Envelope power function. Let S() be the class of all level- tests of a hypothesis H, and

Envelope power function. Let S(α) be the class of all level-α tests of a hypothesis H, and let β∗

α(θ) be the envelope power function, defined by

β∗

α(θ) = sup

φ∈S(α)

βφ(θ), where βφ denotes the power function of φ. If the problem of testing H is invariant under a group G, then β∗

α(θ) is invariant under the induced group G¯ .