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# 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|STAT505 The Multivariate Normal Likelihood

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## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Multivariate Normal Likelihood

Let us assume that the $p \times 1$ vectors $\mathbf{X}{1}, \mathbf{X}{2}, \ldots, \mathbf{X}{n}$ represent a random sample from a multivariate normal population with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. Since $\mathbf{X}{1}, \mathbf{X}{2}, \ldots, \mathbf{X}{n}$ are mutually independent and each has distribution $N_{p}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, the joint density function of all the observations is the product of the marginal normal densities:

\begin{aligned} \left{\begin{array}{c} \text { Joint density } \ \text { of } \mathbf{X}{1}, \mathbf{X}{2}, \ldots, \mathbf{X}{n} \end{array}\right} &=\prod{j=1}^{n}\left{\frac{1}{(2 \pi)^{p / 2}|\mathbf{\Sigma}|^{1 / 2}} e^{-\left(\mathbf{x}{j}-\mu\right)^{\prime} \mathbf{\Sigma}^{-1}\left(\mathbf{x}{j}-\mu\right) / 2}\right} \ &=\frac{1}{(2 \pi)^{n p / 2}} \frac{1}{|\mathbf{\Sigma}|^{n / 2}} e^{-\sum_{j=1}^{n}\left(\mathbf{x}{j}-\mu\right)^{\prime} \mathbf{\Sigma}^{-1}\left(\mathbf{x}{j}-\mu\right) / 2} \end{aligned}
When the numerical values of the observations become available, they may be substituted for the $\mathbf{x}{j}$ in Equation (4-11). The resulting expression, now considered as a function of $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ for the fixed set of observations $\mathbf{x}{1}, \mathbf{x}{2}, \ldots, \mathbf{x}{n}$, is called the likelihood.

Many good statistical procedures employ values for the population parameters that “best” explain the observed data. One meaning of best is to select the parameter values that maximize the joint density evaluated at the observations. This technique is called maximum likelihood estimation, and the maximizing parameter values are called maximum likelihood estimates.

At this point, we shall consider maximum likelihood estimation of the parameters $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ for a multivariate normal population. To do so, we take the observations $\mathbf{x}{1}, \mathbf{x}{2}, \ldots, \mathbf{x}_{n}$ as fixed and consider the joint density of Equation (4-11) evaluated at these values. The result is the likelihood function. In order to simplify matters, we rewrite the likelihood function in another form. We shall need some additional properties for the trace of a square matrix. (The trace of a matrix is the sum of its diagonal elements, and the properties of the trace are discussed in Definition 2A.28 and Result 2A.12.)

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

The repeat construct is less frequently used, but adds flexibility as termination will always depend on a call to break(), which can be located anywhere within the compound statement that forms the body of the loop. To achieve conditional end of iteration, function break() must be called, as otherwise, iteration in a repeat loop will not stop.

approach of adding print() statements, as described on page $101 .$
Although repeat loop constructs are easier to read if they have a single condition resulting in termination of iteration, it is allowed by the R language for the compound statement in the body of a loop to contain more than one call to break(), each within a different if or else statement.

## 统计代写|多元统计分析代写Multivariate Statistical Analysis代考|The Multivariate Normal Likelihood

\left 的分隔符缺失或无法识别\left 的分隔符缺失或无法识别

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

repeat 构造的使用频率较低，但增加了灵活性，因为终止将始终依赖于对 break() 的调用，它可以位于构成循环体的复合语句中的任何位置。要实现迭代的条件结束，必须调用函数break()，否则重复循环中的迭代将不会停止。

## 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。