Question: Let \(\left(X_{1}, \ldots, X_{n}\right)\) be a random sample from uniform \((0, \theta)\), where \(\theta>0\).

1. One wishes to test \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta>\theta_{0}\), where \(\theta_{0}\) is a given positive constant. Noticing that \(\theta\) is the upper bound of the support of uniform \((0, \theta)\), one considers a test that rejects \(H_{0}\) when \(X_{(n)}\) is too large, where \(X_{(n)}\) is the maximum order statistic. Find a \(c\) such that one has a size- \(\alpha\) test that rejects \(H_{0}\) when \(X_{(n)} \geq c\), where \(\alpha \in(0,1)\)
2. Suppose under \(H_{1}\) one has \(\theta=\theta_{1}\), where \(\theta_{1}\) is a constant larger than \(\theta_{0}\). Derive the power of the test you obtain in part (a).
3. Based on the test in part (a), provide a \(100(1-\alpha) \%\) confidence interval for \(\theta\). (Note that a confidence interval can have one bound being negative or positive infinity.)
4. Consider testing \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta \neq \theta_{0}\), where \(\theta_{0}\) is a given positive constant. Derive the size- \(\alpha\) generalized likelihood ratio test.
5. Based on the test in part (d), provide a \(100(1-\alpha) \%\) confidence interval for \(\theta\).
6. Suppose, thinking as a Bayesian, one views that, given \(\theta, X_{1}, \ldots, X_{n}\) are independent and identically distributed according to uniform \((0, \theta)\); and \(\theta\) follows a Pareto distribution
   with the probability density function given by
   \[f_{\theta}(\theta ; \gamma, \beta)=\beta \gamma^{\beta} \theta^{-(\beta+1)} I(\theta \geq \gamma)\]
   where \(\gamma\) and \(\beta\) are hyperparameters, and \(I(\cdot)\) is the indicator function. Show that Pareto distribution is a conjugate prior in this case.
7. In the same context as that in part (f), provide a Bayes estimator of \(\theta\).

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document