Design of the Fuzzy Rank Tests Package

denote the critical function of a randomized test having significance level α and point null hypothesis θ, that is, the randomized test rejects the null hypothesis θ = θ0 at level α when the observed data are x with probability φ(x, α, θ0). The requirement that φ(x, α, θ) be a probability restricts it to being between zero and one (inclusive). The requirement that the test have its nominal level is Eθ{φ(X,α, θ)} = α, for all θ and α. (1)