Locally uniformly most powerful tests. If the sample space is finite and independent of , the test

Locally uniformly most powerful tests. If the sample space is finite and independent of θ, the test ϕ0 of Problem 8.2(i) is not only LMP but also locally uniformly most powerful (LUMP) in the sense that there exists a value

∆ > 0 such that ϕ0 maximizes βϕ(θ) for all θ with 0 < θ − θ0 < ∆.

[See the argument following (6.21) of Section 6.9.]