统计代写|生物统计代写biostatistics代考|STA 310

如果你也在 怎样代写生物统计biostatistics这个学科遇到相关的难题,请随时右上角联系我们的24/7代写客服。


statistics-lab™ 为您的留学生涯保驾护航 在代写生物统计biostatistics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。我们的专家在代写生物统计biostatistics代写方面经验极为丰富,各种生物统计biostatistics相关的作业也就用不着说。

我们提供的生物统计biostatistics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|生物统计代写biostatistics代考|STA 310

统计代写|生物统计代写biostatistics代考|Numerical Aspects and Some Illustrations

Since, on the computational side, we shall base our work the R package sn, described by Azzalini (2019), it is appropriate to describe some key aspects of this package. There exists a comprehensive function for model fitting, called selm, but the actual numerical work in case of an ST model is performed by functions st. mple and mst. mple, in the univariate and the multivariate case, respectively. To numerical efficiency, we shall be using these functions directly, rather than via selm. As their names suggest, st. mple and mst. mple perform MPLE, but they can be used for classical MLE as well, just by omitting the penalty function. The rest of the description refers to st. mple, but mst. mple follows a similar scheme.
In the univariate case, denote by $\theta=(\xi, \omega, \alpha, \nu)^{\top}$ the parameters to be cstimatcd, or possibly $\theta=\left(\beta^{\top}, w, \alpha, v\right)^{\top}$ when a lincar regrcssion mudel is introduced for the location parameter, in which case $\beta$ is a vector of $p$ regression coefficients. Denote by $\log L(\theta)$ the log-likelihood function at point $\theta$. If no starting values are supplied, the first operation of st.mple is to fit a linear model to the available explanatory variables; this reduces to the constant covariate value 1 if $p=1$. For the residuals from this linear fit, sample cumulants of order up to four are computed, hence including the sample variance. An inversion from these

values to $\theta$ may or may not be possible, depending on whether the third and fourth sample cumulants fall in the feasible region for the ST family. If the inversion is successful, initial values of the parameters are so obtained; if not, the final two components of $\theta$ are set at $(\alpha, v)=(0,10)$, retaining the other components from the linear fit. Starting from this point, MLE or MPLE is searched for using a general numerical optimization procedure. The default procedure for performing this step is the $\mathrm{R}$ function nlminb, supplied with the score functions besides the log-likelihood function. We shall refer, comprehensively, to this currently standard procedure as ‘method M0’.

In all our numerical work, method M0 uses st. mple, and the involved function nlminb, with all tuning parameters kept at their default values. The only activated option is the one switching between MPLE and MLE, and even this only for the work of the present section. Later on, we shall always use MPLE, with penalty function Openalty which implements the method proposed in Azzalini and Arellano-Valle (2013).

We start our numerical work with some illustrations, essentially in graphical form, of the log-likelihood generated by some simulated datasets. The aim is to provide a direct perception, although inevitably limited, of the possible behaviour of the log-likelihood and the ensuing problems which it poses for MLE search and other inferential procedures. Given this aim, we focus on cases which are unusual, in some way or another, rather than on ‘plain cases’.

The type of graphical display which we adopt is based on the profile loglikelihood function of $(\alpha, v)$, denoted $\log L_{p}(\alpha, v)$. This is obtained, for any given $(\alpha, v)$, by maximizing $\log L(\theta)$ with respect to the remaining parameters. To simplify readability, we transform $\log L_{p}(\alpha, v)$ to the likelihood ratio test statistic, also called ‘deviance function’:
D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}
where $\log L_{p}(\hat{\alpha}, \hat{v})$ is the overall maximum value of the log-likelihood, equivalent to $\log L(\hat{\theta})$. The concept of deviance applies equally to the penalized log-likelihood.
The plots in Fig. 2 displays, in the form of contour level plots, the behaviour of $D(\alpha, v)$ for two artificially generated samples, with $v$ expressed on the logarithmic scale for more convenient readability. Specifically, the top plots refer to a sample of size $n=50$ drawn from the $\operatorname{ST}(0,1,1,2)$; the left plot, refers to the regular log-likelihood, while the right plot refers to the penalized log-likelihood. The plots include marks for points of special interest, as follows:
$\Delta$ the true parameter point;
o the point having maximal (penalized) log-likelihood on a $51 \times 51$ grid of points spanning the plotted area;

  • the MLE or MPLE point selected by method M0;
  • the preliminary estimate to be introduced in Sect. 3.2, later denoted M1;
    $\times$ the MLE or MPLE point selected by method M2 presented later in the text.

