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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 10

Question

Math 6010 • Linear Models Spring 2016 • Question 10

Let \(X_1,\ldots ,X_n\) be iid Uniform\((0,\theta )\) with \(\theta >0\), and \(X_{(n)}\) be the largest order statistic.

(a) Find the probability that the random interval \((X_{(n)},2X_{(n)})\) contains \(\theta \).

(b) Find the constant \(c\) such that \((X_{(n)},cX_{(n)})\) is a \(100(1-\alpha )\%\) confidence interval for \(\theta \).

Raw LaTeX

Let \(X_1,\ldots,X_n\) be iid Uniform\((0,\theta)\) with \(\theta>0\),
and \(X_{(n)}\) be the largest order statistic.

\textbf{(a)} Find the probability that the random interval
\((X_{(n)},2X_{(n)})\) contains \(\theta\).

\textbf{(b)} Find the constant \(c\) such that \((X_{(n)},cX_{(n)})\) is a
\(100(1-\alpha)\%\) confidence interval for \(\theta\).

Topic Tags

Classical Probability and Inclusion-Exclusion Estimator Distributions Simultaneous Confidence Intervals Uniform Integrability and Truncation Methods