7. Let (X, Y) be distributed according to the exponential family dP8r-82( X, y) = C( °1 , 02)e8.x+82Y d",(x, y). The only unbiased test for testing H : 01 s

a, 02 s b against K : 01 > a or 02 > b or both is t/I(x, y) == a . "For counterexamples when the conditions of the problem are not satisfied, see Kallenberg (1984). [Take a = b = 0, and let fJ(°\,°2 ) be the power function of any level-a test. Unbiasedness implies fJ(O, ° 2) = a for 02 < 0 and hence for all ° 2, since fJ(O, 02)is an analytic function of 02 ' For fixed 02 > 0, fJ(O\, 02) considered as a function of 0\ therefore has a minimum at 0\ = 0, so that afJ(0\,02)/ao\ vanishes at 0\ = 0 for all positive ° 2 , and hence for all ° 2 , By considering alternatively positive and negative values of 02 and using the fact that the partial derivatives of all orders of fJ(O\, 02) with respect to 0\ are analytic, one finds that for each fixed 02 these derivatives all vanish at 0\ = 0 and hence that the function fJ must be a constant. Because of the completeness of (X, Y), fJ(O\, 02) == a implies 4'(x, y) == a.]