# 统计代写|广义线性模型代写generalized linear model代考|BIOS6940

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• Statistical Inference 统计推断
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• (Generalized) Linear Models 广义线性模型
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## 统计代写|广义线性模型代写generalized linear model代考|Factorial Experiments

Example 4.5.2 Consider the same two-way layout as in Example 4.5.1 except now let $S$ and $T$ both be random factors. The model is
$$Y_{i j k}=\alpha+S_i+T_j+S T_{i j}+R(S T){(i j)_k}$$ where $\alpha$ is a constant representing the overall mean effect; $s_i$ are random variables representing the effect of the first random factor; $T_j$ are the random variables representing the second random factor; $S T{i j}$ are random variables representing the interaction between $S$ and $T$; and $R(S T){(i j) k}$ are random variable defined as in Example 4.5.1. Assume the $s$ random variables $S_i \sim$ iid $\mathrm{N}_1\left(0, \sigma_S^2\right)$; the $t$ random variables $T_j \sim$ iid $\mathrm{N}_1\left(0, \sigma_T^2\right)$; the st random variables $S T{i j} \sim$ iid $\mathrm{N}1\left(0, \sigma{S T}^2\right)$; and the $\operatorname{str}$ random variables $R(S T){(i j) k} \sim$ iid $\mathrm{N}_1\left(0, \sigma{R(S T)}\right)$. Furthermore, assume that $\left{S_i, i=1, \ldots, s\right},\left{T_j, j=1, \ldots, t\right},\left{S T_{i j}, i=1, \ldots, s, j=1 \ldots, t\right}$, and $\left{R(S T){(i j) k}, i=1, \ldots, s, j=1, \ldots, t, k=1, \ldots, r\right}$ are mutually independent sets of random variables. Therefore, the $\operatorname{str} \times 1$ random vector $\mathbf{Y} \sim \mathbf{N}{s t r}(\boldsymbol{\mu}, \mathbf{\Sigma})$ where the str $\times 1$ mean vector
$$\boldsymbol{\mu}=\mathrm{E}\left(Y_{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime}=\alpha \mathbf{1}s \otimes \mathbf{1}_t \otimes \mathbf{1}_r$$ and, by the covariance algorithm, the str $\times$ str covariance matrix \begin{aligned} \Sigma=& \sigma_S^2\left[\mathbf{I}_s \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right]+\sigma_T^2\left[\mathbf{J}_s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right] \ &+\sigma{S T}^2\left[\mathbf{I}s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right]+\sigma{R(S T)}^2\left[\mathbf{I}s \otimes \mathbf{I}_t \otimes \mathbf{I}_r\right] . \end{aligned} The sum of squares matrices are not dependent on whether the factors are fixed or random. Therefore, the sum of squares $\mathbf{Y}^{\prime} \mathbf{A}_m \mathbf{Y}$ for $m=1, \ldots, 5$ are the same as those given in Example 4.5,1. Furthermore, $\mathbf{A}_m \boldsymbol{\Sigma}=c_m \mathbf{A}_m$ for $m=1, \ldots, 5$ where \begin{aligned} &c_1=\operatorname{tr} \sigma_S^2+s r \sigma_T^2+r \sigma{S T}^2+\sigma_{R(S T)}^2 \ &c_2=\operatorname{tr} \sigma_S^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_3=s r \sigma_T^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_4=r \sigma_{S T}^2+\sigma_{R(S T)}^2 \ &c_5=\sigma_{R(S T)}^2 \end{aligned}

