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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|THE ^ 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.

Definition 3.3.3 Noncentral Corrected $F$ Random Variable: Let the random variable $U_1 \sim c_1 \chi_{n_1}^2(\lambda)$ and the random variable $U_2 \sim c_2 \chi_{n_2}^2(0)$. If $U_1$ and $U_2$ are independent, then the random variable $F_c=\left(c_2 / c_1\right)\left[\left(U_1 / n_1\right) /\left(U_2 / n_2\right)\right] \sim$ $F_{n_1, n_2}(\lambda)$ is called a corrected $F$ random variable where the ratio $c_2 / c_1$ is the correction factor.

In practice, we often encounter independent random variables $U_1$ and $U_2$, which are distributed as multiples of chi-square random variables $\left(U_2\right.$ being a multiple of a central chi square). The random variable $F=\left(U_1 / n_1\right) /\left(U_2 / n_2\right)$ in this case will be distributed as a noncentral $F$ random variable if and only if $c_1=c_2$ (i.e., $c_2 / c_1=1$ ). Generally, $c_1$ and $c_2$ will be linear combinations of unknown variance parameters.

In the following examples a number of central and noncentral $t$ and $F$ random variables are derived.

## 统计代写|广义线性模型代写generalized linear model代考|BHATS LEMMA

The following lemma by Bhat (1962) is applicable in many ANOVA and regression problems. The lemma provides necessary and sufficient conditions for sums of squarcs to be distributed as multiples of independent chi-square random variables.
Lemma 3.4.1 Let $k$ and $n$ denote fixed positive integers such that $1 \leq k \leq n$. Suppose $\mathbf{I}n=\sum{i=1}^k \mathbf{A}i$, where each $\mathbf{A}_i$ is an $n \times n$ symmetric matrix of rank $n_i$ with $\sum{i=1}^k n_i=n$. If the $n \times 1$ random vector $\mathbf{Y} \sim \mathrm{N}n(\mu, \Sigma)$ and the sum of squares $S_i^2=\mathbf{Y}^{\prime} \mathbf{A}_i \mathbf{Y}$ for $i=1, \ldots, k$, then (a) $S_i^2 \sim c_i \chi{n_i}^2\left(\lambda_i=\boldsymbol{\mu}^{\prime} \mathbf{A}i \boldsymbol{\mu} /\left(2 c_i\right)\right)$ and (b) $\left{S_i^2, i=1, \ldots, k\right}$ are mutually independent if and only if $\Sigma=\sum{i=1}^k c_i \mathbf{A}i$ where $c_i>0$. Proof: This proof is due to Scariano et al. (1984). Assume that the quadratic forms $S_i^2$ satisfy (a) and (b) given in Lemma 3.4.1. By Theorems 3.1.2 and 3.2.1, (i) the matrices $\left(1 / c_i\right) \mathbf{A}_i \boldsymbol{\Sigma}$ are idempotent for $i=1, \ldots, k$ and (ii) $\mathbf{A}_i \boldsymbol{\Sigma} \mathbf{A}_j=\mathbf{0}{n \times n}$ for $i \neq j, i, j=1, \ldots, k$. Furthermore, by Theorem 1.1.7, $\mathbf{A}_i=\mathbf{A}_i^2$ and

$\mathbf{A}i \mathbf{A}_j=\mathbf{0}{n \times n}$ for $i \neq j, i, j=1, \ldots, k$. But (i) and (ii) imply that $\sum_{i=1}^k\left(1 / c_i\right) \mathbf{A}i \mathbf{\Sigma}$ is idempotent of rank $n$ and thus equal to $\mathbf{I}_n$. Hence, $\Sigma=\left[\sum{i=1}^k\left(1 / c_i\right) \mathbf{A}i\right]^{-1}=$ $\sum{i=1}^k c_i \mathbf{A}i$. Conversely, assume $\Sigma=\sum{i=1}^k c_i \mathbf{A}i$. But $\mathbf{A}_i=\mathbf{A}_i^2$ and $\mathbf{A}_i \mathbf{A}_j=\mathbf{0}{n \times n}$, so (i) and (ii) hold. Therefore, by Theorems 3.1.2 and 3.2.1, (a) and (b) hold.

In the next example, Bhat’s lemma is applied to the two-way cross classification described in Example 2.3.2

# 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|^和f分布

$\left.F_{n_1, n_2}(\lambda=0)\right]$ 有一个中心 $F$ 分配。 近年来，史密斯和刘易斯(1980, 1982)，Pavur 和 Lewis (1983)、Scariano、Neill 和 Davenport (1984) 以及 Scariano 和 Davenport (1984) 发展了校正理论 $F$ 随机变 隹 量。修正的定义 $F$ 接下来给出随机变量。

## 统计代写|广义线性模型代写generalized linear model代考|巴特引理

Bhat (1962) 的以下引理适用于许多方差分析和回归问题。引理为平方和作为独立卡方随机变量的倍数分布提供了充分必要条件。

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## MATLAB代写

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