(i) Show that if Y is any random variable with c.d.f. G(·), then P{G(Y ) ≤ u} ≤ u for all 0 ≤ u ≤ 1 .

If G−(t) = P{Y < t}, then show P{1 − G−(Y ) ≤ u} ≤ u for all 0 ≤ u ≤ 1 .

(ii) In Example 3.3.3, show that Fθ0 (T ) and 1 − F−

θ0

(T ) are both valid p-values, in the sense that (3.13) holds.