Paradoxes and Improvements in Interval Estimation

Cases where confidence sets are empty or include every possible parameter value are an embarrassment to standard theory and difficult to explain to students. To alleviate this problem, and as a convenient way of showing how each parameter value is ranked in view of the data, we propose to plot the confidence set for each possible level, called a confidence curve. Different confidence procedures then lead to different confidence curves. In standard situations involving distributions with a monotone likelihood ratio, we suggest using the confidence curve based on Spjøtvoll's acceptability function. For discrete distributions, this leads to an improvement over usual “exact” confidence intervals.