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

统计代写|广义线性模型代写generalized linear model代考|BEST LINEAR UNBIASED ESTIMATORS

In many problems it is of interest to estimate linear combinations of $\beta_0, \ldots, \beta_{p-1}$, say, $\mathbf{t}^{\prime} \boldsymbol{\beta}$, where $\mathbf{t}$ is any nonzero $p \times 1$ vector of known constants. In the next definition the “best” linear unbiased estimator of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ is identified.

Definition 5.2.1 Best Linear Unbiased Estimator $(B L U E)$ of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ : The best linear unbiased estimator of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ is
(i) a linear function of the observed vector $\mathbf{Y}$, that is, a function of the form $\mathbf{a}^{\prime} \mathbf{Y}+a_0$ where $\mathbf{a}$ is an $n \times 1$ vector of constants and $a_0$ is a scalar and
(ii) the unbiased estimator of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ with the smallest variance.
In the next important theorem $\mathbf{t}^{\prime} \hat{\boldsymbol{\beta}}=\mathbf{t}^{\prime}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y}$ is shown to be the BLUE of $\mathbf{t}^{\prime} \beta$ when $\mathrm{E}(\mathbf{E})=\mathbf{0}$ and $\operatorname{cov}(\mathbf{E})=\sigma^2 \mathbf{I}_n$. The theorem is called the Gauss-Markov theorem.

Theorem 5.2.1 Let $\mathbf{Y}=\mathbf{X} \beta+\mathbf{E}$ where $\mathrm{E}(\mathbf{E})=\mathbf{0}$ and $\operatorname{cov}(\mathbf{E})=\sigma^2 \mathbf{I}_n$. Then the least-squares estimator of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ is given by $\mathbf{t}^{\prime} \hat{\boldsymbol{\beta}}=\mathbf{t}^{\prime}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{Y}$ and $\mathbf{t}^{\prime} \hat{\boldsymbol{\beta}}$ is the $B L U E$ of $\mathbf{t}^{\prime} \boldsymbol{\beta}$.

Proof: First, the least-squares estimator of $\mathbf{t}^{\prime} \boldsymbol{\beta}$ is shown to be $\mathbf{t}^{\prime} \hat{\boldsymbol{\beta}}$. Let $\mathbf{T}$ be a $p \times p$ nonsingular matrix such that $\mathbf{T}=\left(\mathbf{t} \mid \mathbf{T}_0\right)$ where $\mathbf{t}$ is a $p \times 1$ vector and $\mathbf{T}_0$ is a $p \times(p-1)$ matrix. If $\mathbf{R}=\mathbf{T}^{-1}$ then
\begin{aligned} \mathbf{Y} &=\mathbf{X} \boldsymbol{\beta}+\mathbf{E} \ &=\mathbf{X R T}^{\prime} \boldsymbol{\beta}+\mathbf{E} \ &=\mathbf{U} \boldsymbol{\omega}+\mathbf{E} \end{aligned}
where $\mathbf{U}=\mathbf{X R}$ and
$$\omega=\mathbf{T}^{\prime} \boldsymbol{\beta}=\left[\begin{array}{c} \mathbf{t}^{\prime} \boldsymbol{\beta} \ \mathbf{T}_0^{\prime} \boldsymbol{\beta} \end{array}\right]$$
The least-squares estimate of $\omega$ is given by
\begin{aligned} \hat{\boldsymbol{\omega}} &=\left(\mathbf{U}^{\prime} \mathbf{U}\right)^{-1} \mathbf{U}^{\prime} \mathbf{Y} \ &=\left(\mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \mathbf{R}\right)^{-1} \mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{Y} \ &=\mathbf{R}^{-1}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{R}^{\prime-1} \mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{Y} \end{aligned}

统计代写|广义线性模型代写generalized linear model代考|ANOVA TABLE FOR THE ORDINARY

An ANOVA table can be constructed that partitions the total sum of squares into the sum of squares due to the overall mean, the sum of squares due to $\beta_1, \ldots, \beta_{p-1}$, and the sum of squares due to the residual. The ANOVA table for this model is given in Table 5.3.1. The sum of squares under the column “SS” can be applied to any form of the $n \times p$ matrix $\mathbf{X}$. The sum of squares under the column “SS Centered” can be applied to centered matrices $\mathbf{X}=\left[\mathbf{1}n \mid \mathbf{X}_c\right]$ where $\mathbf{1}_n^{\prime} \mathbf{X}_c=\mathbf{0}{1 \times(p-1)}$.

