(8.115b)

Proof. Under H10 ?i=1, i=0, 1, 2. Hence ?i=0, i=1, 2. Under H1?,

and ?2=0. Thus

Now using Neyman-Pearson Lemma we get the theorem. Q.E.D.

Theorem 8.5.6. The likelihood ratio test of H20 against H21 is UMP invariant

among all test based on satisfying

Proof. Under H2?,

Hence

Hence has a monotone likelihood ratio in Now using Lehmann (1939) we get the theorem. Q.E.D.