Under the setup of Lemma 3.3.1, suppose the rejection regions are defined by R = {X :

Under the setup of Lemma 3.3.1, suppose the rejection regions are defined by Rα = {X : T (X) ≥ k(α)} (3.47)

for some real-valued statistic T (X) and k(α) satisfying sup

θ∈H Pθ{T (X) ≥ k(α)} = α .

Then, show pˆ = sup

θ∈H P{T (X) ≥ t} , where t is the observed value of T (X).