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Assume the data are generated from the GN-SLR setting, Y₁ = B₁X₁1 + €i, i=1,..., n where €₂~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (£₁1, y₁),…., (Xn1, Yn). We want to estimate ₁ and ² with these data using the Maximum Likelihood Method. (a) Show that the likelihood function is L(Bo, ß1, 0²) = (2π0²)-¹/² n i=1 (b) Compute the negative log-likelihood function MLE = 1(Bo, B1,02) = -log L(Bo, B1,0²) (c) Show that the MLE estimator for ₁ is exp{-22 (Yi - B₁x1)²} exp{- Σï=1 XiYi Σ²_₁x² (d) Show that the MLE estimator for o² is Σï-1(Yi (62) MLE - i-1(3₁ - ÂMLET¡1)² = n

Assume the data are generated from the GN-SLR setting, Y₁ = B₁X₁1 + €i, i=1,..., n where €₂~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (£₁1, y₁),…., (Xn1, Yn). We want to estimate ₁ and ² with these data using the Maximum Likelihood Method. (a) Show that the likelihood function is L(Bo, ß1, 0²) = (2π0²)-¹/² n i=1 (b) Compute the negative log-likelihood function MLE = 1(Bo, B1,02) = -log L(Bo, B1,0²) (c) Show that the MLE estimator for ₁ is exp{-22 (Yi - B₁x1)²} exp{- Σï=1 XiYi Σ²_₁x² (d) Show that the MLE estimator for o² is Σï-1(Yi (62) MLE - i-1(3₁ - ÂMLET¡1)² = n

MATLAB: An Introduction with Applications
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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Assume the data are generated from the GN-SLR setting, Y = B1Xin + ti i = 1, (a) Show that the likelihood function is where €; ~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (x₁1,₁),..., (Xn1, Yn). We want to estimate B₁ and o² with these data using the Maximum Likelihood Method. n i=1 1 L(Bo, B₁,0²) = (2ño²)−¹/² exp{-22 (Yi - B₁x₁)²} (b) Compute the negative log-likelihood function. (c) Show that the MLE estimator for ₁ is ., n 9.9 1(Bo, B₁,0²) = -log L(Bo, B₁, 0²) ÂMLE _ Σi=1 X₁Yi = An 2 (d) Show that the MLE estimator for o2 is i=1 - (62) MLE _ Σi=1(Yi – ÎMLEƑ¡1)² n
Transcribed Image Text:Assume the data are generated from the GN-SLR setting, Y = B1Xin + ti i = 1, (a) Show that the likelihood function is where €; ~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (x₁1,₁),..., (Xn1, Yn). We want to estimate B₁ and o² with these data using the Maximum Likelihood Method. n i=1 1 L(Bo, B₁,0²) = (2ño²)−¹/² exp{-22 (Yi - B₁x₁)²} (b) Compute the negative log-likelihood function. (c) Show that the MLE estimator for ₁ is ., n 9.9 1(Bo, B₁,0²) = -log L(Bo, B₁, 0²) ÂMLE _ Σi=1 X₁Yi = An 2 (d) Show that the MLE estimator for o2 is i=1 - (62) MLE _ Σi=1(Yi – ÎMLEƑ¡1)² n
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