Improved Buehler limits based on refined designated statistics

The Buehler 1-α upper confidence limit is as small as possible, subject to the constraints that (a) its coverage probability never falls below 1-α and (b) it is a non-decreasing function of a designated statistic. The designated statistic may have ties among its possible values. We prove that breaking such ties by a sufficiently small modification can never increase the Buehler limit. We also prove that, under commonly satisfied conditions, breaking ties by a sufficiently small modification will result in an improved, i.e. smaller Buehler limit. We conclude that designated statistics should not contain ties, apart from ties justified by symmetry requirements. In particular, this principle excludes the most commonly suggested designated statistics in the literature.