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。

• 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
assignmentutor™您的专属作业导师

## 统计代写|生物统计代写biostatistics代考|Model 1-JLSM: Under Student’s $t$ Distribution

It is common for observations to come from a population that has a heavy-tailed distribution. Student’s t distribution is one of the basic models for describing heavytailed data, which is common in a variety of applications.

Let $y_{i} \in \mathbb{R}$, for $i=1,2, \ldots, n$, be independently distributed random variables and assume that for each $i, y_{i}$ has the student $\mathrm{t}$ distribution $\left(y_{i} \sim t\left(\mu_{i}, \sigma_{i}^{2}, v\right)\right)$ with the following probability density function (pdf)
$$f\left(y_{i} ; \mu_{i}, \sigma_{i}, v\right)=\frac{c_{v}}{\sigma_{i}}\left(v+\frac{\left(y_{i}-\mu_{i}\right)^{2}}{\sigma_{i}^{2}}\right)^{-\frac{v+1}{2}}$$
where $\mu_{i} \in \mathbb{R}$ and $\sigma_{i}>0$ are the location and scale parameters, respectively. Here $v$ is the degrees of freedom that can be seen as a robustness parameter and down weights the effect of the outliers. As $v$ tends to infinity, this model reduces the JLSM under normal distribution. In this study, the parameter $v$ is regarded as known and we take $v=3$ to achieve the robustness (see Lange et al. 1989; Arslan and Genç 2003). Figure 1 shows the plots of the pdf of the Student $t$ distribution for different degrees of the freedom.
Now, let us consider the JLSM of the student $t$ distribution given by
$$\left{\begin{array}{c} y_{i} \sim t\left(\mu_{i}, \sigma_{i}^{2}, v\right) \ \mu_{i}=\boldsymbol{x}{i}^{T} \boldsymbol{\beta} \ \log \sigma{i}^{2}=z_{i}^{T} \boldsymbol{\gamma} \ i=1,2, \ldots, n \end{array}\right.$$
where $\boldsymbol{x}{i}=\left[x{i 1}, x_{i 2}, \ldots, x_{i p}\right]^{T}$ and $z_{i}=\left[z_{i 1}, z_{i 2}, \ldots, z_{i q}\right]^{T}$ are the observed covariates corresponding to the response $y_{i}$, and $\beta=\left[\beta_{1}, \beta_{2}, \ldots, \beta_{p}\right]^{T} \in \mathbb{R}^{p}$ and $\gamma=\left[\gamma_{1}, \gamma_{2}, \ldots, \gamma_{q}\right]^{T} \in \mathbb{R}^{q}$ are the unknown parameter vectors in the location and scale models, respectively. We will assume that $n>p+q$. Note that, although we use two different sets of explanatory variables to model location and scale parameters, there may be only one set of explanatory variables to model location and scale parameters in some problems.

## 统计代写|生物统计代写biostatistics代考|Under Skew-t Distribution

In addition to heavy tailedness, there can be a presence of high skewness in the data. To accommodate skewness and heavy tailed data together, the construction of flexible parametric skew distributions has received considerable attention in recent years. Numerous authors have developed various classes of these distributions. In this study, we will use the skew-t distribution, which is proposed by Azzalini (2003). To provide a wide and flexible family of modeling data that account for skewness and heavy tail, Azzalini (2003) have proposed skew-t distribution by introducing a generalization of the Student’s t distribution.

Let $y_{i} \in \mathbb{R}$, for $i=1,2, \ldots, n$, be independently distributed random variables and assume that for each $i, y_{i}$ has the skew-t distribution $\left(y_{i} \sim S t\left(\mu_{i}, \sigma_{i}, \lambda i, v\right)\right)$ with the following pdf
$$f_{S_{t}, v}\left(y_{i} ; \mu_{i}, \sigma_{i}, \lambda_{i}, v\right)=\frac{2}{\sigma_{i}} t_{v}\left(y_{i 0}, v\right) T_{v+1}\left(\lambda y_{i 0} \sqrt{\frac{v+1}{v+y_{i 0}^{2}}}\right)$$
where $\mu_{i} \in \mathbb{R}, \sigma_{i}>0$ and $\lambda_{i} \in \mathbb{R}$ are the location, scale and skewness parameters, respectively. Here $y_{i 0}=\left(y_{i}-\mu_{i}\right) / \sigma_{i}, t_{v}(\cdot)$ denotes the pdf of Student t distribution with $v$ degrees of freedom and $T_{v+1}(\cdot)$ denotes the cumulative distribution function (cdf) of Student $t$ distribution with $v+1$ degrees of freedom (Azzalini 2003). Figure 2 shows the plots of the pdf of the skew-t distribution for different values of $\lambda$ and $v$.

