Suppose X has density (with respect to some measure ) p(x) = C() exp[T (x)]h(x) , for

Suppose X has density (with respect to some measure μ)

pθ(x) = C(θ) exp[θT (x)]h(x) , for some real-valued θ. Assume the distribution of T (X) is continuous under θ (for any θ). Consider the problem of testing θ = θ0 versus θ = θ0. If the null hypothesis is rejected, then a decision is to be made as to whether θ > θ0 or θ < θ0. We say that a Type 3 (or directional) error is made when it is declared that θ > θ0 when in fact θ < θ0 (or vice versa). Consider a level α test that rejects the null hypothesis if T < C1 or T > C2 for constants C1 < C2. Further suppose that it is declared that θ < θ0 if T < C1 and θ > θ0 if T > C2.
(i) If the constants are chosen so that the test is UMPU, show that the Type 3 error is controlled in the sense that sup θ=θ0 Pθ{Type 3 error is made} ≤ α . (4.25)
(ii) If the constants are chosen so that the test is equi-tailed in the sense Pθ0 {T (X) < C1} = Pθ0 {T (X) > C2} = α/2 , then show (4.25) holds with α replaced by α/2.
(iii) Give an example where the UMPU level α test has the left side of (4.25) strictly > α/2. [Confidence intervals for θ after rejection of a two-sided test are discussed in Finner (1994).]