Suppose \(X_{1}, \ldots, X_{n}\) is a random sample from an ExPOnEntial location family of densities of the form \(f(x)=\exp [-(x-\theta)] \delta\{x ;[\theta, \infty)\}\), where \(\theta \in \Omega=\mathbb{R}\).

a. Let \(X_{(1)}\) be the first order-statistic of the sample \(X_{1}, \ldots, X_{n}\). That is \(X_{(1)}=\min \left\{X_{1}, \ldots, X_{n}ight\}\). Prove that

\[C(\alpha, \boldsymbol{\omega})=\left[X_{(1)}+n^{-1} \log \left(1-\omega_{U}ight), X_{(1)}+n^{-1} \log \left(1-\omega_{L}ight)ight]\]

is a \(100 \alpha \%\) confidence interval for \(\theta\) when \(\omega_{U}-\omega_{L}=\alpha\) where \(\omega_{L} \in[0,1]\) and \(\omega_{U} \in[0,1]\). Hint: Use the fact that the density of \(X_{(1)}\) is \(f\left(x_{(1)}ight)=\) \(n \exp \left[-n\left(x_{(1)}-\thetaight)ight] \delta\left\{x_{(1)} ;[\theta, \infty)ight\}\).

b. Use the confidence interval given above to derive an observed confidence level for an arbitrary region \(\Psi=\left(t_{L}, t_{U}ight) \subset \mathbb{R}\) where \(t_{L}