For testing the hypothesis H : 1 2(1 2) against the alternatives

For testing the hypothesis H : θ1 ≤ θ ≤ θ2(θ1 ≤ θ2) against the alternatives θ < θ1 or θ > θ2, or the hypothesis θ = θ0 against the alternatives

θ = θ0, in an exponential family or more generally in a family of distributions satisfying the assumptions of Problem 3.58, a UMP test does not exist.

[This follows from a consideration of the UMP tests for the one-sided hypotheses H1 : θ ≥ θ1 and H2 : θ ≤ θ2.]