8. For testing the hypothesis H: °= 00 (00 an interior point of 0) in the one-parameter exponential family of Section 2, let rc be the totality of tests satisfying (3) and (5) for some - 00 s C\ s C2 s 00 and 0 s 1\, 12 s 1. (i) rc is complete in the sense that given any level-a test 4'0 of H there exists 4' E rc such that 4' is uniformly at least as powerful as 4'0' (ii) If 4'\,4'2 E rc, then neither of the two tests is uniformly more powerful than the other. (iii) Let the problem be considered as a two-decision problem, with decisions do and

d, corresponding to acceptance and rejection of H, and with loss function L(O, d;) = L;(O), i = 0,1. Then rc is minimal essentially complete provided L\ (0) < Lo(0) for all °'* ° 0 , (iv) Extend the result of part (iii) to the hypothesis H': 0\ ° °2 , [(i): Let the derivative of the power function of cfIo at 00 be fJ~o(Oo) = p. Then there exists .p E "" such that p;(80 ) = p and .p is UMP among all tests satisfying this condition. (ii) : See Chapter 3, end of Section 7. (iii): See Chapter 3, proof of Theorem 3.]