This is from a homework problem, but I have generalized my question so that it only conceptually applicable.
I am asked to calculate the expected length of a Uniformly Most Accurate (UMA) confidence interval.
The hypothesis being tested is $H_0: \theta=\theta_0$ vs $H_1: \theta<\theta_0$, and I am given a Uniformly Most Powerful (UMP) test with a rejection region of the form $P(T<c)=\alpha$.
My course texts state that I can use the acceptance region from the UMP test, $P(T\geq c)=1-\alpha$, to create a UMA confidence interval and, due to the form of the hypothesis, the interval will be from negative infinity to some upper limit, U, i.e.: $[-\infty,U)$.
It then asks me to calculate the expected length of the interval. My first thought is that the expected length is infinite, but that seemed too trivial. It also seems suspicious that a confidence interval of infinite length would be UMA. Am I misinterpreting what is meant by the expected length of a confidence interval?