Question

10 Generalized Linear Regression In the problems of this section $\mathbf{x}^T \mathbf{\beta} = \beta_0 + \sum_{i=1}^p \beta_i x_i$ Problem 10.4. Consider the normal multiple least squares model, $\mathbf{Y}^* = \mathbf{x}^T \mathbf{\beta} + \epsilon$, (134) where $\mathbf{x}^T = (1, x_1, x_2, \dots, x_p)$, $\mathbf{\beta}^T = (\beta_0, \beta_1, \dots, \beta_p)$ and $\epsilon \sim N(0, 1)$, and is independent of $\mathbf{x}$. a) Explain why we have $P(-\epsilon \le x) = P(\epsilon \le x)$. (135) Aid: Let $\Phi(x)$ be the cumulative distribution function of $N(0, 1)$ and $\phi(x)$ be the corresponding pdf. Since $\phi(x) = \phi(-x)$, we have $\Phi(-x) = 1 - \Phi(x)$. b) Construct Verify that $Y = \begin{cases} 1 & \text{if } Y^* > 0, \\ 0 & \text{otherwise.} \end{cases}$ $P(Y = 1 | \mathbf{x}) = \Phi(\mathbf{x}^T \mathbf{\beta})$, You may use (135) even if you have failed to show it. c) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The way to construct a GLM goes via a link function. Present and justify the link function in this assignment.

          10 Generalized Linear Regression
In the problems of this section
$\mathbf{x}^T \mathbf{\beta} = \beta_0 + \sum_{i=1}^p \beta_i x_i$
Problem 10.4.
Consider the normal multiple least squares model,
$\mathbf{Y}^* = \mathbf{x}^T \mathbf{\beta} + \epsilon$,
(134)
where $\mathbf{x}^T = (1, x_1, x_2, \dots, x_p)$, $\mathbf{\beta}^T = (\beta_0, \beta_1, \dots, \beta_p)$ and $\epsilon \sim N(0, 1)$, and is independent of $\mathbf{x}$.
a) Explain why we have
$P(-\epsilon \le x) = P(\epsilon \le x)$.
(135)
Aid: Let $\Phi(x)$ be the cumulative distribution function of $N(0, 1)$ and $\phi(x)$ be the corresponding pdf. Since $\phi(x) = \phi(-x)$, we have
$\Phi(-x) = 1 - \Phi(x)$.
b) Construct
Verify that
$Y = \begin{cases} 1 & \text{if } Y^* > 0, \\ 0 & \text{otherwise.} \end{cases}$
$P(Y = 1 | \mathbf{x}) = \Phi(\mathbf{x}^T \mathbf{\beta})$,
You may use (135) even if you have failed to show it.
c) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The way to construct a GLM goes via a link function. Present and justify the link function in this assignment.

10 Generalized Linear Regression
In the problems of this section
𝐱^T β = β0 + ∑i=1^p xi
Problem 10.4.
Consider the normal multiple least squares model,
𝐘^* = 𝐱^T β + ϵ,
(134)
where 𝐱^T = (1, x1, x2, …, xp), β^T = (β0, β1, …, ) and ϵ∼ N(0, 1), and is independent of 𝐱.
a) Explain why we have
P(-ϵ≤ x) = P(ϵ≤ x).
(135)
Aid: Let Φ(x) be the cumulative distribution function of N(0, 1) and ϕ(x) be the corresponding pdf. Since ϕ(x) = ϕ(-x), we have
Φ(-x) = 1 - Φ(x).
b) Construct
Verify that
Y = 1 if Y^* > 0,
0 otherwise.
P(Y = 1 | 𝐱) = Φ(𝐱^T β),
You may use (135) even if you have failed to show it.
c) The r.v. Y in this assignment is a special case of generalized linear regression (GLM). The way to construct a GLM goes via a link function. Present and justify the link function in this assignment.

Added by Amanda M.

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