A “Paradox” in Confidence Interval Construction Using Sufficient Statistics

Statistical inference about parameters should depend on raw data only through sufficient statistics—the well known sufficiency principle. In particular, inference should depend on minimal sufficient statistics if these are simpler than the raw data. In this article, we construct one-sided confidence intervals for a proportion which: (i) depend on the raw binary data, and (ii) are uniformly shorter than the smallest intervals based on the binomial random variable—a minimal sufficient statistic. In practice, randomized confidence intervals are seldom used. The proposed intervals violate the aforementioned principle if the search of optimal intervals is restricted within the class of nonrandomized confidence intervals. Similar results occur for other discrete distributions