Lecture 10 slides: Uniformly most powerful tests

Let Θ = Θ0 ∪Θ1 be a parameter space. Consider a parametric family {f(x|θ), θ ∈ Θ}. Suppose we want to test the null hypothesis, H0, that θ ∈ Θ0 against the alternative, Ha, that θ ∈ Θ1. Let C be some critical set. Then the probability that the null hypothesis is rejected is given by β(θ) = Pθ{X ∈/ C}. Recall that the test based on C has level α if α ≥ supθ Θ0 β(θ). The restriction of β(·) on Θ1 is called the power of the ∈ test. Let C ′ be another critical set. Denote the power of the test based on C ′ by β′(θ). Suppose that both tests have level α. Then the test based on C is more powerful than the test based on C ′ if β(θ) ≥ β′(θ) for all θ ∈ Θ1. Any test which is more powerful than any other test in some class G will be called uniformly the most powerful test in the class G (UMP test). As follows from the theorem below, the UMP test exists if both the null and the alternative are simple.