′′(X)=−rX−2+2吨+X−3. mle,作为解决方案获得l′(X)=0, 是X=吨+/r, 其方差近似为−l′′(X)−1=X2/r(附录 B)。

考虑一个来自指数分布的随机样本,均值X. 然而,这一次我们只观察到r最小的吨一世s,在哪里r是一个预先确定的数字:这被称为 Type-II 审查。让吨(1),…,吨(n)是样本订单统计量(吨一世s按升序排列)。为了计算似然函数,我们使用 (a) 密度X−1和−F/X为了吨(1),…,吨(r)(因为观察到它们的值)和(b)幸存者函数和−吨/X评价为吨=吨(r)为了吨(r+1),…,吨(n)(因为我们只知道它们的值超过吨(r))。对数似然函数现在是
l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi} \right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}l(\xi)=\log \left{\prod_{i=1}^{r}\left(\xi^{-1} \mathrm{e}^{-t_{(i)} / \xi} \right) \times \prod_{i=r+1}^{n}\left(\mathrm{e}^{-t_{(r)} / \xi}\right)\right}=-r \log \xi-\xi^{-1} t_{+}
尽管现在看起来与 Type-I 审查非常相似r是非随机的并且吨+=∑一世=1r吨(一世)+(n−r)吨(r). 评分函数为l′(X)=−rX−1+ 吨+X−2信息函数为−l′′(X)=−rX−2+2吨+X−3. 米是X^=吨+/r, 其方差近似为−l′′(X^)−1=X^2/r.

统计代写|多元统计分析代写Multivariate Statistical Analysis代考 请认准statistics-lab™

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





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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