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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|广义线性模型代写generalized linear model代考|INDEPENDENCE

The independence of two quadratic forms is examined in the next theorem.
Theorem 3.2.1 Let $\mathbf{A}$ and $\mathbf{B}$ be $n \times n$ constant matrices. Let the $n \times 1$ random vector $\mathbf{Y} \sim \mathrm{N}_n(\mu, \Sigma)$. The quadratic forms $\mathbf{Y}^{\prime} \mathbf{A} \mathbf{Y}$ and $\mathbf{Y}^{\prime} \mathbf{B} \mathbf{Y}$ are independent if and only if $\mathbf{A} \Sigma \mathbf{B}=\mathbf{0}$ (or $\mathbf{B} \Sigma \mathbf{A}=\mathbf{0}$ ).

Proof: The matrices $\mathbf{A}, \Sigma$, and $\mathbf{B}$ are symmetric. Therefore, $\mathbf{A} \Sigma \mathbf{B}=\mathbf{0}$ is equivalent to $\mathbf{B} \Sigma \boldsymbol{\Sigma}=\mathbf{0}$. Assume $\mathbf{A} \boldsymbol{\Sigma} \mathbf{B}=\mathbf{0}$. Since $\boldsymbol{\Sigma}$ is positive definite, by

Theorem 1.1.5, there exists an $n \times n$ nonsingular matrix $\mathbf{S}$ such that $\mathbf{S} \Sigma \mathbf{S}^{\prime}=\mathbf{I}n$. Then $\mathbf{Z}=\mathbf{S Y} \sim \mathrm{N}_n\left(\mathbf{S} \boldsymbol{\mu}, \mathbf{I}_n\right)$. Let $\mathbf{G}=\left(\mathbf{S}^{-1}\right)^{\prime} \mathbf{A} \mathbf{S}^{-1}$ and $\mathbf{H}=\left(\mathbf{S}^{-1}\right)^{\prime} \mathbf{B S}^{-1}$. Therefore, $\mathbf{Y}^{\prime} \mathbf{A Y}=\mathbf{Z}^{\prime} \mathbf{G Z}, \mathbf{Y}^{\prime} \mathbf{B Y}=\mathbf{Z}^{\prime} \mathbf{H Z}$, and $\mathbf{A} \mathbf{\Sigma B}=\mathbf{S}^{\prime} \mathbf{G S} \mathbf{\Sigma} \mathbf{S}^{\prime} \mathbf{H S}=\mathbf{S}^{\prime} \mathbf{G H S}$. Thus, the statement $\mathbf{A} \mathbf{\Sigma} \mathbf{B}=\mathbf{0}$ implies $\mathbf{Y}^{\prime} \mathbf{A Y}$ and $\mathbf{Y}^{\prime} \mathbf{B Y}$ are independent and the statement $\mathbf{G H}=\mathbf{0}$, implies $\mathbf{Z}^{\prime} \mathbf{G Z}$ and $\mathbf{Z}^{\prime} \mathbf{H Z}$ are independent, are equivalent. since $\mathbf{G}$ is symmetric, there exists an orthogonal matrix $\mathbf{P}$ such that $\mathbf{G}=\mathbf{P} \mathbf{D P}$, where $a=\operatorname{rank}(\mathbf{A})=\operatorname{rank}(\mathbf{G})$ and $\mathbf{D}$ is a diagonal matrix with $a$ nonzero diagonal elements. Without loss of generality, assume that $$\mathbf{D}=\left[\begin{array}{cc} \mathbf{D}_a & \mathbf{0} \ \mathbf{0} & \mathbf{0} \end{array}\right]$$ where $\mathbf{D}_a$ is the $a \times a$ diagonal matrix containing the nonzero elements of $\mathbf{D}$. Let $\mathbf{X}=\mathbf{P Z} \sim \mathrm{N}_n\left(\mathbf{P S} \boldsymbol{\mu}, \mathbf{I}_n\right)$ and partition $\mathbf{X}$ as $\left[\mathbf{X}_1^{\prime}, \mathbf{X}_2^{\prime}\right]^{\prime}$, where $\mathbf{X}_1$ is $a \times 1$. Note that $\mathbf{X}_1$ and $\mathbf{X}_2$ are independent. Then $\mathbf{Z}^{\prime} \mathbf{G Z}=\mathbf{Z}^{\prime} \mathbf{P}^{\prime} \mathbf{D P Z}=\mathbf{X}^{\prime} \mathbf{D} \mathbf{X}=\mathbf{X}_1^{\prime} \mathbf{D}_a \mathbf{X}_1$ and $\mathbf{Z}^{\prime} \mathbf{H Z}=\mathbf{X}^{\prime} \mathbf{P H P}^{\prime} \mathbf{X}=\mathbf{X}^{\prime} \mathbf{C X}$ where the symmetric matrix $\mathbf{C}=\mathbf{P H P}^{\prime}$. If $\mathbf{G H}=\mathbf{0}$ then $\mathbf{P}^{\prime} \mathbf{D P P} \mathbf{C P}^{\prime}=\mathbf{0}$, which implies $\mathbf{D C}=\mathbf{0}$. Partitioning $\mathbf{C}$ to conform with $\mathbf{D}$, $$\mathbf{0}=\mathbf{D C}=\left[\begin{array}{cc} \mathbf{D}_a & \mathbf{0} \ \mathbf{0} & \mathbf{0} \end{array}\right]\left[\begin{array}{ll} \mathbf{C}{11} & \mathbf{C}{12} \ \mathbf{C}{12}^{\prime} & \mathbf{C}{22} \end{array}\right]=\left[\begin{array}{cc} \mathbf{D}_a \mathbf{C}{11} & \mathbf{D}a \mathbf{C}{12} \ \mathbf{0} & \mathbf{0} \end{array}\right],$$
which implies $\mathbf{C}{11}=\mathbf{0}$ and $\mathbf{C}{12}=\mathbf{0}$. Therefore,
$$\mathbf{C}=\left[\begin{array}{cc} \mathbf{0} & \mathbf{0} \ \mathbf{0} & \mathbf{C}{22} \end{array}\right],$$ which implies $\mathbf{X}^{\prime} \mathbf{C X}=\mathbf{X}_2^{\prime} \mathbf{C}{22} \mathbf{X}_2$, which is independent of $\mathbf{X}_1^{\prime} \mathbf{D}_a \mathbf{X}_1$. Therefore, $\mathbf{Z}^{\prime} \mathbf{G Z}$ and $\mathbf{Z}^{\prime} \mathbf{H Z}$ are independent. The proof of the converse statement is supplied by Searle (1971).

