Let F0 be a family of probability measures over (X , A), and let C be a class of transformations of the space X . Define a class F1 of distributions by F1 ∈ F1 if there exists F0 ∈ F0 and f ∈ C such that the distribution of f(X)

is F1 when that of X is F0. If φ is any test satisfying

(a) EF0 φ(X) = α for all F0 ∈ F0, and

(b) φ(x) ≤ φ[f(x)] for all x and all f ∈ C, then φ is unbiased for testing F0 against F1