Let X1,..., X, be a random sample from Beta(0, 1) distribution, where 0 > 0 (a) Show that Ô = is the mle of 0. 2=1 In X; (b) Find the distribution of Y = - In X (using section 1.7) (c) Use Part (b) to show that W = -E In X; has gamma distribution IT(n, 1/0). (d) show that 20W has x2(2n) distribution.
Let X1,..., X, be a random sample from Beta(0, 1) distribution, where 0 > 0 (a) Show that Ô = is the mle of 0. 2=1 In X; (b) Find the distribution of Y = - In X (using section 1.7) (c) Use Part (b) to show that W = -E In X; has gamma distribution IT(n, 1/0). (d) show that 20W has x2(2n) distribution.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
100%
Please see attached mathematical statistics question below.
part(d)
How to show or prove that 2θW has χ2(2n) distribution?
Please do not reject question if you do not know how to solve this problem. Allow another expert a chance to answer this question.
![Consider the scenario where \( X_1, \dots, X_n \) is a random sample from the Beta(\(\theta, 1\)) distribution, where \(\theta > 0\).
**Task Descriptions:**
(a) **Maximum Likelihood Estimation (MLE):**
Show that the estimator
\[
\hat{\theta} = \frac{n}{\sum_{i=1}^n \ln X_i}
\]
is the MLE of \(\theta\).
(b) **Distribution Identification:**
Find the distribution of \( Y = - \ln X \) by referring to section 1.7.
(c) **Gamma Distribution Verification:**
Utilize the result from part (b) to demonstrate that
\[
W = - \sum_{i=1}^n \ln X_i
\]
follows a gamma distribution \(\Gamma(n, 1/\theta)\).
(d) **Chi-Squared Distribution Validation:**
Establish that \( 2\theta W \) follows a chi-squared distribution \(\chi^2(2n)\).
(e) **Confidence Interval Derivation:**
Using the result from part (d), identify \( c_1, c_2 \) such that
\[
P\left( c_1 < \frac{2\theta n}{\hat{\theta}} < c_2 \right) = 1 - \alpha,
\]
for \(\alpha \in (0,1)\). Subsequently, derive a \((1-\alpha)100\%\) confidence interval for \(\theta\).
(f) **Interval Length Comparison:**
For \(\alpha = 0.02\) and \( n = 15 \), compare the length of this interval with that obtained in Example 6.2.6.
**Instructions for Using This Material:**
- Follow each part sequentially, using previous results to build understanding.
- Apply theoretical understanding of probability distributions to derive results.
- Extend concepts learned to related problems and examples for deeper comprehension.
- Employ statistical software or tables as necessary to compute specific probabilities or quantiles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15feef40-32fb-4401-88ed-794608a5d767%2F571f17d2-bd4a-4c58-8cbd-fa6646247e48%2Fljjlvg9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the scenario where \( X_1, \dots, X_n \) is a random sample from the Beta(\(\theta, 1\)) distribution, where \(\theta > 0\).
**Task Descriptions:**
(a) **Maximum Likelihood Estimation (MLE):**
Show that the estimator
\[
\hat{\theta} = \frac{n}{\sum_{i=1}^n \ln X_i}
\]
is the MLE of \(\theta\).
(b) **Distribution Identification:**
Find the distribution of \( Y = - \ln X \) by referring to section 1.7.
(c) **Gamma Distribution Verification:**
Utilize the result from part (b) to demonstrate that
\[
W = - \sum_{i=1}^n \ln X_i
\]
follows a gamma distribution \(\Gamma(n, 1/\theta)\).
(d) **Chi-Squared Distribution Validation:**
Establish that \( 2\theta W \) follows a chi-squared distribution \(\chi^2(2n)\).
(e) **Confidence Interval Derivation:**
Using the result from part (d), identify \( c_1, c_2 \) such that
\[
P\left( c_1 < \frac{2\theta n}{\hat{\theta}} < c_2 \right) = 1 - \alpha,
\]
for \(\alpha \in (0,1)\). Subsequently, derive a \((1-\alpha)100\%\) confidence interval for \(\theta\).
(f) **Interval Length Comparison:**
For \(\alpha = 0.02\) and \( n = 15 \), compare the length of this interval with that obtained in Example 6.2.6.
**Instructions for Using This Material:**
- Follow each part sequentially, using previous results to build understanding.
- Apply theoretical understanding of probability distributions to derive results.
- Extend concepts learned to related problems and examples for deeper comprehension.
- Employ statistical software or tables as necessary to compute specific probabilities or quantiles.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman