The Power of Confidence Intervals

We consider the power to reject false values of the parameter in Frequentist methods for the calculation of confidence intervals. We connect the power with the physical significance (reliability) of confidence intervals for a parameter bounded to be non-negative. We show that the confidence intervals (upper limits) obtained with a (biased) method that near the boundary has large power in testing the parameter against larger alternatives and small power in testing the parameter against smaller alternatives are physically more significant. Considering the recently proposed methods with correct coverage, we show that the physical significance of upper limits is smallest in the Unified Approach and highest in the Maximum Likelihood Estimator method. We illustrate our arguments in the specific cases of a bounded Gaussian distribution and a Poisson distribution with known background.