Additionally, for 6.1.3, let 0MLE denote your MLE of 0, and show that (a) MLE is a consistent estimator of 0, i.e., LE 0, as n →x. [Hints will be given in class on this.] (b) Consider the MLE given by MLE distribution of Zn. = = Yn − ½ and put Zn = n(0 – ÔMLE). Find the limiting
Additionally, for 6.1.3, let 0MLE denote your MLE of 0, and show that (a) MLE is a consistent estimator of 0, i.e., LE 0, as n →x. [Hints will be given in class on this.] (b) Consider the MLE given by MLE distribution of Zn. = = Yn − ½ and put Zn = n(0 – ÔMLE). Find the limiting
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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![Additionally, for 6.1.3, let \(\hat{\theta}_{\text{MLE}}\) denote your MLE of \(\theta\), and show that
(a) \(\hat{\theta}_{\text{MLE}}\) is a consistent estimator of \(\theta\), i.e., \(\hat{\theta}_{\text{MLE}} \xrightarrow{p} \theta\), as \(n \to \infty\). [Hints will be given in class on this.]
(b) Consider the MLE given by \(\hat{\theta}_{\text{MLE}} = Y_n - \frac{1}{2}\) and put \(Z_n = n(\theta - \hat{\theta}_{\text{MLE}})\). Find the limiting distribution of \(Z_n\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F64387d7c-e9ac-4d98-a514-66477ae8240a%2F6fuj6o1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Additionally, for 6.1.3, let \(\hat{\theta}_{\text{MLE}}\) denote your MLE of \(\theta\), and show that
(a) \(\hat{\theta}_{\text{MLE}}\) is a consistent estimator of \(\theta\), i.e., \(\hat{\theta}_{\text{MLE}} \xrightarrow{p} \theta\), as \(n \to \infty\). [Hints will be given in class on this.]
(b) Consider the MLE given by \(\hat{\theta}_{\text{MLE}} = Y_n - \frac{1}{2}\) and put \(Z_n = n(\theta - \hat{\theta}_{\text{MLE}})\). Find the limiting distribution of \(Z_n\).
![6.1.3. Let \( Y_1 < Y_2 < \cdots < Y_n \) be the order statistics of a random sample from a distribution with pdf \( f(x; \theta) = 1 \), \( \theta - \frac{1}{2} \leq x \leq \theta + \frac{1}{2} \), \(-\infty < \theta < \infty\), zero elsewhere. This is a nonregular case. Show that every statistic \( u(X_1, X_2, \ldots, X_n) \) such that
\[
Y_n - \frac{1}{2} \leq u(X_1, X_2, \ldots, X_n) \leq Y_1 + \frac{1}{2}
\]
is a mle of \( \theta \). In particular, \( (4Y_1 + 2Y_n + 1)/6 \), \( (Y_1 + Y_n)/2 \), and \( (2Y_1 + 4Y_n - 1)/6 \) are three such statistics. Thus, uniqueness is not, in general, a property of mles.
---
6.1. Maximum Likelihood Estimation
Page 361
### Explanation:
The text discusses properties of maximum likelihood estimators (MLE) in a statistical context. The main focus is on demonstrating that within a specific distribution, multiple statistics can serve as MLEs of the parameter \(\theta\), highlighting the non-uniqueness of MLEs in this case.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F64387d7c-e9ac-4d98-a514-66477ae8240a%2Fxvxlkyj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6.1.3. Let \( Y_1 < Y_2 < \cdots < Y_n \) be the order statistics of a random sample from a distribution with pdf \( f(x; \theta) = 1 \), \( \theta - \frac{1}{2} \leq x \leq \theta + \frac{1}{2} \), \(-\infty < \theta < \infty\), zero elsewhere. This is a nonregular case. Show that every statistic \( u(X_1, X_2, \ldots, X_n) \) such that
\[
Y_n - \frac{1}{2} \leq u(X_1, X_2, \ldots, X_n) \leq Y_1 + \frac{1}{2}
\]
is a mle of \( \theta \). In particular, \( (4Y_1 + 2Y_n + 1)/6 \), \( (Y_1 + Y_n)/2 \), and \( (2Y_1 + 4Y_n - 1)/6 \) are three such statistics. Thus, uniqueness is not, in general, a property of mles.
---
6.1. Maximum Likelihood Estimation
Page 361
### Explanation:
The text discusses properties of maximum likelihood estimators (MLE) in a statistical context. The main focus is on demonstrating that within a specific distribution, multiple statistics can serve as MLEs of the parameter \(\theta\), highlighting the non-uniqueness of MLEs in this case.
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