For testing h0 1 versus 1 show that the problem is

Question

Let X₁ and X₂ be independently distributed as the exponential distributions E(0, θ_i), i = 1, 2, respectively. Define θ = θ₁/θ₂.

(a) For testing H₀; θ ≤ 1 versus θ > 1, show that the problem is invariant under the group of transformations gc(x₁, x₂) = (cx₁, cx₂), c > 0, and that a UMPI test of size α rejects H₀ when X₂/X₁ > (1 - α)/α.

(b) For testing H₀; θ = 1 versus θ ≠ 1, show that the problem is invariant under the group of transformations in (a) and g(x₁, x₂) = (x₂, x₁), and that a UMPI test of size α rejects H₀ when X₁/X₂ > (2 - α)/α and X₂/X₁ > (2 - α)/α.