Posted on Categories:Linear Model, 数据科学代写, 线性模型代写, 统计代写, 统计代考

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## 统计代写|线性模型代写Linear Model代考|Multivariate density functions

In considering $n$ random variables $X_{1}, X_{2}, \ldots, X_{n}$, for which $x_{1}, x_{2}, \ldots$, $x_{n}$ represents a set of realized values we write the cumulative density function as
$$\operatorname{Pr}\left(X_{1} \leq x_{1}, X_{2} \leq x_{2}, \ldots, X_{n} \leq x_{n}\right)=F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$
Then the density function is
$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{\partial^{n}}{\partial x_{1} \partial x_{2} \ldots \partial x_{n}} F\left(x_{1}, x_{2}, \ldots, x_{n}\right) \text {. }$$
Conditions which $f\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ must satisfy are and
$$\begin{gathered} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \geq 0 \text { for }-\infty<x_{i}<\infty \text { for all } i \ \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}=1 \end{gathered}$$

## 统计代写|线性模型代写Linear Model代考|Moments

The $k$ th moment about zero of the $i$ th variable is $E\left(x_{i}^{k}\right)$, the expected value of the $k$ th power of $x_{i}$ :
$$\mu_{x_{i}}^{(k)}=E\left(x_{i}^{k}\right)=\int_{-\infty}^{\infty} x_{i}^{k} g\left(x_{i}\right) d x_{i}$$
and on substituting from (7) for $g\left(x_{i}\right)$ this gives
$$\mu_{x_{i}}^{(k)}=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} x_{i}^{k} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n} .$$
In particular, when $k=1$, the superscript $(k)$ is usually omitted and $\mu_{i}$ is written for $\mu_{i}^{(1)}$.
The covariance between the $i$ th and $j$ th variables for $i \neq j$ is
\begin{aligned} \sigma_{i j} &=E\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) \ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) g\left(x_{i}, x_{j}\right) d x_{i} d x_{j} \ &=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}, \end{aligned}
and similarly the variance of the $i$ th variable is
\begin{aligned} \sigma_{i i} \equiv \sigma_{i}^{2} &=E\left(x_{i}-\mu_{i}\right)^{2} \ &=\int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} g\left(x_{i}\right) d x_{i} \ &=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n} \end{aligned}

## 统计代写|线性模型代写Linear Model代考|Multivariate density functions

$$\operatorname{Pr}\left(X_{1} \leq x_{1}, X_{2} \leq x_{2}, \ldots, X_{n} \leq x_{n}\right)=F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$

$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\frac{\partial^{n}}{\partial x_{1} \partial x_{2} \ldots \partial x_{n}} F\left(x_{1}, x_{2}, \ldots, x_{n}\right)$$

$$f\left(x_{1}, x_{2}, \ldots, x_{n}\right) \geq 0 \text { for }-\infty<x_{i}<\infty \text { for all } i \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}=1$$

## 统计代写|线性模型代写Linear Model代考|Moments

$$\mu_{x_{i}}^{(k)}=E\left(x_{i}^{k}\right)=\int_{-\infty}^{\infty} x_{i}^{k} g\left(x_{i}\right) d x_{i}$$

$$\mu_{x_{i}}^{(k)}=\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} x_{i}^{k} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} d x_{2} \ldots d x_{n}$$

$$\sigma_{i j}=E\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) \quad=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) g\left(x_{i}, x_{j}\right) d x_{i} d x_{j}=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)\left(x_{j}-\mu_{j}\right) f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$

$$\sigma_{i i} \equiv \sigma_{i}^{2}=E\left(x_{i}-\mu_{i}\right)^{2} \quad=\int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} g\left(x_{i}\right) d x_{i}=\int_{-\infty}^{\infty} \ldots \int_{-\infty}^{\infty}\left(x_{i}-\mu_{i}\right)^{2} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) d x_{1} \ldots d x_{n}$$

## 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。

Posted on Categories:Linear Model, 数据科学代写, 线性模型代写, 统计代写, 统计代考

## avatest™帮您通过考试

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## 统计代写|线性模型代写Linear Model代考|Properties of a generalized inverse

