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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|Multivariate Normal Distribution

In the following examples mean vectors and covariance matrices are derived for a few common problems.

Example 2.1.3 Let $Y_1, \ldots, Y_n$ be independent, identically distributed $\mathrm{N}_1\left(\alpha, \sigma^2\right)$ random variables. By Theorem 2.1.4 $\operatorname{cov}\left(Y_i, Y_j\right)=0$ for $i \neq j$. Furthermore, $\mathrm{E}\left(Y_i\right)=\alpha$ and the $\operatorname{var}\left(Y_i\right)=\sigma^2$ for all $i=1, \ldots, n$. Therefore, the $n \times 1$ random vector $\mathbf{Y}=\left(Y_1, \ldots, Y_n\right)^{\prime} \sim \mathrm{N}_n\left(\alpha \mathbf{1}_n, \sigma^2 \mathbf{I}_n\right)$.

Example 2.1.4 Consider the one-way classification described in Example 1.2.10. Let $Y_{i j}$ be a random variable representing the $j^{\text {th }}$ replicate observation in the $i^{\text {th }}$ level of the fixed factor for $i=1, \ldots, t$ and $j=1, \ldots, r$. Let the $\operatorname{tr} \times 1$ random vector $\left.\mathbf{Y}=Y_{11}, \ldots, Y_{1 r}, \ldots, Y_{t 1}, \ldots, Y_{t r}\right)^{\prime}$ where the $Y_{i j}$ ‘s are assumed to be independent, normally distributed random variables with $\mathrm{E}\left(Y_{i j}\right)=\mu_i$ and $\operatorname{var}\left(Y_{i j}\right)=\sigma^2$. This experiment can be characterized with the model
$$Y_{i j}=\mu_i+R(T){(i) j}$$ where the $R(T){(i) j}$ are independent, identically distributed normal random variables with mean 0 and variance $\sigma^2$. The letter $R$ signifies replicates and the letter $T$ signifies the fixed factor or fixed treatments. Therefore, $R(T)$ represents the effect of the random replicates nested in the fixed treatment levels. The parentheses around $T$ identify the nesting. By Theorems 2.1.2 and 2.1.4, $\mathbf{Y} \sim \mathrm{N}{t r}(\mu, \boldsymbol{\Sigma})$ where the $t r \times 1$ mean vector $\mu$ is given by \begin{aligned} \boldsymbol{\mu} &=\left[\mathrm{E}\left(Y{11}\right), \ldots, \mathrm{E}\left(Y_{1 r}\right), \ldots, \mathrm{E}\left(Y_{t 1}\right), \ldots, \mathrm{E}\left(Y_{t r}\right)\right]^{\prime} \ &=\left[\mu_1, \ldots, \mu_1, \ldots, \mu_t, \ldots, \mu_t\right]^{\prime} \ &=\left[\mu_1 \mathbf{1}r^{\prime}, \ldots, \mu_t \mathbf{1}_r^{\prime}\right]^{\prime} \ &=\left(\mu_1, \ldots, \mu_t\right)^{\prime^* \otimes \mathbf{1}_r} \end{aligned} and using Definition 1.3.3 the elements of the $\operatorname{tr} \times \operatorname{tr}$ covariance matrix $\mathbf{\Sigma}$ are \begin{aligned} \operatorname{cov}\left(Y{i j}, Y_{i^{\prime} j^{\prime}}\right) &=\mathrm{E}\left[\left(Y_{i j}-\mathrm{E}\left(Y_{i j}\right)\right)\left(Y_{i^{\prime} j^{\prime}}-\mathrm{E}\left(Y_{i^{\prime} j^{\prime}}\right)\right)\right] \ &=\mathrm{E}\left[\left(\mu_i+R(T){(i) j}-\mu_i\right)\left(\mu{i^{\prime}}+R(T){\left(i^{\prime}\right) j^{\prime}}-\mu{i^{\prime}}\right)\right] \ &=\mathrm{E}\left[R(T){(i) j} R(T){\left(i^{\prime}\right) j^{\prime}}\right] \ &= \begin{cases}\sigma^2 & \text { if } i=i^{\prime} \text { and } j=j^{\prime} \ 0 & \text { otherwise. }\end{cases} \end{aligned}
That is, $\boldsymbol{\Sigma}=\sigma^2 \mathbf{I}_t \otimes \mathbf{I}_r$.

