In the situation of the previous problem there exists no test for testing H : ξ = 0 at level α, which for all σ has power ≥ β>α against the alternatives (ξ, σ) with ξ = ξ1 > 0.

[Let β(ξ1, σ) be the power of any level α test of H, and let β(σ) denote the power of the most powerful test for testing ξ = 0 against ξ = ξ1 when σ is known.

Then infσ β(ξ1, σ) ≤ infσ β(σ) = α.]