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• 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|BILINEAR FORMS

Knowing the distributional properties of quadratic forms of normal variables enables us to discuss properties of bilinear forms. We consider the general bilinear form $\mathbf{x}1^{\prime} \mathbf{A}{12} \mathbf{x}2$ where $\mathbf{x}_1$ and $\mathbf{x}_2$ are of order $n_1$ and $n_2$, distributed as $N\left(\boldsymbol{\mu}_1, \mathbf{C}{11}\right)$ and as $N\left(\boldsymbol{\mu}2, \mathbf{C}{22}\right)$ respectively, with the matrix of covariances between $\mathbf{x}1$ and $\mathbf{x}_2$ being $\mathbf{C}{12}$ of order $n_1 \times n_2$; i.e.,
$$E\left(\mathbf{x}1-\mu_1\right)\left(\mathbf{x}_2-\mu_2\right)^{\prime}=\mathbf{C}{12} .$$
Properties of the bilinear form are readily derived from those of quadratic forms because $\mathbf{x}1^{\prime} \mathbf{A}{12} \mathbf{x}2$ can be expressed as a quadratic form: $$\mathbf{x}_1^{\prime} \mathbf{A}{12} \mathbf{x}2=\frac{1}{2}\left[\begin{array}{ll} \mathbf{x}_1^{\prime} & \mathbf{x}_2^{\prime} \end{array}\right]\left[\begin{array}{cc} 0 & \mathbf{A}{12} \ \mathbf{A}{21} & 0 \end{array}\right]\left[\begin{array}{l} \mathbf{x}_1 \ \mathbf{x}_2 \end{array}\right] \quad \text { with } \quad \mathbf{A}{21}=\left(\mathbf{A}{12}\right)^{\prime} .$$ Hence where $$\mathbf{B}=\mathbf{B}^{\prime}=\left[\begin{array}{cc} \mathbf{0} & \mathbf{A}{12} \ \mathbf{A}{21} & 0 \end{array}\right] \quad \text { with } \quad \mathbf{A}{21}=\left(\mathbf{A}_{12}\right)^{\prime} \text {, }$$ and $\quad \mathbf{y}$ is $N(\mu, \mathbf{V}) \quad$ with $\quad \mu=\left[\begin{array}{l}\mu_1 \ \mu_2\end{array}\right], \quad \mathbf{V}=\left[\begin{array}{ll}\mathbf{C}{11} & \mathbf{C}{12} \ \mathbf{C}{21} & \mathbf{C}{22}\end{array}\right]$ and
$$\mathbf{C}{21}=\left(\mathbf{C}{12}\right)^{\prime} \text {. }$$
Thus properties of $\mathbf{x}1^{\prime} \mathbf{A}{12} \mathbf{x}2$ are equivalent to those of $\frac{1}{2}\left(\mathbf{y}^{\prime} \mathbf{B y}\right)$ which, for some purposes, is better viewed as $\mathbf{y}^{\prime}\left(\frac{1}{2} \mathbf{B}\right) \mathbf{y}$. Similar to Theorem 1, we have the mean value of $\mathbf{x}_1^{\prime} \mathbf{A}{12} \mathbf{x}2$ : whether the distribution of the $x$ ‘s is normal or not, $$E\left(\mathbf{x}_1^{\prime} \mathbf{A}{12} \mathbf{x}2\right)=\operatorname{tr}\left(\mathbf{A}{12} \mathbf{C}{21}\right)+\mu_1^{\prime} \mathbf{A}{12} \mu_2 .$$

## 统计代写|应用线性模型代写Applied Linear Models代考|THE SINGULAR NORMAL DISTRIBUTION

Up to this point we have assumed that $\mathbf{V}$ is non-singular when $\mathbf{x}$ is $N(\boldsymbol{\mu}, \mathbf{V})$. We now consider the situation when $V$ is singular. A simple example of this is the variance-covariance matrix of three random variables $X_1, X_2$ and $X_1-X_2$.

If
then $\quad \mathbf{V} \equiv \operatorname{var}\left[\begin{array}{c}X_1 \ X_2 \ X_1-X_2\end{array}\right]=\left[\begin{array}{ccc}\sigma_1^2 & \sigma_{12} & \sigma_1^2-\sigma_{12} \ \sigma_{12} & \sigma_2^2 & \sigma_{12}-\sigma_2^2 \ \sigma_1^2-\sigma_{12} & \sigma_{12}-\sigma_2^2 & \sigma_1^2+\sigma_2^2-2 \sigma_{12}\end{array}\right]$
with $\mathbf{V}$ being singular. For such variables being normally distributed we emphasize the singularity of $\mathbf{V}$ by writing, in general, $\mathbf{x} \sim S N(\mu, \mathbf{V})$.

Because $\mathbf{V}^{-1}$ does not exist, the density function of the $S N(\mu, \mathbf{V})$ distribution cannot be written down. However, its characteristic function (m.g.f. using $i t$ in place of $t$ ) does exist; it is $e^{i t^{\prime} \mu-\frac{1}{2} t^{\prime} \mathrm{v} t}$. Therefore, by the continuity theorem for characteristic functions [see, for example, Cramer (1951, p. 312) and Anderson (1958, p. 25)], we are guaranteed that the density function exists, even though it cannot be written explicitly.

The general characterization of the $S N(\mu, \mathbf{V})$ distribution given by Anderson (1958, p. 25) is useful. Suppose $\mathbf{y}$ is a vector having the $N(\mathbf{0}, \mathbf{I})$ distribution. Then variables obtained by the transformation $\mathbf{x}=\mu+\mathbf{L y}$ have the $S N\left(\mu, \mathbf{L L}^{\prime}\right)$ distribution, when $\mathbf{L L}^{\prime}$ is not of full rank. Situations arise in linear models that are similar to this, when we develop equations $\mathbf{X}^{\prime} \mathbf{X} \mathbf{b}^o=$ $\mathbf{X}^{\prime} \mathbf{y}$ that have a solution $\mathbf{b}^o=\mathbf{G X}^{\prime} \mathbf{y}$ where $\mathbf{X}^{\prime} \mathbf{X}$ is singular. Then, if $\mathbf{y}$ has a normal distribution, $\mathbf{b}^{\circ}$ will also, but its variance-covariance matrix will be singular. Discussion of the singular normal distribution is therefore pertinent. We consider five theorems, 1s-5s, analogues of those for non-singular $\mathbf{V}$ in Sec. 5. Although they are stated as applying to the $S N(\mu, \mathbf{V})$ distribution, we henceforth take this to be either the singular or the non-singular normal distribution; i.e., $\mathbf{V}$ is to be considered as being either singular or non-singular. In the case that $\mathbf{V}$ is non-singular, Theorems 1s-5s reduce to Theorems 1-5 respectively.

# 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|BILINEAR FORMS

$$\mathbf{C} 21=(\mathbf{C} 12)^{\prime} \text {. }$$

$$E\left(\mathbf{x}_1^{\prime} \mathbf{A} 12 \mathbf{x} 2\right)=\operatorname{tr}(\mathbf{A} 12 \mathbf{C} 21)+\mu_1^{\prime} \mathbf{A} 12 \mu_2 .$$

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