## 统计代写|广义线性模型代写generalized linear model代考|CONDITIONAL DISTRIBUTIONS OF MULTIVARIATE NORMAL

In this section conditional multivariate normal distributions are discussed.
Theorem 2.2.1 Let the $n \times 1$ random vector $\mathbf{Y}=\left(\mathbf{Y}1^{\prime}, \mathbf{Y}_2^{\prime}\right)^{\prime} \sim \mathrm{N}_n(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ where $\boldsymbol{\mu}=\left(\boldsymbol{\mu}_1^{\prime}, \boldsymbol{\mu}_2^{\prime}\right)^{\prime}$ is the $n \times 1$ mean vector, $$\boldsymbol{\Sigma}=\left[\begin{array}{ll} \boldsymbol{\Sigma}{11} & \boldsymbol{\Sigma}{12} \ \boldsymbol{\Sigma}{21} & \boldsymbol{\Sigma}{22} \end{array}\right]$$ is the $n \times n$ positive definite covariance matrix, $\mathbf{Y}_i$ and $\mu_i$ are $n_i \times 1$ vectors, $\boldsymbol{\Sigma}{i j}$ is an $n_i \times n_j$ matrix for $i, j=1,2$ and $n=n_1+n_2$. The conditional distribution of the $n_1 \times 1$ random vector $\mathbf{Y}1$ given the $n_2 \times 1$ vector of constants $\mathbf{Y}_2=\mathbf{c}_2$ is $\mathbf{N}{n_1}\left[\boldsymbol{\mu}1+\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1}\left(\mathbf{c}_2-\boldsymbol{\mu}_2\right), \boldsymbol{\Sigma}{11}-\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \boldsymbol{\Sigma}{21}\right]$ Proof: Define the $n_1 \times 1$ vector $\mathbf{V}_1$ and the $n_2 \times 1$ vector $\mathbf{V}_2$ as $$\left[\begin{array}{l} \mathbf{V}_1 \ \mathbf{V}_2 \end{array}\right]=\left[\begin{array}{c} \mathbf{Y}_1-\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \mathbf{Y}_2 \ \mathbf{Y}_2 \end{array}\right]=\left[\begin{array}{cc} \mathbf{I}{n_1} & -\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \ \mathbf{0} & \mathbf{I}{n_2} \end{array}\right] \mathbf{Y} .$$ By Theorem 2.1.2 with the $n \times 1$ vector $\mathbf{b}=\mathbf{0}{n \times 1}$ and the $n \times n$ matrix
$$\mathbf{B}=\left[\begin{array}{cc} \mathbf{I}{n_1} & -\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \ \mathbf{0} & \mathbf{I}{n_2} \end{array}\right] .$$
the $n \times 1$ random vector $\left(\mathbf{V}1^{\prime}, \mathbf{V}_2^{\prime}\right)^{\prime} \sim \mathbf{N}_n\left(\boldsymbol{\mu}^, \boldsymbol{\Sigma}^\right)$ where the $n \times 1$ mean vector $\boldsymbol{\mu}^$ is $$\boldsymbol{\mu}^=\mathbf{B}\left[\begin{array}{l} \boldsymbol{\mu}_1 \ \boldsymbol{\mu}_2 \end{array}\right]=\left[\begin{array}{c} \boldsymbol{\mu}_1-\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \boldsymbol{\mu}_2 \ \boldsymbol{\mu}_2 \end{array}\right] .$$ and the $n \times n$ covariance matrix $\boldsymbol{\Sigma}^$ is \begin{aligned} \boldsymbol{\Sigma}^ &=\mathbf{B} \boldsymbol{\Sigma} \mathbf{B}^{\prime} \ &=\left[\begin{array}{cc} \mathbf{I}{n_1} & -\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \ \mathbf{0} & \mathbf{I}{n_2} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{\Sigma}{11} & \boldsymbol{\Sigma}{12} \ \boldsymbol{\Sigma}{21} & \boldsymbol{\Sigma}{22} \end{array}\right]\left[\begin{array}{cc} \mathbf{I}{n_1} & \mathbf{0} \ \boldsymbol{\Sigma}{22}^{-1} \boldsymbol{\Sigma}{21} & \mathbf{I}{n_2} \end{array}\right] \ &=\left[\begin{array}{cc} \boldsymbol{\Sigma}{11}-\boldsymbol{\Sigma}{12} \boldsymbol{\Sigma}{22}^{-1} \boldsymbol{\Sigma}{21} & \mathbf{0} \ \mathbf{0} & \boldsymbol{\Sigma}{22} \end{array}\right] . \end{aligned}

