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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 8

Question

Math 6010 • Linear Models 2004 • Question 8

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta )= \begin {cases} 0, & \text {if } t\leq \theta ,\\ e^{-(t-\theta )}, & \text {if } t>\theta . \end {cases} \] (a) Find two \(1-\alpha \) equal-tail confidence intervals for \(\theta \). One should be based on the smallest order statistic and the other should be based on the sample mean.

(b) Compare the expected lengths of the confidence intervals.

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with density function
\[
f(t,\theta)=
\begin{cases}
0, & \text{if } t\leq\theta,\\
e^{-(t-\theta)}, & \text{if } t>\theta.
\end{cases}
\]
(a) Find two \(1-\alpha\) equal-tail confidence intervals for \(\theta\). One should be based on the smallest order statistic and the other should be based on the sample mean.

(b) Compare the expected lengths of the confidence intervals.

Topic Tags

Covariance and Independence Estimator Distributions Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods