Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 0-¹ if 0
Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 0-¹ if 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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![**Random Sample and Estimator Concepts from Uniform Distribution**
Consider a random sample \(X_1, \cdots, X_n\) of size \(n\) from the \(U(0, \theta)\) distribution, where \(\theta > 0\) is an unknown parameter. The probability density function (pdf) of the \(U(0, \theta)\) distribution is given by:
\[
f(x) =
\begin{cases}
\theta^{-1} & \text{if } 0 < x < \theta \\
0 & \text{otherwise}
\end{cases}
\]
It is important to note that the information about \(\theta\) contained in the random sample \(X_1, \cdots, X_n\) equals the information about \(\theta\) contained in the statistic:
\[
T = \max(X_1, \cdots, X_n).
\]
**Understanding the Statistic \(T\):**
To understand why this is the case, consider obtaining the random sample in a sequential manner. You start with \(X_1\) and pause before obtaining \(X_2\). The value \(X_1\) gives you information that \(\theta > X_1\). Once you have \(X_1\), you receive \(X_2\). If \(X_2 > X_1\), then you gain further information about \(\theta\), specifically \(\theta > X_2\). If \(X_2 \leq X_1\), it doesn't add anything beyond confirming what you already know from \(X_1\). Therefore, the insight you gain continues to be that \(\theta\) exceeds the maximum of the observed values, which is \(T\).
**Tasks:**
(a) Demonstrate that \(T\) is the maximum likelihood estimator of \(\theta\). Recall that the likelihood function \(\mathcal{L}\) of \(\theta\), given a data sample \(x_1, \cdots, x_n\), is the product of \(f(x_1), \cdots, f(x_n)\).
(b) Develop an unbiased estimator of \(\theta\) that is not a function of \(T\) and compute its variance. Begin by calculating the expected value and the variance of the \(U(0, \theta)\) distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b1ffeba-917b-4621-8d66-7fb659f36c04%2F95506618-7165-41a2-862a-594b579402b1%2F1cxoi0s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Random Sample and Estimator Concepts from Uniform Distribution**
Consider a random sample \(X_1, \cdots, X_n\) of size \(n\) from the \(U(0, \theta)\) distribution, where \(\theta > 0\) is an unknown parameter. The probability density function (pdf) of the \(U(0, \theta)\) distribution is given by:
\[
f(x) =
\begin{cases}
\theta^{-1} & \text{if } 0 < x < \theta \\
0 & \text{otherwise}
\end{cases}
\]
It is important to note that the information about \(\theta\) contained in the random sample \(X_1, \cdots, X_n\) equals the information about \(\theta\) contained in the statistic:
\[
T = \max(X_1, \cdots, X_n).
\]
**Understanding the Statistic \(T\):**
To understand why this is the case, consider obtaining the random sample in a sequential manner. You start with \(X_1\) and pause before obtaining \(X_2\). The value \(X_1\) gives you information that \(\theta > X_1\). Once you have \(X_1\), you receive \(X_2\). If \(X_2 > X_1\), then you gain further information about \(\theta\), specifically \(\theta > X_2\). If \(X_2 \leq X_1\), it doesn't add anything beyond confirming what you already know from \(X_1\). Therefore, the insight you gain continues to be that \(\theta\) exceeds the maximum of the observed values, which is \(T\).
**Tasks:**
(a) Demonstrate that \(T\) is the maximum likelihood estimator of \(\theta\). Recall that the likelihood function \(\mathcal{L}\) of \(\theta\), given a data sample \(x_1, \cdots, x_n\), is the product of \(f(x_1), \cdots, f(x_n)\).
(b) Develop an unbiased estimator of \(\theta\) that is not a function of \(T\) and compute its variance. Begin by calculating the expected value and the variance of the \(U(0, \theta)\) distribution.
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