Prove Theorem 9.1.3.
In Theorem 9.1.3.
One-Sided Confidence Intervals from One-Sided Tests. Let X = (X1, . . . , Xn) be a random sample from a distribution that depends on a parameter θ. Let g(θ) be a real-valued function, and suppose that for each possible value g0 of g(θ), there is a level α0 test δg0 of the hypotheses (9.1.19). For each possible value x of X, define ω(x) by Eq. (9.1.14). Let γ = 1− α0. Then the random set ω(X) satisfies Eq. (9.1.15) for all θ0 ∈ Ω.