  We study exact confidence intervals and two-sided hypothesis tests for
univariate parameters of stochastically increasing discrete distributions, such
as the binomial and Poisson distributions. It is shown that several popular
methods for constructing short intervals lack strict nestedness, meaning that
accepting a lower confidence level not always will lead to a shorter confidence
interval. These intervals correspond to a class of tests that are shown to
assign differing $p$-values to indistinguishable models. Finally, we show that
among strictly nested intervals, fiducial intervals, including the
Clopper-Pearson interval for a binomial proportion and the Garwood interval for
a Poisson mean, are optimal.
