Let X₁, ..., Xn € [X], where X r.v. with pdf 0ï³−¹1(0,1)(x) w.r.t. the unknown parameter 0 > 0. Find the m.l.e. and MLE of 0. Hint. The likelihood function of the sample is 6* 3
Let X₁, ..., Xn € [X], where X r.v. with pdf 0ï³−¹1(0,1)(x) w.r.t. the unknown parameter 0 > 0. Find the m.l.e. and MLE of 0. Hint. The likelihood function of the sample is 6* 3
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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![(We drop \( I_{(0,1)}-(x_1, \ldots, x_n) \) for convenience). Thus
\[
\frac{\partial}{\partial \theta} L(\theta) = \text{set it equal to } 0
\]
Finding \(\theta = -\frac{n}{\sum_{i=1}^{n} \ln x_i}\) as a critical point of \(L\). (Explain why \(\theta\) is positive.) Show that \( -\frac{n}{\sum_{i=1}^{n} \ln x_i}\) is a maximum point of \(L\) by checking the signs of \(L\) on the left (positive) and right (negative) of \(-\frac{n}{\sum_{i=1}^{n} \ln x_i}\). Therefore
\[
\hat{\theta} = -\frac{n}{\sum_{i=1}^{n} \ln x_i}
\]
and the MLE is
\[
\hat{\theta} = -\frac{n}{\sum_{i=1}^{n} \ln X_i}.
\]
[End of Text]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2Fa5defa94-1762-4653-8be9-fcdadc97e1f3%2Fiv8cn9w_processed.png&w=3840&q=75)
Transcribed Image Text:(We drop \( I_{(0,1)}-(x_1, \ldots, x_n) \) for convenience). Thus
\[
\frac{\partial}{\partial \theta} L(\theta) = \text{set it equal to } 0
\]
Finding \(\theta = -\frac{n}{\sum_{i=1}^{n} \ln x_i}\) as a critical point of \(L\). (Explain why \(\theta\) is positive.) Show that \( -\frac{n}{\sum_{i=1}^{n} \ln x_i}\) is a maximum point of \(L\) by checking the signs of \(L\) on the left (positive) and right (negative) of \(-\frac{n}{\sum_{i=1}^{n} \ln x_i}\). Therefore
\[
\hat{\theta} = -\frac{n}{\sum_{i=1}^{n} \ln x_i}
\]
and the MLE is
\[
\hat{\theta} = -\frac{n}{\sum_{i=1}^{n} \ln X_i}.
\]
[End of Text]
![**Problem Statement:**
6*. Let \( X_1, \ldots, X_n \in [X] \), where \( X \) is a random variable with probability density function \( \theta x^{\theta - 1} 1_{(0,1)}(x) \) with respect to the unknown parameter \( \theta > 0 \). Find the maximum likelihood estimator (m.l.e.) and maximum likelihood estimate (MLE) of \( \theta \).
*Hint*: The likelihood function of the sample is
\[ f_n(x_1, \ldots, x_n | \theta) = \theta^n \left( \prod_{i=1}^{n} x_i \right)^{\theta - 1} 1_{(0,1)}(x_1, \ldots, x_n). \]
Denote \( L(\theta) = \ln f_n(x_1, \ldots, x_n | \theta) \). Then,
\[ L(\theta) = n \ln \theta + (\theta - 1) \sum_{i=1}^{n} \ln x_i. \]
(We drop \( 1_{(0,1)}(x_1, \ldots, x_n) \) for convenience). Thus, ...](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2Fa5defa94-1762-4653-8be9-fcdadc97e1f3%2Fmxpe4rb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
6*. Let \( X_1, \ldots, X_n \in [X] \), where \( X \) is a random variable with probability density function \( \theta x^{\theta - 1} 1_{(0,1)}(x) \) with respect to the unknown parameter \( \theta > 0 \). Find the maximum likelihood estimator (m.l.e.) and maximum likelihood estimate (MLE) of \( \theta \).
*Hint*: The likelihood function of the sample is
\[ f_n(x_1, \ldots, x_n | \theta) = \theta^n \left( \prod_{i=1}^{n} x_i \right)^{\theta - 1} 1_{(0,1)}(x_1, \ldots, x_n). \]
Denote \( L(\theta) = \ln f_n(x_1, \ldots, x_n | \theta) \). Then,
\[ L(\theta) = n \ln \theta + (\theta - 1) \sum_{i=1}^{n} \ln x_i. \]
(We drop \( 1_{(0,1)}(x_1, \ldots, x_n) \) for convenience). Thus, ...
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