For testing the hypothesis H : 1 2(1 2) against the alternatives 2, or the hypothesis = 0 against

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 3.53, 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.]