For testing 0 versus n, let n be a test satisfying lim sup n E0 ( n ) = < and En

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 15.3.3, any LAUMP level-α
test sequence φ∗
n satisfies Eθ0 (φ∗
n ) → α.