Similar to the Student’s t distribution case, the JLSM under skew-t distribution is defined as follows.
$$\left{\begin{array}{c} y_{i} \sim S t\left(\mu_{i}, \sigma_{i}^{2}, \lambda, v\right) \ \mu_{i}=\boldsymbol{x}{i}^{T} \boldsymbol{\beta} \ \log \sigma{i}^{2}=z_{i}^{T} \boldsymbol{\gamma} \ i=1,2, \ldots, n \end{array}\right.$$
Note that, the skewness parameter $\lambda$ has no variability in this model. When $\lambda$ is equal to zero, this model reduces the JLSM of Student $t$ distribution. Moreover when $\lambda=0$ and $\nu \rightarrow \infty$, this model reduces the JLSM of normal distribution. Here we will assume that $n>p+q+1$. Similar to Student’s $\mathrm{t}$ distribution case, the parameter $v$ is taken 3 to achieve the robustness and regarded as known.

We first obtain the ML estimates of the parameters of JLSM of skew-t distribution. Let $\theta=\left(\theta_{1}, \theta_{2}, \ldots, \theta_{s_{2}}\right)=\left(\beta^{T}, \gamma^{T}\right)$ with $s_{2}=p+q+1$ be the combined vector of unknown parameters. Given independent observations $y_{1}, y_{2}, \ldots, y_{n}$ the log likslihood function of $\theta$ corrcsponding to thc JLSM of the skcw t distribution can be written as follows.
$$\ell(\boldsymbol{\theta} \mid y, \boldsymbol{x}, \boldsymbol{z})=n \log \left(c_{v}\right)-\frac{1}{2} \sum_{i=1}^{n} z_{i}^{T} \boldsymbol{\gamma}-\frac{v+1}{2} \sum_{i=1}^{n} \log \left(\nu+\frac{\left(y_{i}-\boldsymbol{x}{i}^{T} \boldsymbol{\beta}\right)^{2}}{\boldsymbol{e}^{z{i}^{T} \gamma}}\right)$$

## 统计代写|生物统计代写biostatistics代考|Model-3 JLSSM: Under Skew-t Distribution

JLSMs of Student’s $t$ and skew-t distributions are limited in addressing only the heteroscedasticity. However, the skewness parameter is at least as important as the other parameters to model the data and it may be different for each observation and depend on some of the covariates. Because of this case, modeling the skewness may also be required. Since our main concern is to provide the best modeling of all parameters and to obtain the best modeling of the data, we also consider the skewness model in addition to location and scale. For this purpose, JLSM under skew-t distribution can be extended to JLSSM under skew-t distribution in order to allow modeling the skewness of the data. In this subsection, we consider the JLSSM under skew-t distribution to take into account the variability of skewness parameter.
Let $y_{i} \in \mathbb{R}$, for $i=1,2, \ldots, n$, be independently distributed with $S t(\mu, \sigma, \lambda, v)$ In some cases, in addition to $\mu$ and $\sigma^{2}$, the skewness parameter $\lambda$ may also be different for each $y_{i}, i=1,2, \ldots, n$, and may also be related to a number of variables. Then, the JLSSM under skew-t distribution is defined as follows.
$$\left{\begin{array}{c} y_{i} \sim S t\left(\mu_{i}, \sigma_{i}^{2}, \lambda_{i}, v\right) \ \mu_{i}=\boldsymbol{x}{i}^{T} \boldsymbol{\beta} \ \log \sigma{i}^{2}=z_{i}^{T} \boldsymbol{\gamma} \ \lambda_{i}=v_{i}^{T} \alpha \ i=1,2, \ldots, n \end{array}\right.$$
where $v_{t}=\left[v_{t 1}, v_{i 2}, \ldots, v_{i r}\right]^{T}$ denote the observed covariates and $\alpha=$ $\left[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}\right]^{T} \in \mathbb{R}^{r}$ is the unknown parameter vector in the skewness model. We will assume that $n>p+q+r$.

