For testing the hypothesis H : = 0, (0 an interior point of ) in the

For testing the hypothesis H : θ = θ0, (θ0 an interior point of ) in the one-parameter exponential family of Section 4.2, let C be the totality of tests satisfying (4.3) and (4.5) for some −∞ ≤ C1 ≤ C2 ≤ ∞ and 0 ≤ γ1, γ2 ≤ 1.

(i) C is complete in the sense that given any level-α test φ0 of H there exists φ ∈ C such that φ is uniformly at least as powerful as φ0.

(ii) If φ1, φ2 ∈ C, then neither of the two tests is uniformly more powerful than the other.

(iii) Let the problem be considered as a two-decision problem, with decisions d0 and d1 corresponding to acceptance and rejection of H and with loss function L(θ, di) = Li(θ),i = 0, 1. Then C is minimal essentially complete provided L1(θ) < L0(θ) for all θ = θ0.

(iv) Extend the result of part (iii) to the hypothesis H : θ1 ≤ θ ≤ θ2. (For more general complete class results for exponential families and beyond (see Brown and Marden (1989)).

[(i): Let the derivative of the power function of φ0 at θ0 be β

φ0

(θ0) = ρ. Then there exists φ ∈ C such that β

φ(θ0) = ρ and φ is UMP among all tests satisfying this condition.

(ii): See the end of Section 3.7.

(iii): See the proof of Theorem 3.4.2.]