统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

We have seen in Sect. 2 the ST log-likelihood function can be problematic; it is then advisable to select carefully the starting point for the MLE search. While contrasting the risk of landing on a local maximum, a connected aspect of interest is to reduce the overall computing time. Here are some preliminary considerations about the stated target.

Since these initial estimates will be refined by a subsequent step of log-likelihood maximization, there is no point in aiming at a very sophisticate method. In addition, we want to keep the involved computing header as light as possible. Therefore, we want a method which is simple and quick to compute; at the same time, it should be reasonably reliable, hopefully avoiding nonsensical outcomes.

Another consideration is that we cannot work with the methods of moments, or some variant of it, as this would impose a condition $v>4$, bearing in mind the constraints recalled in Sect. 1.2. Since some of the most interesting applications of ST-based models deal with very heavy tails, hence with low degrees of freedom, the condition $v>4$ would be unacceptable in many important applications. The implication is that we have to work with quantiles and derived quantities.

To ease exposition, we begin by presenting the logic in the basic case of independent observations from a common univariate distribution $\mathrm{ST}\left(\xi, \omega^{2}, \lambda, v\right)$. The first step is to select suitable quantile-based measures of location, scale,

asymmetry and tail-weight. The following list presents a set of reasonable choices; these measures can be equally referred to a probability distribution or to a sample, depending on the interpretation of the terms quantile, quartile and alike.

Location The median is the obvious choice here; denote it by $q_{2}$, since it coincides with the second quartile.

Scale A commonly used measure of scale is the semi-interquartile difference, also called quartile deviation, that is
where $q_{j}$ denotes the $j$ th quartile; see for instance Kotz et al. (2006, vol. 10, p. 6743).

Asymmetry A classical non-parametric measure of asymmetry is the so-called Bowley’s measure
G=\frac{\left(q_{3}-q_{2}\right)-\left(q_{2}-q_{1}\right)}{q_{3}-q_{1}}=\frac{q_{3}-2 q_{2}+q_{1}}{2 d_{q}}
see Kotz et al. (2006, vol. 12, p. 7771-3). Since the same quantity, up to an inessential difference, had previously been used by Galton, some authors attribute to him its introduction. We shall refer to $G$ as the Galton-Bowley measure.

Kurtosis A relatively more recent proposal is the Moors measure of kurtosis, presented in Moors (1988),
where $e_{j}$ denotes the $j$ th octile, for $j=1, \ldots, 7$. Clearly, $e_{2 j}=q_{j}$ for $j=$ $1,2,3$

统计代写|生物统计代写biostatistics代考|Inversion of Quantile-Based Measures to ST Parameters

For the inversion of the parameter set $Q=\left(q_{2}, d_{q}, G, M\right)$ to $\theta=(\xi, \omega, \lambda, v)$, the first stage considers only the components $(G, M)$ which are to be mapped to $(\lambda, v)$, exploiting the invariance of $G$ and $M$ with respect to location and scale. Hence, at this stage, we can work assuming that $\xi=0$ and $\omega=1$.

Start by computing, for any given pair $(\lambda, v)$, the set of octiles $e_{1}, \ldots, e_{7}$ of $\mathrm{ST}(0,1, \lambda, v)$, and from here the corresponding $(G, M)$ values. Operationally, we have computed the ST quantiles using routine qst of package sn. Only nonnegative values of $\lambda$ need to be considered, because a reversal of the $\lambda$ sign simply reverses the sign of $G$, while $M$ is unaffected, thanks to the mirroring property of the ST quantiles when $\lambda$ is changed to $-\lambda$.

