Let X1, X2,..., Xm denote a random sample from the exponential density with mean 1 and

Chapter 10, Problem 10.109

(choose chapter or problem)

Let \(X_{1}, X_{2}, \ldots, X_{m}\) denote a random sample from the exponential density with mean \(\theta_{1}\) and let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote an independent random sample from an exponential density with mean \(\theta_{2}\).

a Find the likelihood ratio criterion for testing \(H_{0}: \theta_{1}=\theta_{2}\) versus \(H_{a}: \theta_{1} \neq \theta_{2}\).

b Show that the test in part (a) is equivalent to an exact F test [Hint: Transform \(\sum X_{i}\) and \(\sum Y_{j}\) to \(\chi^{2}\) random variables.]