The expected mean squares for each effect are calculated using Theorem 1.3.2:
\begin{aligned} \text { EMS (overall mean) } &=\mathrm{E}\left(\mathbf{Y}^{\prime} \frac{1}{n} \mathbf{J}n \mathbf{Y}\right) \ &=\operatorname{tr}\left[\left(\sigma^2 / n\right) \mathbf{J}_n\right]+\boldsymbol{\beta}^{\prime} \mathbf{X}^{\prime} \frac{1}{n} \mathbf{J}_n \mathbf{X} \boldsymbol{\beta} \ &=\sigma^2+\boldsymbol{\beta}^{\prime}\left[\mathbf{1}_n \mid \mathbf{X}_c\right]^{\prime} \frac{1}{n} \mathbf{J}_n\left[\mathbf{1}_n \mid \mathbf{X}_c\right] \boldsymbol{\beta} \ &=\sigma^2+\boldsymbol{\beta}^{\prime}\left[\begin{array}{c} \mathbf{1}_n^{\prime} \mathbf{1}_n \ \mathbf{X}_c^{\prime} \mathbf{1}_n \end{array}\right]\left[\mathbf{1}_n^{\prime} \mathbf{1}_n \mid \mathbf{1}_n^{\prime} \mathbf{X}_c\right] \boldsymbol{\beta} / n \ &=\sigma^2+\boldsymbol{\beta}^{\prime}\left[\begin{array}{cc} n^2 & \mathbf{0} \ \mathbf{0} & \mathbf{0} \end{array}\right] \boldsymbol{\beta} / n \ &=\sigma^2+n \beta_0^{* 2} \end{aligned} $\mathrm{EMS}$ (Regression) $=E\left[\mathbf{Y}^{\prime} \mathbf{X}_c\left(\mathbf{X}_c^{\prime} \mathbf{X}_c\right)^{-1} \mathbf{X}_c^{\prime} \mathbf{Y} /(p-1)\right]$ \begin{aligned} =&\left{\operatorname{tr}\left[\sigma^2 \mathbf{X}_c\left(\mathbf{X}_c^{\prime} \mathbf{X}_c\right)^{-1} \mathbf{X}_c^{\prime}\right]\right.\ &\left.+\boldsymbol{\beta}^{\prime} \mathbf{X}^{\prime} \mathbf{X}_c\left(\mathbf{X}_c^{\prime} \mathbf{X}_c\right)^{-1} \mathbf{X}_c^{\prime} \mathbf{X} \boldsymbol{\beta}\right} /(p-1) \ =&\left{\sigma^2(p-1)\right.\ &\left.+\boldsymbol{\beta}^{\prime}\left[\begin{array}{c} \mathbf{1}_n^{\prime} \ \mathbf{X}_c^{\prime} \end{array}\right] \mathbf{X}_c\left(\mathbf{X}_c^{\prime} \mathbf{X}_c\right)^{-1} \mathbf{X}_c^{\prime}\left[\mathbf{1}_n \mid \mathbf{X}_c\right] \boldsymbol{\beta}\right} /(p-1) \ =& \sigma^2+\left(\beta_1, \ldots, \beta{p-1}\right) \mathbf{X}c^{\prime} \mathbf{X}_c\left(\beta_1, \ldots, \beta{p-1}\right)^{\prime} /(p-1) \end{aligned}
and EMS (residual) $=\mathrm{E}\left(\hat{\sigma}^2\right)=\sigma^2$ as derived in Section 5.1. The ANOVA table for the fuel, speed, grade data set is provided in the next example.

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|BEST LINEAR UNBIASED ESTIMATORS

(i) 观察向量的线性函数 $\mathbf{Y}$ ，即形式的函数 $\mathbf{a}^{\prime} \mathbf{Y}+a_0$ 在哪里 $\mathbf{a}$ 是一个 $n \times 1$ 常数向量和 $a_0$ 是一个标量并且
(ii) 的无偏估计量 $\mathbf{t}^{\prime} \boldsymbol{\beta}$ 具有最小的方差。

$$\mathbf{Y}=\mathbf{X} \boldsymbol{\beta}+\mathbf{E} \quad=\mathbf{X R T} \mathbf{T}^{\prime} \boldsymbol{\beta}+\mathbf{E}=\mathbf{U} \boldsymbol{\omega}+\mathbf{E}$$

$$\omega=\mathbf{T}^{\prime} \boldsymbol{\beta}=\left[\mathbf{t}^{\prime} \boldsymbol{\beta} \mathbf{T}_0^{\prime} \boldsymbol{\beta}\right]$$

$$\hat{\boldsymbol{\omega}}=\left(\mathbf{U}^{\prime} \mathbf{U}\right)^{-1} \mathbf{U}^{\prime} \mathbf{Y} \quad=\left(\mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \mathbf{R}\right)^{-1} \mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{Y}=\mathbf{R}^{-1}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{R}^{\prime-1} \mathbf{R}^{\prime} \mathbf{X}^{\prime} \mathbf{Y}$$

统计代写|广义线性模型代写generalized linear model代考|ANOVA TABLE FOR THE ORDINARY

$\mathrm{EMS}$ (overall mean) $=\mathrm{E}\left(\mathbf{Y}^{\prime} \frac{1}{n} \mathbf{J} n \mathbf{Y}\right) \quad=\operatorname{tr}\left[\left(\sigma^2 / n\right) \mathbf{J}_n\right]+\boldsymbol{\beta}^{\prime} \mathbf{X}^{\prime} \frac{1}{n} \mathbf{J}_n \mathbf{X} \boldsymbol{\beta}=\sigma^2+\boldsymbol{\beta}^{\prime}\left[\mathbf{1}_n \mid \mathbf{X}_c\right]^{\prime} \frac{1}{n} \mathbf{J}_n$ $\operatorname{EMS}($ 回归 $)=E\left[\mathbf{Y}^{\prime} \mathbf{X}_c\left(\mathbf{X}_c^{\prime} \mathbf{X}_c\right)^{-1} \mathbf{X}_c^{\prime} \mathbf{Y} /(p-1)\right]$

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