## 统计代写|广义线性模型代写generalized linear model代考|ORDINARY LEAST-SQUARES ESTIMATION

We begin with a simple example. An engineer wants to relate the fuel consumption of a new type of automobile to the speed of the vehicle and the grade of the road traveled. He has a fleet of $n$ vehicles. Each vehicl0e is assigned to operate at a constant speed (in miles per hour) on a specific grade (in percent grade) and the fuel consumption (in $\mathrm{ml} / \mathrm{sec}$ ) is recorded. The engineer believes that the expected fuel consumption is a linear function of the speed of the vehicle and the speed of the vehicle times the grade of the road. Let $Y_i$ be a random variable that represents the observed fuel consumption of the $i^{\text {th }}$ vehicle, operating at a fixed speed, on a road with a constant grade. Let $x_{i 1}$ represent the speed of the $i^{\text {th }}$ vehicle and let $x_{i 2}$ represent the speed times the grade of the $i^{\text {th }}$ vehicle. The expected fuel consumption of the $i^{\text {th }}$ vehicle can be represented by
$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}$$
where $\beta_0, \beta_1$, and $\beta_2$ are unknown parameters. Due to qualities intrinsic to each vehicle, the observed fuel consumptions differ somewhat from the expected fuel consumptions. Therefore, the observed fuel consumption of the $i^{\text {th }}$ vehicle is represented by
$$Y_i=\mathrm{E}\left(Y_i\right)+E_i$$
or
$$Y_i=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+E_i$$
where $E_i$ is a random variable representing the difference between the observed fuel consumption and the expected fuel consumption of the $i^{\text {th }}$ vehicle. An example data set for this fuel, speed, grade experiment is provided in Table 5.1.1. In a more general setting consider a problem where the expected value of a random variable $Y_i$ is assumed to be a linear combination of $p-1$ different variables $x_{i 1}, x_{i 2}, \ldots, x_{i, p-1}$. That is,
$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1} .$$
Adding a component of error, $E_i$, to represent the difference between the observed value of $Y_i$ and the expected value of $Y_i$ we obtain
$$Y_i=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1}+E_i$$

# 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|析因实验

$$Y_{i j k}=\alpha+S_i+T_j+S T_{i j}+R(S T)(i j)k$$ 在哪里 $\alpha$ 是代表整体平均效应的常数; $s_i$ 是代表第一个随机因塐影响的随机变量; $T_j$ 是代表 第二个随机因子的随机变量； $S T i j$ 是代表之间相互作用的随机恋量 $S$ 和 $T$; 和 $R(S T)(i j) k$ 是如示例 $4.5 .1$ 中定义的随机变量。假设 $s$ 随机变量 $S_i \sim$ 独立同居 $\mathrm{N}_1\left(0, \sigma_S^2\right) ;$ 这 $t$ 随机变量 $T_j \sim$ 独立同居 $\mathrm{N}_1\left(0, \sigma_T^2\right) ;$ st 随机变量 $S T i j \sim$ 独立同居 $\mathrm{N} 1\left(0, \sigma S T^2\right)$; 和str随机变量 $R(S T)(i j) k \sim$ 独立同居 $\mathrm{N}_1(0, \sigma R(S T))$. 此外，假设 集。因此， $\operatorname{str} \times 1$ 随机向量 $\mathbf{Y} \sim \mathbf{N} \operatorname{str}(\mu, \boldsymbol{\Sigma}) \mathrm{str}$ 在哪里 $\times 1$ 平均向量 $$\boldsymbol{\mu}=\mathrm{E}\left(Y{111}, \ldots, Y_{11 r}, \ldots, Y_{s t 1}, \ldots, Y_{s t r}\right)^{\prime}=\alpha \mathbf{1} s \otimes \mathbf{1}t \otimes \mathbf{1}_r$$ 并且，通过协方差算法， str $\times$ str 协方差矩阵 $$\Sigma=\sigma_S^2\left[\mathbf{I}_s \otimes \mathbf{J}_t \otimes \mathbf{J}_r\right]+\sigma_T^2\left[\mathbf{J}_s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right] \quad \quad+\sigma S T^2\left[\mathbf{I} s \otimes \mathbf{I}_t \otimes \mathbf{J}_r\right]+\sigma R(S T)^2$$ 平方和矩阵不取决于因子是固定的还是随机的。因此，平方和 $\mathbf{Y}^{\prime} \mathbf{A}_m \mathbf{Y}$ 为了 $m=1, \ldots, 5$ 与例 4.5,1 中给出的相同。此外， $\mathbf{A}_m \mathbf{\Sigma}=c_m \mathbf{A}_m$ 为了 $m=1, \ldots, 5$ 在哪里 $$c_1=\operatorname{tr} \sigma_S^2+s r \sigma_T^2+r \sigma S T^2+\sigma{R(S T)}^2 \quad c_2=\operatorname{tr} \sigma_S^2+r \sigma_{S T}^2+\sigma_{R(S T)}^2 c_3=s r \sigma_T^2$$

## 统计代写|广义线性模型代写generalized linear model代考|普通最小二乘估计

$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}$$

$$Y_i=\mathrm{E}\left(Y_i\right)+E_i$$

$$Y_i=\beta_0+\beta_1 x_{i 1}+\beta_2 x_{i 2}+E_i$$

$$\mathrm{E}\left(Y_i\right)=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1} .$$

$$Y_i=\beta_0+\beta_1 x_{i 1}+\cdots+\beta_{p-1} x_{i, p-1}+E_i$$

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