For testing 0 versus n, let n be a test satisfying lim sup n E0 (

For testing θ0 versus θn, let φ∗

n be a test satisfying lim sup n Eθ0 (φ∗

n) = α∗ < α

and Eθn (φ∗

n) → β∗.

(i) Show there exists a test sequence ψn satisfying lim supn Eθ0 (ψn) = α and a number β such that lim Eθn (ψn) = β ≥ β∗ , and this last inequality is strict unless β∗ = 1.

(ii) Hence, show that, under the conditions of Theorem 13.3.3, any LAUMP level

α test sequence φ∗

n satisfies Eθ0 (φ∗

n) → α.