51. Let Xl " ' " X, be i.i.d. with density Po or PI' so that the MP level-a test of H : Po rejects when Il;'= I r( Xi) c,,, where r( X,) = PI ( X,)/Po( Xi)' or equivalently when (34) 1 In O)ogr(x,) - Eo[Jogr(x,)]} i ; (i) It follows from the central limit theorem (Chapter 5, Theorem 3) that under H the left side of (34) tends in law to N(O, 0 2) with 0 2 = Varo[log r(X,)] provided 0 2 < 00 . (ii) From (i) it follows that k; --+ OUa where cII(ua ) = 1 -

a. (iii) The power of the test (34) agaisnt PI tends to 1 as n --+ 00. [(iii): Problem 14(iv).] ]