6.2.3. Given the pdf 1 f(x; 0) = a1 + (x −0)²] -∞<<∞, -∞ <<∞, show that the Rao-Cramér lower bound is 2/n, where n is the size of a random sam- ple from this Cauchy distribution. What is the asymptotic distribution of √(1-0) if is the mle of 0?
6.2.3. Given the pdf 1 f(x; 0) = a1 + (x −0)²] -∞<<∞, -∞ <<∞, show that the Rao-Cramér lower bound is 2/n, where n is the size of a random sam- ple from this Cauchy distribution. What is the asymptotic distribution of √(1-0) if is the mle of 0?
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
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Chapter1: Starting With Matlab
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![### 6.2.3. Problem Statement
Given the probability density function (pdf):
\[
f(x; \theta) = \frac{1}{\pi [1 + (x - \theta)^2]}, \quad -\infty < x < \infty, \quad -\infty < \theta < \infty,
\]
show that the Rao–Cramér lower bound is \(2/n\), where \(n\) is the size of a random sample from this Cauchy distribution.
What is the asymptotic distribution of \(\sqrt{n}(\hat{\theta} - \theta)\) if \(\hat{\theta}\) is the maximum likelihood estimator (MLE) of \(\theta\)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F9039d145-b039-4e39-9808-4a78a5737ba1%2F97x2te_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 6.2.3. Problem Statement
Given the probability density function (pdf):
\[
f(x; \theta) = \frac{1}{\pi [1 + (x - \theta)^2]}, \quad -\infty < x < \infty, \quad -\infty < \theta < \infty,
\]
show that the Rao–Cramér lower bound is \(2/n\), where \(n\) is the size of a random sample from this Cauchy distribution.
What is the asymptotic distribution of \(\sqrt{n}(\hat{\theta} - \theta)\) if \(\hat{\theta}\) is the maximum likelihood estimator (MLE) of \(\theta\)?
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