### 统计代写|广义线性模型代写generalized linear model代考|STA517

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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 Systematic Structure

Let us now turn our attention to the systematic part of the model. Suppose that we have data on $p$ predictors $x_1, \ldots, x_p$ which take values $x_{i 1}, \ldots, x_{i p}$ for the $i$-th unit. We will assume that the expected response depends on these predictors. Specifically, we will assume that $\mu_i$ is as linear function of the predictors
$$\mu_i=\beta_1 x_{i 1}+\beta_2 x_{i 2}+\ldots+\beta_p x_{i p}$$
for some unknown coefficients $\beta_1, \beta_2, \ldots, \beta_p$. The coefficients $\beta_j$ are called regression coefficients and we will devote considerable attention to their interpretation.

This equation may be written more compactly using matrix notation as
$$\mu_i=\mathbf{x}_i^{\prime} \boldsymbol{\beta},$$
where $\mathbf{x}_i^{\prime}$ is a row vector with the values of the $p$ predictors for the $i$-th unit and $\boldsymbol{\beta}$ is a column vector containing the $p$ regression coefficients. Even more compactly, we may form a column vector $\boldsymbol{\mu}$ with all the expected responses and then write
$$\boldsymbol{\mu}=\mathbf{X} \boldsymbol{\beta},$$
where $\mathbf{X}$ is an $n \times p$ matrix containing the values of the $p$ predictors for the $n$ units. The matrix $\mathbf{X}$ is usually called the model or design matrix. Matrix notation is not only more compact but, once you get used to it, it is also easier to read than formulas with lots of subscripts.

The expression $\mathbf{X} \boldsymbol{\beta}$ is called the linear predictor, and includes many special cases of interest. Later in this chapter we will show how it includes simple and multiple linear regression models, analysis of variance models and analysis of covariance models.

The simplest possible linear model assumes that every unit has the same expected value, so that $\mu_i=\mu$ for all $i$. This model is often called the null model, because it postulates no systematic differences between the units. The null model can be obtained as a special case of Equation $2.3$ by setting $p=1$ and $x_i=1$ for all $i$. In terms of our example, this model would expect fertility to decline by the same amount in all countries, and would attribute all observed differences between countries to random variation.

## 统计代写|广义线性模型代写generalized linear model代考|Estimation of the Parameters

The likelihood principle instructs us to pick the values of the parameters that maximize the likelihood, or equivalently, the logarithm of the likelihood function. If the observations are independent, then the likelihood function is a product of normal densities of the form given in Equation 2.1. Taking logarithms we obtain the normal log-likelihood
$$\log L\left(\boldsymbol{\beta}, \sigma^2\right)=-\frac{n}{2} \log \left(2 \pi \sigma^2\right)-\frac{1}{2} \sum\left(y_i-\mu_i\right)^2 / \sigma^2,$$
where $\mu_i=\mathbf{x}_i^{\prime} \boldsymbol{\beta}$. The most important thing to notice about this expression is that maximizing the log-likelihood with respect to the linear parameters $\boldsymbol{\beta}$ for a fixed value of $\sigma^2$ is exactly equivalent to minimizing the sum of squared differences between observed and expected values, or residual sum of squares
$$\operatorname{RSS}(\boldsymbol{\beta})=\sum\left(y_i-\mu_i\right)^2=(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{\prime}(\mathbf{y}-\mathbf{X} \boldsymbol{\beta}) .$$
In other words, we need to pick values of $\boldsymbol{\beta}$ that make the fitted values $\mu_i=\mathbf{x}_i^{\prime} \boldsymbol{\beta}$ as close as possible to the observed values $y_i$.

Taking derivatives of the residual sum of squares with respect to $\boldsymbol{\beta}$ and setting the derivative equal to zero leads to the so-called normal equations for the maximum-likelihood estimator $\hat{\boldsymbol{\beta}}$
$$\mathbf{X}^{\prime} \mathbf{X} \hat{\boldsymbol{\beta}}=\mathbf{X}^{\prime} \mathbf{y} \text {. }$$
If the model matrix $\mathbf{X}$ is of full column rank, so that no column is an exact linear combination of the others, then the matrix of cross-products $\mathbf{X}^{\prime} \mathbf{X}$ is of full rank and can be inverted to solve the normal equations. This gives an explicit formula for the ordinary least squares (OLS) or maximum likelihood estimator of the linear parameters.

# 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 Systematic Structure

$$\mu_i=\beta_1 x_{i 1}+\beta_2 x_{i 2}+\ldots+\beta_p x_{i p}$$

$$\mu_i=\mathbf{x}_i^{\prime} \boldsymbol{\beta},$$

$$\boldsymbol{\mu}=\mathbf{X} \boldsymbol{\beta},$$

## 统计代写|广义线性模型代写generalized linear model代考|Estimation of the Parameters

$$\log L\left(\boldsymbol{\beta}, \sigma^2\right)=-\frac{n}{2} \log \left(2 \pi \sigma^2\right)-\frac{1}{2} \sum\left(y_i-\mu_i\right)^2 / \sigma^2,$$

$$\operatorname{RSS}(\boldsymbol{\beta})=\sum\left(y_i-\mu_i\right)^2=(\mathbf{y}-\mathbf{X} \boldsymbol{\beta})^{\prime}(\mathbf{y}-\mathbf{X} \boldsymbol{\beta}) .$$

$$\mathbf{X}^{\prime} \mathbf{X} \hat{\boldsymbol{\beta}}=\mathbf{X}^{\prime} \mathbf{y}$$

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## MATLAB代写

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