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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 3

Question

Math 6010 • Linear Models 2000 • Question 3

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed uniform \((0,\theta )\) random variables with common density function \[ f(t)= \begin {cases} 0, & \text {if } t\notin [0,\theta ],\\ 1/\theta , & \text {if } t\in [0,\theta ]. \end {cases} \] (a) Find the maximum likelihood estimator for \(\theta \).

(b) Find a \(1-\alpha \) equal-tail confidence interval for \(\theta \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed uniform \((0,\theta)\) random variables with common density function
\[
f(t)=
\begin{cases}
0, & \text{if } t\notin [0,\theta],\\
1/\theta, & \text{if } t\in [0,\theta].
\end{cases}
\]
(a) Find the maximum likelihood estimator for \(\theta\).

(b) Find a \(1-\alpha\) equal-tail confidence interval for \(\theta\).

Topic Tags

Estimator Distributions Maximum Likelihood Estimation Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods