Let X be distributed according to P, , and let T be sufficient for .

Let X be distributed according to Pθ, θ ∈ Ω, and let T be sufficient for θ. If ϕ(X) is any test of a hypothesis concerning θ, then ψ(T) given by

ψ(t) = E[ϕ(X) | t] is a test depending on T only, an its power function is identical with that of ϕ(X).