It is important to stress that the JLSSM under skew-t distribution includes the previous models given in Eqs. (2) and (11) as special cases. If the skewness parameter does not have variability, then JLSSM under skew-t distribution reduces to the JLSM under skew-t distribution. If the skewness parameter is equal to zero, the model reduces the JLSM under Student t distribution. In addition when $v \rightarrow \infty$, the JLSSM under skew-t distribution reduces the JLSSM under skewnormal distribution. The advantage of the JLSSM under skew-t distribution is that it may give a better fit for heavy-tailed and/or asymmetric data sets.

## 统计代写|生物统计代写biostatistics代考|Model 1-JLSM: Under Student’s 吨分配

$$f\left(y_{i} ; \mu_{i}, \sigma_{i}, v\right)=\frac{c_{v}}{\sigma_{i}}\left(v+\frac{\left(y_{i}-\mu_{i}\right)^{2}}{\sigma_{i}^{2}}\right)^{-\frac{v+1}{2}}$$

$\$ \$$Veft { 给出的分布$$
y_{i} \sim t\left(\mu_{i}, \sigma_{i}^{2}, v\right) \mu_{i}=\boldsymbol{x} i^{T} \boldsymbol{\beta} \log \sigma i^{2}=z_{i}^{T} \gamma i=1,2, \ldots, n
$$正确的。 \ \$$ $\gamma=\left[\gamma_{1}, \gamma_{2}, \ldots, \gamma_{q}\right]^{T} \in \mathbb{R}^{q}$ 分别是位置模型和尺度模型中的末知参数向量。我们将假设 $n>p+q$. 请注意，虽然我们使用两组不同的解 释变量来模拟位置和尺度参数，但在某些问题中可能只有一组解释变量来模拟位置和尺度参数。

## 统计代写|生物统计代写biostatistics代考|Under Skew-t Distribution

$$f_{S_{t, v}}\left(y_{i} ; \mu_{i}, \sigma_{i}, \lambda_{i}, v\right)=\frac{2}{\sigma_{i}} t_{v}\left(y_{i 0}, v\right) T_{v+1}\left(\lambda y_{i 0} \sqrt{\frac{v+1}{v+y_{i 0}^{2}}}\right)$$

$\$ \$$左 {$$
y_{i} \sim S t\left(\mu_{i}, \sigma_{i}^{2}, \lambda, v\right) \mu_{i}=\boldsymbol{x} i^{T} \boldsymbol{\beta} \log \sigma i^{2}=z_{i}^{T} \gamma i=1,2, \ldots, n
$$正确的。 \ \$$

$$\ell(\boldsymbol{\theta} \mid y, \boldsymbol{x}, \boldsymbol{z})=n \log \left(c_{v}\right)-\frac{1}{2} \sum_{i=1}^{n} z_{i}^{T} \gamma-\frac{v+1}{2} \sum_{i=1}^{n} \log \left(\nu+\frac{\left(y_{i}-\boldsymbol{x} i^{T} \boldsymbol{\beta}\right)^{2}}{\boldsymbol{e}^{z i^{T} \gamma}}\right)$$

## 统计代写|生物统计代写biostatistics代考|Model-3 JLSSM: Under Skew-t Distribution

$\$ \$$左 {$$
y_{i} \sim S t\left(\mu_{i}, \sigma_{i}^{2}, \lambda_{i}, v\right) \mu_{i}=\boldsymbol{x} i^{T} \boldsymbol{\beta} \log \sigma i^{2}=z_{i}^{T} \boldsymbol{\gamma} \lambda_{i}=v_{i}^{T} \alpha i=1,2, \ldots, n


\$\$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。