2. Locally most powerful tests. Let d be a measure of the distance of an alternative 8 from a given hypothesis H. A level-a test CPo is said to be locally most powerful (LMP) if, given any other level-a test cP , there exists A such that (37) P"'o( 8) P",( 8) for all 8 with 0 < d(8) < A. Suppose that 8 is real-valued and that the power function of every test is continuously differentiable at 80 ,

(i) If there exists a unique level-a test CPo of H : (J = (Jo maximizingP; «(Jo), then CPo is the unique LMP level-a test of H against (J > (Jo for d«(J) ... (J - (Jo . (ii) To see that (i) is not correct without the uniqueness assumption, let X take on the values 0 and 1 with probabilities Po(O) = t - (J3, Po(l) ... t + (J3, - t < (J3 < t, and consider testing H: (J = 0 against K : () > O. Then every test cp of size a maximizes P;(O), but not every such test is LMP. [Kallenberg et al. (1984).] (iii) The following- is another counterexample to (i) without uniqueness, in which in fact no LMP test exists. Let X take on the values 0,1 ,2 with probabilities Po ( x) = a + ([() + () 2Sin(i)] Po(O) = 1 - po(l) - Po(2) , for x = 1,2, where -1 S () s 1 and ( is a sufficientlysmall number. Then a test cp at level a maximizes /3'(0) provided cp(l) + cp(2) = 1; but no LMP test exists. (iv) A unique LMP test maximizes the minimum power locally provided its power function is bounded away from a for every set of alternatives which is bounded away from H. (v) Let Xl "'" Xn be a sample from a Cauchy distribution with unknown location parameter (J, so that the joint density of the X's is 'l'T- nn 7_dl + (Xi - (J)2 r 1. The LMP test for testing (J = 0 against (J > 0 at level a < t is not unbiased and hence does not maximize the minimum power locally. [(iii): The unique most powerful test against (J is { cp(l) cp(2) = 1 if Sin( ) sin( ) , and each of these inequalities holds at values of (J arbitrarily close to O. (v): There exists M so large that any point with X i M for all i = 1, .. . , n lies in the acceptance region of the LMP test. Hence the power of the test tends to zero as (J tends to infinity.]