Chapter 10.11, Problem 109E

Let X₁, X₂,…, X_m denote a random sample from the exponential density with mean θ₁ and let Y₁, Y₂,…, Y_n denote an independent random sample from an exponential density with mean θ₂.

1. a Find the likelihood ratio criterion for testing H₀ : θ₁ = θ₂ versus H_a : θ₁ ≠ θ₂.
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 χ² random variables.]