Complete the proof of Theorem 8.5.3 by dealing with the case in which r(v, x) is strictly decreasing in v for each x.
In Theorem 8.5.3
Let X = (X1, . . . ,Xn) be a random sample from a distribution that depends on a parameter (or vector of parameters) θ. Suppose that a pivotal V exists. Let G be the c.d.f. of V, and assume that G is continuous. Assume that a function r exists as in Eq. (8.5.7), and assume that r(v, x) is strictly increasing in v for each x. Let 0 < γ < 1 and let γ2 > γ1 be such that γ2 − γ1= γ. Then the following statistics are the endpoints of an exact coefficient γ confidence interval for g(θ):
A = r(G−1(γ1), X),
B = r(G−1(γ2), X).