Initially, our numerical exploration of the inversion process examined the contour level plots of $G$ and $M$ as functions of $\lambda$ and $v$, as this appeared to be the more natural approach. Unfortunately, these plots turned out not to be useful, because of the lack of a sufficiently regular pattern of the contour curves. Therefore these plots are not even displayed here.

A more useful display is the one adopted in Fig. 3, where the coordinate axes are now $G$ and $M$. The shaded area, which is the same in both panels, represents the set of feasible $(G, M)$ points for the ST family. In the first plot, each of the black lines indicates the locus of points with constant values of $\delta$, defined by (4), when $v$ spans the positive half-line; the selected $\delta$ values are printed at the top of the shaded area, when feasible without clutter of the labels. The use of $\delta$ instead of $\lambda$ simply yields a better spread of the contour lines with different parameter values, but it is conceptually irrelevant. The second plot of Fig. 3 displays the same admissible region with superimposed a different type of loci, namely those corresponding to specified values of $v$, when $\delta$ spans the $[0,1]$ interval; the selected $v$ values are printed on the left side of the shaded area.

Details of the numerical calculations are as follows. The Galton-Bowley and the Moors measures have been evaluated over a $13 \times 25$ grid of points identified by the selected values
\delta^{}=&(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95,0.99,1) \ v^{}=&(0.30,0.32,0.35,0.40,0.45,0.50,0.60,0.70,0.80,0.90,1,1.5,2\
&3,4,5,7,10,15,20,30,40,50,100, \infty)

统计代写|生物统计代写biostatistics代考|STA 310


统计代写|生物统计代写biostatistics代考|Numerical Aspects and Some Illustrations

由于在计算方面,我们的工作将基于 Azzalini (2019) 描述的 R 包 sn,因此描述该包的一些关键方面是合适的。存在一个用于模型拟合的综合函数,称为 selm,但在 ST 模型的情况下,实际数值工作由函数 st 执行。mple 和 mst。mple,分别在单变量和多变量情况下。为了数值效率,我们将直接使用这些函数,而不是通过 selm。正如他们的名字所暗示的那样,圣。mple 和 mst。mple 执行 MPLE,但它们也可以用于经典 MLE,只需省略惩罚函数。其余的描述指的是圣。mple,但mst。mple 遵循类似的方案。
在单变量情况下,表示为θ=(X,ω,一个,ν)⊤要 cstimatcd 的参数,或者可能θ=(b⊤,在,一个,在)⊤当为位置参数引入 lincar rercssion mudel 时,在这种情况下b是一个向量p回归系数。表示为日志⁡大号(θ)点的对数似然函数θ. 如果没有提供起始值,则 st.mple 的第一个操作是将线性模型拟合到可用的解释变量;这减少到常数协变量值 1 如果p=1. 对于这种线性拟合的残差,计算最多四阶的样本累积量,因此包括样本方差。从这些倒置

值到θ可能或不可能,取决于第三个和第四个样本累积量是否落在 ST 系列的可行区域内。如果反演成功,则得到参数的初始值;如果没有,最后两个组件θ设置在(一个,在)=(0,10),保留线性拟合中的其他分量。从这一点开始,使用一般数值优化程序搜索 MLE 或 MPLE。执行此步骤的默认程序是R函数 nlminb,除了对数似然函数外,还提供分数函数。我们将把这个目前的标准程序统称为“方法 M0”。

在我们所有的数值工作中,方法 M0 使用 st。mple 和涉及的函数 nlminb,所有调整参数都保持在默认值。唯一激活的选项是在 MPLE 和 MLE 之间切换,甚至这仅适用于本节的工作。稍后,我们将始终使用带有惩罚函数 Openalty 的 MPLE,它实现了 Azzalini 和 Arellano-Valle (2013) 中提出的方法。

我们从一些模拟数据集生成的对数似然的插图开始我们的数值工作,基本上以图形形式。目的是提供对数似然的可能行为以及它为 MLE 搜索和其他推理过程带来的后续问题的直接感知,尽管不可避免地受到限制。鉴于这一目标,我们专注于以某种方式不寻常的案件,而不是“普通案件”。