Four useful properties of a generalized inverse of $\mathbf{X}^{\prime} \mathbf{X}$ are contained in the following theorem.
Theorem 7. When $\mathbf{G}$ is a generalized inverse of $\mathbf{X}^{\prime} \mathbf{X}$, then
(i) $\mathbf{G}^{\prime}$ is also a generalized inverse of $\mathbf{X}^{\prime} \mathbf{X}$;
(ii) $\mathbf{X G} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}$; i.e., $\mathbf{G} \mathbf{X}^{\prime}$ is a generalized inverse of $\mathbf{X}$;
(iii) $\mathbf{X G X}^{\prime}$ is invariant to $\mathbf{G}$;
(iv) $\mathbf{X G X} \mathbf{X}^{\prime}$ is symmetric, whether $\mathbf{G}$ is or not.
Proof. By definition, $\mathbf{G}$ satisfies
$$\mathbf{X}^{\prime} \mathbf{X} \mathbf{G X} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}^{\prime} \mathbf{X} \text {. }$$
Transposing gives $\mathbf{X}^{\prime} \mathbf{X G} \mathbf{G}^{\prime} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}^{\prime} \mathbf{X}$, and so (i) is established; and applying Lemma 3 yields (ii). To substantiate (iii) suppose that $\mathbf{F}$ is some other generalized inverse, different from $\mathbf{G}$. Then (ii) gives $\mathbf{X G X}^{\prime} \mathbf{X}=\mathbf{X F X}^{\prime} \mathbf{X}$ and the use of Lemma 3 then yields $\mathbf{X G} \mathbf{X}^{\prime}=\mathbf{X F} \mathbf{X}^{\prime}$; i.e., $\mathbf{X G X ^ { \prime }}$ is the same for all generalized inverses of $\mathbf{X}^{\prime} \mathbf{X}$. Finally, to prove (iv) consider $\mathbf{S}$ as a symmetric generalized inverse of $\mathbf{X}^{\prime} \mathbf{X}$. Then $\mathbf{X S X}^{\prime}$ is symmetric. But $\mathbf{X S} \mathbf{X}^{\prime}=\mathbf{X G} \mathbf{X}^{\prime}$ and therefore $\mathbf{X G X}^{\prime}$ is symmetric. Hence the theorem is proved.

Corollary. Applying part (i) of the theorem to its other parts shows that $\mathbf{X G}^{\prime} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}, \quad \mathbf{X}^{\prime} \mathbf{X} \mathbf{G} \mathbf{X}^{\prime}=\mathbf{X}^{\prime} \quad$ and $\mathbf{X}^{\prime} \mathbf{X} \mathbf{G}^{\prime} \mathbf{X}^{\prime}=\mathbf{X}^{\prime} ;$ $\mathbf{X G}^{\prime} \mathbf{X}^{\prime}=\mathbf{X G X ^ { \prime }} ;$ and $\mathbf{X G} \mathbf{G}^{\prime} \mathbf{X}^{\prime}$ is symmetric.

## 统计代写|线性模型代写Linear Model代考|Two methods of derivation

In addition to the methods given in Sec. 1, two methods discussed by John (1964) are sometimes pertinent to linear models. They depend on the regular inverse of a non-singular matrix:
$$\mathbf{S}^{-1}=\left[\begin{array}{cc} \mathbf{X}^{\prime} \mathbf{X} & \mathbf{H}^{\prime} \ \mathbf{H} & \mathbf{0} \end{array}\right]^{-1}=\left[\begin{array}{cc} \mathbf{B}{11} & \mathbf{B}{12} \ \mathbf{B}{21} & \mathbf{B}{22}=\mathbf{0} \end{array}\right] .$$
$\mathbf{H}$ used here, in keeping with John’s notation, is not the matrix GA used earlier. Where $\mathbf{X}^{\prime} \mathbf{X}$ has order $p$ and rank $p-m(m>0)$, the matrix $\mathbf{H}$ is any matrix of order $m \times p$ that is of full row rank with its rows also LIN of those of $\mathbf{X}^{\prime} \mathbf{X}$. [The existence of such a matrix is assured by considering $m$ vectors of order $p$ that are LIN of any set of $p-m$ LIN rows of $\mathbf{X}^{\prime} \mathbf{X}$. Furthermore, if these rows constitute $\mathbf{H}$ in such a way that the $m$ LIN rows of $\mathbf{H}$ correspond in $\mathbf{S}$ to the $m$ rows of $\mathbf{X}^{\prime} \mathbf{X}$ that are linear combinations of that set of $p-m$ rows, then $\mathbf{S}^{-1}$ of (30) exists.] With (30) existing the two matrices
$\mathbf{B}_{11}$ and $\left(\mathbf{X}^{\prime} \mathbf{X}+\mathbf{H}^{\prime} \mathbf{H}\right)^{-1}$ are generalized inverses of $\mathbf{X}^{\prime} \mathbf{X}$.
Three useful lemmas help in proving these results.

