Locally most powerful tests. 6 Let d be a measure of the distance of an alternative θ from a given hypothesis H. A level-α test ϕ0 is said to be locally most powerful (LMP) if, given any other level-α test ϕ, there exists  such that

βϕ0 (θ ) ≥ βϕ(θ ) for all θ with 0 < d(θ ) < . (8.44)

Suppose that θ is real-valued and that the power function of every test is continuously differentiable at θ0.

(i) If there exists a unique level-α test ϕ0 of H : θ = θ0, maximizing β

ϕ(θ0), then

ϕ0 is the unique LMP level-α test of H against θ>θ0 for d(θ ) = θ − θ0.

(ii) To see that (i) is not correct without the uniqueness assumption, let X take on the values 0 and 1 with probabilities Pθ (0) = 1 2 − θ 3, Pθ (1) = 1 2 + θ 3, −1 2 <

θ 3 < 1 2 , and consider testing H : θ = 0 against K : θ > 0. Then every test ϕ

of size α maximizes β

ϕ(0), but not every such test is LMP. [Kallenberg et al.

(1984).]

(iii) The following7 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 Pθ (x) = α +

θ + θ 2 sin  x θ
 for x = 1, 2, Pθ (0) = 1 − pθ (1) − pθ (2), where −1 ≤ θ ≤ 1 and is a sufficiently small number. Then a test ϕ at level α maximizes β
(0) provided ϕ(1) + ϕ(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 α for every set of alternatives which is bounded away from H.
(v) Let X1,..., Xn be a sample from a Cauchy distribution with unknown location parameter θ, so that the joint density of the X’s is π−n n i=1[1 + (xi − θ )2]
−1.
The LMP test for testing θ = 0 against θ > 0 at level α < 1 2 is not unbiased and hence does not maximize the minimum power locally.
[(iii): The unique most powerful test against θ is 
ϕ(1)
ϕ(2) = 1 if sin 1 θ

>=
, and each of these inequalities holds at values of θ arbitrarily close to 0.
(v): There exists M so large that any point with xi ≥ 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 θ tends to infinity.]