我们采用的图形显示类型是基于轮廓对数似然函数(一个,在), 表示日志⁡大号p(一个,在). 这是获得的,对于任何给定的(一个,在),通过最大化日志⁡大号(θ)关于其余参数。为了简化可读性,我们将日志⁡大号p(一个,在)似然比检验统计量,也称为“偏差函数”:

D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}
在哪里日志⁡大号p(一个^,在^)是对数似然的整体最大值,相当于日志⁡大号(θ^). 偏差的概念同样适用于惩罚对数似然。
图 2 中的图以等高线水平图的形式显示了D(一个,在)对于两个人工生成的样本,在以对数刻度表示,以便于阅读。具体来说,上面的图是指一个大小的样本n=50取自英石⁡(0,1,1,2); 左图是指常规对数似然,而右图是指惩罚对数似然。这些图包括特殊兴趣点的标记,如下所示:
o 在 a 上具有最大(惩罚)对数似然的点51×51跨越绘图区域的点网格;

  • 方法 M0 选择的 MLE 或 MPLE 点;
  • 将在 Sect 中介绍的初步估计。3.2,后面记为M1;
    ×文中稍后介绍的方法 M2 选择的 MLE 或 MPLE 点。

统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

我们在教派中见过。2 ST 对数似然函数可能有问题;然后建议仔细选择 MLE 搜索的起点。在对比着陆到局部最大值的风险时,感兴趣的一个相关方面是减少整体计算时间。以下是关于既定目标的一些初步考虑。


另一个考虑是,我们不能使用矩量方法或它的一些变体,因为这会强加一个条件在>4,牢记 Sect 中回忆的约束。1.2. 由于基于 ST 的模型的一些最有趣的应用处理非常重的尾部,因此具有低自由度,条件在>4在许多重要应用中是不可接受的。这意味着我们必须使用分位数和派生数量。

为了便于说明,我们首先介绍来自共同单变量分布的独立观察的基本情况下的逻辑小号吨(X,ω2,λ,在). 第一步是选择合适的基于分位数的位置、规模、


位置 中位数是这里的明显选择;表示为q2,因为它与第二个四分位数一致。

尺度 常用的尺度度量是半四分位差,也称为四分位差,即

在哪里qj表示j第四分位数;例如,参见 Kotz 等人。(2006 年,第 10 卷,第 6743 页)。

不对称 不对称的经典非参数度量是所谓的鲍利度量

参见 Kotz 等人。(2006 年,第 12 卷,第 7771-3 页)。由于高尔顿之前使用了相同的数量,但存在无关紧要的差异,因此一些作者将其归因于他的介绍。我们将参考G作为 Galton-Bowley 度量。

峰度 一个相对较新的建议是 Moors 峰度测量,在 Moors (1988) 中提出,

在哪里和j表示j八分位数,对于j=1,…,7. 清楚地,和2j=qj为了j= 1,2,3

统计代写|生物统计代写biostatistics代考|Inversion of Quantile-Based Measures to ST Parameters


从计算开始,对于任何给定的对(λ,在), 八分位数的集合和1,…,和7的小号吨(0,1,λ,在),从这里对应的(G,米)价值观。在操作上,我们使用包 sn 的例程 qst 计算了 ST 分位数。只有非负值λ需要考虑,因为逆转λ符号只是反转符号G, 尽管米不受影响,这要归功于 ST 分位数的镜像属性,当λ改为−λ.


更有用的显示是图 3 中采用的显示,其中坐标轴现在是G和米. 两个面板中相同的阴影区域表示可行的集合(G,米)ST家族的积分。在第一个图中,每条黑线表示具有常数值的点的轨迹d,由 (4) 定义,当在跨越正半线;被选中的d值打印在阴影区域的顶部,如果可行,标签不会混乱。指某东西的用途d代替λ简单地产生具有不同参数值的等高线的更好分布,但它在概念上无关紧要。图 3 的第二个图显示了相同的允许区域,其中叠加了不同类型的基因座,即对应于指定值的基因座在, 什么时候d跨越[0,1]间隔; 被选中的在值打印在阴影区域的左侧。

数值计算的细节如下。Galton-Bowley 和 Moors 措施已经过评估13×25由所选值标识的点网格

统计代写|生物统计代写biostatistics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。







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。