## 统计代写|线性模型代写Linear Model代考|Properties of a generalized inverse

(i) $\mathbf{G}^{\prime}$ ‘也是一个广义逆 $\mathbf{X}^{\prime} \mathbf{X} \mathbf{X}$
(二) $\mathbf{X G X} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}$; $\mathrm{IE}{\text {。 }}$ G $\mathbf{X}^{\prime}$ 是的广义逆 $\mathbf{X}$; (E) $\mathbf{X G X}$ ‘不㚆 $\mathbf{G}{\mathbf{\prime}}$
(四) $\mathbf{X G X X}$ ‘是对称的，是否 $\mathbf{G}$ 是与否。

$$\mathbf{X}^{\prime} \mathbf{X G X} \mathbf{X}^{\prime} \mathbf{X}=\mathbf{X}^{\prime} \mathbf{X} .$$

## 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。

Posted on Categories:Linear Model, 数据科学代写, 线性模型代写, 统计代写, 统计代考

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 统计代写|线性模型代写Linear Model代考|Definition and existence

A generalized inverse of a matrix $\mathbf{A}$ is defined, in this book, as any matrix $\mathbf{G}$ that satisfies the equation
$$\mathbf{A} \mathbf{G A}=\mathbf{A} .$$
The name “generalized inverse” for matrices $\mathbf{G}$ defined by (1) is unfortunately not universally accepted, although it is used quite widely. Names such as “conditional inverse”, “pseudo inverse” and ” $g$-inverse” are also to be found in the literature, sometimes for matrices defined as is $\mathbf{G}$ of (1) and sometimes for matrices defined as variants of $\mathbf{G}$. However, throughout this book the name “generalized inverse” of $\mathbf{A}$ is used exclusively for any matrix $\mathbf{G}$ satisfying (1).

Notice that (1) does not define $\mathbf{G}$ as “the” generalized inverse of $\mathbf{A}$ but as “a” generalized inverse. This is because $\mathbf{G}$, for a given matrix $\mathbf{A}$, is not unique. As shown below, there is an infinite number of matrices $\mathbf{G}$ that satisfy (1) and so we refer to the whole class of them as generalized inverses of $\mathbf{A}$.

One way of illustrating the existence of $\mathbf{G}$ and its non-uniqueness starts with the equivalent diagonal form of $\mathbf{A}$. If $\mathbf{A}$ has order $p \times q$ the reduction to this diagonal form can be written as
$$\mathbf{P}{p \times p} \mathbf{A}{p \times Q} \mathbf{Q}{q \times Q}=\Delta{p \times a} \equiv\left[\begin{array}{ll} \mathbf{D}{r \times r} & \mathbf{0}{r \times(q-r)} \ \mathbf{0}{(p-r) \times r} & \mathbf{0}{(p-r) \times(a-r)} \end{array}\right]$$
or, more simply, as
$$\mathbf{P A Q}=\Delta=\left[\begin{array}{cc} \mathbf{D}_{r} & 0 \ 0 & 0 \end{array}\right]$$

## 统计代写|线性模型代写Linear Model代考|Consistent equations

A convenient starting point from which to develop the solution of linear equations using a generalized inverse is the definition of consistent equations.
Definition. The linear equations $\mathbf{A x}=\mathbf{y}$ are defined as being consistent if any linear relationships existing among the rows of $\mathbf{A}$ also exist among the corresponding elements of $\mathbf{y}$.

As a simple example, the equations
$$\left[\begin{array}{ll} 1 & 2 \ 3 & 6 \end{array}\right]\left[\begin{array}{l} x_{1} \ x_{2} \end{array}\right]=\left[\begin{array}{r} 7 \ 21 \end{array}\right]$$
are consistent: in the matrix on the left the second row is thrice the first, and this is also true of the elements on the right. But the equations
$$\left[\begin{array}{ll} 1 & 2 \ 3 & 6 \end{array}\right]\left[\begin{array}{l} x_{1} \ x_{2} \end{array}\right]=\left[\begin{array}{l} 7 \ 24 \end{array}\right]$$
are not consistent. Further evidence of this is seen by writing them in full:
$$x_{1}+2 x_{2}=7 \quad \text { and } \quad 3 x_{1}+6 x_{2}=24 .$$
As a consequence of the first, $3 x_{1}+6 x_{2}=21$, which cannot be true if the second is to hold. The equations are therefore said to be inconsistent.

## 统计代写|线性模型代写Linear Model代考|Definition and existence

$$\mathbf{A G A}=\mathbf{A} .$$

inverse”也可以在文献中找到，有时用于定义如下的矩阵 $\mathbf{G}$ (1) 的，有时对于定义为㚆体的矩阵 $\mathbf{G}$. 然而，在整本书中， “义逆” 这个名称 $\mathbf{A}$ 专门用于任何矩阵 $\mathbf{G}$ 满足 (1)。

$$\mathbf{P A Q}=\Delta=\left[\begin{array}{llll} \mathbf{D}{r} & 0 & 0 & 0 \end{array}\right]$$

## 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。