Let f, , denote a family of densities with respect to a measure . (We

Let fθ, θ ∈ , denote a family of densities with respect to a measure

μ. (We assume is endowed with a σ-field so that the densities fθ(x) are jointly measurable in θ and x.) Consider the problem of testing a simple null hypothesis

θ = θ0 against the composite alternatives K = {θ : θ = θ0}. Let  be a probability distribution on K .

(i) As explicitly as possible, find a test φ that maximizes K Eθ(φ)d(θ), subject to it being level α.

(ii) Let h(x) = fθ(x)d(θ). Consider the nonrandomized φ test that rejects if and only if h(x)/ fθ0 (x) > k, and suppose μ{x : h(x) = k fθ(x)} = 0. Then, φ is admissible at level α = Eθ0 (φ) in the sense that it is impossible that there exists another level α test φ such that Eθ(φ

) ≥ Eθ(φ) for all θ.

(iii) Show that the test of