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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写应用线性模型Applied Linear Models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。我们的专家在代写应用线性模型Applied Linear Models代写方面经验极为丰富，各种代写应用线性模型Applied Linear Models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

## 统计代写|应用线性模型代写Applied Linear Models代考|DISTRIBUTIONS AND QUADRATIC FORMS

Analysis of variance techniques involve partitioning a total sum of squares into component sums of squares whose ratios (under appropriate distributional conditions) lead to $F$-statistics suitable for testing certain hypotheses. When discussing linear models generally, especially where unbalanced data (data having unequal subclass numbers) are concerned, it is convenient to think of sums of squares involved in this process as quadratic forms in the observations. In this context very general theorems can be established, of which familiar analyses of variance and associated $F$-tests are then just special cases. An introductory outline ${ }^1$ of the general procedure is easily described.
Suppose $\mathbf{y}{n \times 1}$ is a vector of $n$ observations. Then $\mathbf{y}^{\prime} \mathbf{y}=\sum{i=1}^n y_i^2$ is the total sum of squares of the observations which gets partitioned into component sums of squares in an analysis of variance. Let $\mathbf{P}$ be an orthogonal matrix
$$\mathbf{P P}^{\prime}=\mathbf{P}^{\prime} \mathbf{P}=\mathbf{I},$$
and partition $\mathbf{P}$ row-wise into $k$ sub-matrices $\mathbf{P}i$, of order $n_i \times n$, for $i=$ $1,2, \ldots, k$, with $\sum{i=1}^k n_i=n$; i.e.,
$$\mathbf{P}=\left[\begin{array}{c} \mathbf{P}_1 \ \mathbf{P}_2 \ \cdot \ \cdot \ \cdot \ \mathbf{P}_k \end{array}\right] \quad \text { and } \quad \mathbf{P}^{\prime}=\left[\begin{array}{llll} \mathbf{P}_1^{\prime} & \mathbf{P}_2^{\prime} & \cdots & \mathbf{P}_k^{\prime} \end{array}\right] .$$
${ }^1$ Kindly brought to my notice by D. L. Wēēks̄.

## 统计代写|应用线性模型代写Applied Linear Models代考|SYMMETRIC MATRICES

An expression of the form $\mathbf{x}^{\prime} \mathbf{A y}$ is called a bilinear form. It is a homogeneous second-degree function of the first degree in each of the $x$ ‘s and $y$ ‘s. For example,
\begin{aligned} \mathbf{x}^{\prime} \mathbf{A y} &=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{rr} 4 & 8 \ -2 & 7 \end{array}\right]\left[\begin{array}{l} y_1 \ y_2 \end{array}\right] \ &=4 x_1 y_1+8 x_1 y_2-2 x_2 y_1+7 x_{\mathrm{a}} y_{\mathrm{g}} . \end{aligned}

When $\mathbf{x}$ is used in place of $\mathbf{y}$ the expression becomes $\mathbf{x}^{\prime} \mathbf{A x}$; it is then called a quadratic form and is a quadratic function of the $x$ ‘s:
\begin{aligned} \mathbf{x}^{\prime} \mathbf{A x} &=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{rr} 4 & 8 \ -2 & 7 \end{array}\right]\left[\begin{array}{l} x_1 \ x_2 \end{array}\right] \ &=4 x_1^2+(8-2) x_1 x_2+7 x_2^2 \ &=4 x_1^2+(3+3) x_1 x_2+7 x_2^2 \ &=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{ll} 4 & 3 \ 3 & 7 \end{array}\right]\left[\begin{array}{l} x_1 \ x_2 \end{array}\right] . \end{aligned}
In this way any quadratic form $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}$ can be written as $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}=\mathbf{x}^{\prime} \mathbf{B x}$ where $\mathbf{B}=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}^{\prime}\right)$ is symmetric. Furthermore, whereas any quadratic form can be written as $\mathbf{x}^{\prime} \mathbf{A x}$ for an infinite number of matrices, each can be written in only one way as $\mathbf{x}^{\prime} \mathbf{B x}$ for $\mathbf{B}$ symmetric. For example,
$$4 x_1^2+6 x_1 x_2+7 x_2^2=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{cc} 4 & 3+a \ 3-a & 7 \end{array}\right]\left[\begin{array}{l} x_1 \ x_2 \end{array}\right]$$
for any value of $a$, but only when $a=0$ is the matrix involved symmetric. This means that for any particular quadratic form there is only one, unique matrix such that the quadratic form can be written as $\mathbf{x}^{\prime} \mathbf{A x}$ with $\mathbf{A}$ being symmetric. Because of the uniqueness of this symmetric matrix all further discussion of quadratic forms $\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}$ is confined to the case of $\mathbf{A}$ being symmetric.

# 应用线性模型代考

## 统计代写|应用线性模型代写Applied Linear Models代考|DISTRIBUTIONS AND QUADRATIC FORMS

$$\mathbf{P P}^{\prime}=\mathbf{P}^{\prime} \mathbf{P}=\mathbf{I},$$
$$\mathbf{P}=\left[\begin{array}{lll} \mathbf{P}_1 \mathbf{P}_2 & \cdots \mathbf{P}_k \end{array}\right] \text { and } \mathbf{P}^{\prime}=\left[\begin{array}{llll} \mathbf{P}_1^{\prime} & \mathbf{P}_2^{\prime} & \cdots & \mathbf{P}_k^{\prime} \end{array}\right] \text {. }$$
${ }^1$ 请 DL Wēēks̄ 通知我。

## 统计代写|应用线性模型代写Applied Linear Models代考|SYMMETRIC MATRICES

$$\mathbf{x}^{\prime} \mathbf{A y}=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{lll} 4 & 8-2 & 7 \end{array}\right]\left[y_1 y_2\right] \quad=4 x_1 y_1+8 x_1 y_2-2 x_2 y_1+7 x_{\mathrm{a}} y_{\mathrm{g}} .$$

$$\mathbf{x}^{\prime} \mathbf{A} \mathbf{x}=\left[\begin{array}{lll} x_1 & x_2 \end{array}\right]\left[\begin{array}{lll} 4 & 8-2 & 7 \end{array}\right]\left[x_1 x_2\right] \quad=4 x_1^2+(8-2) x_1 x_2+7 x_2^2=4 x_1^2+(3+3) x_1 x_2+7 x_2^2 \quad=\left[\begin{array}{lll} x_1 & x_2 \end{array}\right]\left[\begin{array}{lll} 4 & 33 & 7 \end{array}\right]\left[\begin{array}{l} x_1 \ x_2 \end{array}\right] .$$

$$4 x_1^2+6 x_1 x_2+7 x_2^2=\left[\begin{array}{ll} x_1 & x_2 \end{array}\right]\left[\begin{array}{lll} 4 & 3+a 3-a & 7 \end{array}\right]\left[\begin{array}{l} x_1 x_2 \end{array}\right]$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师