# 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|Multivariate Normal Distribution

$$Y_{i j}=\mu_i+R(T)(i) j$$

$$\boldsymbol{\mu}=\left[\mathrm{E}(Y 11), \ldots, \mathrm{E}\left(Y_{1 r}\right), \ldots, \mathrm{E}\left(Y_{t 1}\right), \ldots, \mathrm{E}\left(Y_{t r}\right)\right]^{\prime} \quad=\left[\mu_1, \ldots, \mu_1, \ldots, \mu_t, \ldots, \mu_t\right]^{\prime}=\left[\mu_1 \mathbf{1}^{\prime}, \ldots, \mu_t \mathbf{1}r^{\prime}\right]^{\prime} \quad=\left(\mu_1, \ldots, \mu_t\right)^{\boldsymbol{r}^* \otimes 1_r}$$ 并使用定义 1.3.3 的元素 $\operatorname{tr} \times \operatorname{tr}$ 协方差矩阵 $\boldsymbol{\Sigma}$ 是 $$\operatorname{cov}\left(Y i j, Y{i^{\prime} j^{\prime}}\right)=\mathrm{E}\left[\left(Y_{i j}-\mathrm{E}\left(Y_{i j}\right)\right)\left(Y_{i^{\prime} j^{\prime}}-\mathrm{E}\left(Y_{i^{\prime} j^{\prime}}\right)\right)\right] \quad=\mathrm{E}\left[\left(\mu_i+R(T)(i) j-\mu_i\right)\left(\mu i^{\prime}+R(T)\left(i^{\prime}\right) j^{\prime}-\mu i^{\prime}\right)\right]=\mathrm{E}\left[R(T)(i) j R(T)\left(i^{\prime}\right) j^{\prime}\right] \quad={c$$

## 统计代写|广义线性模型代写generalized linear model代考|CONDITIONAL DISTRIBUTIONS OF MULTIVARIATE NORMAL

$$\boldsymbol{\Sigma}=\left[\begin{array}{lll} \boldsymbol{\Sigma} 11 & \boldsymbol{\Sigma} 12 \boldsymbol{\Sigma} 21 & \boldsymbol{\Sigma} 22 \end{array}\right]$$

$$\left[\mathbf{V}_1 \mathbf{V}_2\right]=\left[\mathbf{Y}_1-\boldsymbol{\Sigma} 12 \boldsymbol{\Sigma} 22^{-1} \mathbf{Y}_2 \mathbf{Y}_2\right]=\left[\begin{array}{lll} \mathbf{I} n_1 & -\boldsymbol{\Sigma} 12 \boldsymbol{\Sigma} 22^{-1} \mathbf{0} & \mathbf{I} n_2 \end{array}\right] \mathbf{Y} .$$

$$\mathbf{B}=\left[\begin{array}{lll} \mathbf{I} n_1 & -\boldsymbol{\Sigma} 12 \boldsymbol{\Sigma} 22^{-1} \mathbf{0} & \mathbf{I} n_2 \end{array}\right] .$$

$$\boldsymbol{\mu}^{=} \mathbf{B}\left[\boldsymbol{\mu}_1 \boldsymbol{\mu}_2\right]=\left[\boldsymbol{\mu}_1-\boldsymbol{\Sigma} 12 \boldsymbol{\Sigma} 22^{-1} \boldsymbol{\mu}_2 \boldsymbol{\mu}_2\right] .$$

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