Suppose that Y₁ = 0.5, Y₂ = 0.6, Y3 = 0.2, Y4 = 0.7 and Y5 = 0.6, represents a random sample. Each of these Y's comes from the same population and has as a density of fy;(vi) = (0 + 1)y{; 0 < y < 1, 0> -1 It can be shown that the natural logarithm of the likelihood function is equal to: n* ln(0 + 1) + 0 * Eln(yi) a. Determine the form of the maximum likelihood estimator for 0. b. Use the MLE formula and the data provided to find an estimate for 0.
Suppose that Y₁ = 0.5, Y₂ = 0.6, Y3 = 0.2, Y4 = 0.7 and Y5 = 0.6, represents a random sample. Each of these Y's comes from the same population and has as a density of fy;(vi) = (0 + 1)y{; 0 < y < 1, 0> -1 It can be shown that the natural logarithm of the likelihood function is equal to: n* ln(0 + 1) + 0 * Eln(yi) a. Determine the form of the maximum likelihood estimator for 0. b. Use the MLE formula and the data provided to find an estimate for 0.
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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![### Maximum Likelihood Estimation Problem
Suppose that \( Y_1 = 0.5, Y_2 = 0.6, Y_3 = 0.2, Y_4 = 0.7 \) and \( Y_5 = 0.6 \), represent a random sample. Each of these \( Y \)'s comes from the same population and has a density function given by:
\[ f_y(y_i) = (\theta + 1)y_i^{\theta} \]
where \( 0 < y < 1 \) and \( \theta > -1 \).
It can be shown that the natural logarithm of the likelihood function is:
\[ n \cdot \ln(\theta + 1) + \theta \cdot \sum \ln(y_i) \]
### Tasks:
a. Determine the form of the maximum likelihood estimator for \( \theta \).
b. Use the MLE formula and the data provided to find an estimate for \( \theta \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6f976c9-9a7a-4501-98d3-6f357dfb2f6c%2F62879643-340a-4d8c-be71-f3a2bd084422%2Fu9m6vx_processed.png&w=3840&q=75)
Transcribed Image Text:### Maximum Likelihood Estimation Problem
Suppose that \( Y_1 = 0.5, Y_2 = 0.6, Y_3 = 0.2, Y_4 = 0.7 \) and \( Y_5 = 0.6 \), represent a random sample. Each of these \( Y \)'s comes from the same population and has a density function given by:
\[ f_y(y_i) = (\theta + 1)y_i^{\theta} \]
where \( 0 < y < 1 \) and \( \theta > -1 \).
It can be shown that the natural logarithm of the likelihood function is:
\[ n \cdot \ln(\theta + 1) + \theta \cdot \sum \ln(y_i) \]
### Tasks:
a. Determine the form of the maximum likelihood estimator for \( \theta \).
b. Use the MLE formula and the data provided to find an estimate for \( \theta \).
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