Locally most powerful tests. 5 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) < ∆. (8.38)

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 − θ3, Pθ(1) = 1 2 + θ3,

−1 2 < θ3 < 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 following6 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 + θ2 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



>=< sin 2

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.]