## 统计代写|广义线性模型代写generalized linear model代考|THE t AND F DISTRIBUTIONS

The normal and chi-square distributions were discussed at length in the previous sections. We now examine the distributions of certain functions of chi-square and normal random variables.

Definition 3.3.1 Noncentral t Random Variable: Let the random variable $Y \sim$ $\mathrm{N}_1\left(\alpha, \sigma^2\right)$ and the random variable $U \sim \chi_n^2(0)$. If $Y$ and $U$ are independent, then the random variable $T=(Y / \sigma) / \sqrt{U / n}$ is distributed as a noncentral $t$ random variable with $n$ degrees of freedom and noncentrality parameter $\lambda=\alpha^2 / 2$. Denote this noncentral $t$ random variable as $t_n(\lambda)$.

Definition 3.3.2 Noncentral F Random Variable: Let the random variable $U_1 \sim$ $\chi_{n_1}^2(\lambda)$ and the random variable $U_2 \sim \chi_{n_2}^2(0)$. If $U_1$ and $U_2$ are independent, then the random variable $F=\left(U_1 / n_1\right) /\left(U_2 / n_2\right)$ is distributed as a noncentral $F$ random variable with $n_1$ and $n_2$ degrees of freedom and noncentrality parameter $\lambda$. Denote this noncentral $F$ random variable as $F_{n_1, n_2}(\lambda)$.

A $t$ random variable with $n$ degrees of freedom and a noncentrality parameter equal to zero [i.e., $t_n(\lambda=0)$ ] has a central $t$ distribution. Likewise, an $F$ random variable with $n_1$ and $n_2$ degrees of freedom and a noncentrality parameter equal to zero [i.e., $F_{n_1, n_2}(\lambda=0)$ ] has a central $F$ distribution.

In recent years Smith and Lewis $(1980,1982)$, Pavur and Lewis (1983), Scariano, Neill, and Davenport (1984) and Scariano and Davenport (1984) have developed the theory of the corrected $F$ random variable. The definition of the corrected $F$ random variable is given next.

# 广义线性模型代考

## 统计代写|广义线性模型代写generalized linear model代考|THE t AND F DISTRIBUTIONS

$-$ 个随机变量 $n$ 自由度和等于零的非中心性参数即， $\left.t_n(\lambda=0)\right]$ 有一个中心 $t$ 分配。同样，一个 $F$ 随机变量 $n_1$ 和 $n_2$ 自由度和等于零的非中心性参数[即， $F_{n_1, n_2}(\lambda=0)$ ] 有一个中心 $F$ 分配。

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