*7. In this problem, you'll find MLES for the Gaussian simple linear regression model. Let N (Bo + B1xi, o²) for i = 1,..., n, where x; are fixed 'covariates'. Since the Y;'s indep. Y; are independent and the density of Y; is 1 exp { 1 f (yi; Bo, B1, o²) = %3D V2no2 202 the likelihood function is { 1 L (Bo, B1, 0²; y:) II f (yi; ßo, P1, o²) = (27o²)¯ , { (di – Bo – B1æ:)² exp 202 i=1 i=1 (a) Write the log-likelihood € (Bo, ß1,0²; y;) = log L (Bo, B1,0²;y;). (b) Find e (Bo, B1,0²; y;). (c) Find e (Bo, B1,0²; y;). (d) Find e (Bo, B1,0²; y;). aß1 (e) Solve the system of equations aBo e (Bo, B1, o²; y:) = 0 e (Во, 30, B1, 0²; y;) = 0 e (Bo, B1, o²; y;) = 0 for Bo, B1,0² to obtain Bo = j – B1ữ E=1 TiYi – nyā B, = 1 E(yi – Bo – B1x:)² i=1 Comment: to verify that these solutions jointly specify a maximum, one must show that the matrix of second-order partial derivatives (known as the Hessian) is negative definite at the solutions. This is the multivariate analogue of the second derivative test. You are not expected to know how to do this.
*7. In this problem, you'll find MLES for the Gaussian simple linear regression model. Let N (Bo + B1xi, o²) for i = 1,..., n, where x; are fixed 'covariates'. Since the Y;'s indep. Y; are independent and the density of Y; is 1 exp { 1 f (yi; Bo, B1, o²) = %3D V2no2 202 the likelihood function is { 1 L (Bo, B1, 0²; y:) II f (yi; ßo, P1, o²) = (27o²)¯ , { (di – Bo – B1æ:)² exp 202 i=1 i=1 (a) Write the log-likelihood € (Bo, ß1,0²; y;) = log L (Bo, B1,0²;y;). (b) Find e (Bo, B1,0²; y;). (c) Find e (Bo, B1,0²; y;). (d) Find e (Bo, B1,0²; y;). aß1 (e) Solve the system of equations aBo e (Bo, B1, o²; y:) = 0 e (Во, 30, B1, 0²; y;) = 0 e (Bo, B1, o²; y;) = 0 for Bo, B1,0² to obtain Bo = j – B1ữ E=1 TiYi – nyā B, = 1 E(yi – Bo – B1x:)² i=1 Comment: to verify that these solutions jointly specify a maximum, one must show that the matrix of second-order partial derivatives (known as the Hessian) is negative definite at the solutions. This is the multivariate analogue of the second derivative test. You are not expected to know how to do this.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Title: Maximum Likelihood Estimation for Gaussian Simple Linear Regression**
In this problem, you'll find the Maximum Likelihood Estimates (MLEs) for the Gaussian simple linear regression model. Let \( Y_i \sim \text{indep. } N (\beta_0 + \beta_1 x_i, \sigma^2) \) for \( i = 1, \ldots, n \), where \( x_i \) are fixed 'covariates'. Since the \( Y_i \)'s are independent, the density of \( Y_i \) is
\[
f(y_i; \beta_0, \beta_1, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp \left\{ -\frac{1}{2\sigma^2} (y_i - \beta_0 - \beta_1 x_i)^2 \right\}
\]
The likelihood function is
\[
\mathcal{L} (\beta_0, \beta_1, \sigma^2; y_i) = \prod_{i=1}^{n} f(y_i; \beta_0, \beta_1, \sigma^2) = (2\pi \sigma^2)^{-\frac{n}{2}} \exp \left\{ -\frac{1}{2\sigma^2} \sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2 \right\}
\]
(a) **Log-Likelihood Function:**
Write the log-likelihood \( \ell (\beta_0, \beta_1, \sigma^2; y_i) = \log \mathcal{L} (\beta_0, \beta_1, \sigma^2; y_i) \).
(b) **Partial Derivative with respect to \( \beta_0 \):**
Find \( \frac{\partial}{\partial \beta_0} \ell (\beta_0, \beta_1, \sigma^2; y_i) \).
(c) **Partial Derivative with respect to \( \beta_1 \):**
Find \( \frac{\partial}{\partial \beta_1} \ell (\beta_0, \beta_1, \sigma^2; y_i) \).
(d) **Partial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F544a9f38-3e68-4113-87a7-993914a696cb%2F818af393-629c-4279-a7e2-78f27df5345c%2Fx3f8u4r_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Maximum Likelihood Estimation for Gaussian Simple Linear Regression**
In this problem, you'll find the Maximum Likelihood Estimates (MLEs) for the Gaussian simple linear regression model. Let \( Y_i \sim \text{indep. } N (\beta_0 + \beta_1 x_i, \sigma^2) \) for \( i = 1, \ldots, n \), where \( x_i \) are fixed 'covariates'. Since the \( Y_i \)'s are independent, the density of \( Y_i \) is
\[
f(y_i; \beta_0, \beta_1, \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp \left\{ -\frac{1}{2\sigma^2} (y_i - \beta_0 - \beta_1 x_i)^2 \right\}
\]
The likelihood function is
\[
\mathcal{L} (\beta_0, \beta_1, \sigma^2; y_i) = \prod_{i=1}^{n} f(y_i; \beta_0, \beta_1, \sigma^2) = (2\pi \sigma^2)^{-\frac{n}{2}} \exp \left\{ -\frac{1}{2\sigma^2} \sum_{i=1}^{n} (y_i - \beta_0 - \beta_1 x_i)^2 \right\}
\]
(a) **Log-Likelihood Function:**
Write the log-likelihood \( \ell (\beta_0, \beta_1, \sigma^2; y_i) = \log \mathcal{L} (\beta_0, \beta_1, \sigma^2; y_i) \).
(b) **Partial Derivative with respect to \( \beta_0 \):**
Find \( \frac{\partial}{\partial \beta_0} \ell (\beta_0, \beta_1, \sigma^2; y_i) \).
(c) **Partial Derivative with respect to \( \beta_1 \):**
Find \( \frac{\partial}{\partial \beta_1} \ell (\beta_0, \beta_1, \sigma^2; y_i) \).
(d) **Partial
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