Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be two independent random samples. We assume that \(X_1,X_2,\ldots ,X_n\) are independent identically distributed exponential \((\lambda )\) random variables and \(Y_1,Y_2,\ldots ,Y_m\) are independent identically distributed exponential \((\mu )\) random variables. We want to test \(H_0:\lambda =\mu \) against the alternative \(H_A:\lambda \ne \mu \).
(i) Find \(T_{m,n}\) the likelihood ratio test. (7 points)
(ii) Prove that \(-2\log T_{m,n}\) converges in distribution under the null hypothesis when \(\min (m,n)\to \infty \). (7 points)
(iii) Show that the likelihood ratio test is asymptotically consistent and prove that the power goes to 1 as \(\min (m,n)\to \infty \). (7 points)