Let X and Y be independently distributed according to oneparameter exponential families, so that their joint distribution is given by dPθ1,θ2 (x, y) = C(θ1)e

θ1T (x) dµ(x)K(θ2)e

θ2U(y) dν(y).

Suppose that with probability 1 the statistics T and U each take on at least three values and that

(a,

b) is an interior point of the natural parameter space. Then a UMP unbiased test does not exist for testing H : θ1 =

a, θ2 = b against the alternatives θ1 = a or θ2 = b.

13

[The most powerful unbiased tests against the alternatives θ1 =

a, θ2 = b have acceptance regions C1 < T(x) < C2 and K1 < U(y) < K2 respectively. These tests are also unbiased against the wider class of alternatives K : θ1 = a or θ2 = b or both.]