Construction of Confidence Intervals and Regions for Ordered Binomial Probabilities

In biomedical studies and other areas, there are often situations where parameters are known to be ordered. in these situations, incorporating the order restriction can produce much more efficient estimates than ignoring it. There is much research on point estimation and tests with order restrictions, but little on the construction of confidence intervals. Our particular interest is in the case where two probabilities for binomial random variables can be equal or very close to each other, where difficulty arises and the standard methods for inference no longer apply. We investigate methods for constructing confidence intervals for the ordered probabilities based on appropriate asymptotic distributions and several versions of the bootstrap. Via simulation studies we find that the usual percentile bootstrap and a parametric bootstrap with parameter shrunk to the boundary both have good finite sample properties. We further consider the construction of confidence regions for two ordered probabilities. We propose a small sample test for the probabilities and a method for constructing confidence regions by inverting this test, which yields confidence regions with good coverage rates even in very small samples. Supplemental materials for the technical results and proofs are available online.