A level- test 0 is locally unbiased (loc. unb.) if there exists 0 > 0 such that

A level-α test ϕ0 is locally unbiased (loc. unb.) if there exists 0 > 0 such that βϕ0 (θ ) ≥ α for all θ with 0 < d(θ ) < 0; it is LMP loc. unb. if it is loc.

unb. and if, given any other loc. unb. level-α test ϕ, there exists  such that (8.44)

holds. Suppose that θ is real-valued and that d(θ ) = |θ − θ0|, and that the power function of every test is twice continuously differentiable at θ = θ0.

(i) If there exists a unique test ϕ0 of H : θ = θ0 against K : θ = θ0 which among all loc. unb. tests maximizes β(θ0), then ϕ0 is the unique LMP loc. unb. level-α

test of H against K.

(ii) The test of part (i) maximizes the minimum power locally provided its power function is bounded away from α for every set of alternatives that is bounded away from H.

[(ii): A necessary condition for a test to be locally minimax is that it is loc. unb.]