Asymptotic infimum coverage probability for interval estimation of proportions

In this paper, we discuss asymptotic infimum coverage probability (ICP) of eight widely used confidence intervals for proportions, including the Agresti–Coull (A–C) interval (Am Stat 52:119–126, 1998) and the Clopper–Pearson (C–P) interval (Biometrika 26:404–413, 1934). For the A–C interval, a sharp upper bound for its asymptotic ICP is derived. It is less than nominal for the commonly applied nominal values of 0.99, 0.95 and 0.9 and is equal to zero when the nominal level is below 0.4802. The \(1-\alpha \) C–P interval is known to be conservative. However, we show through a brief numerical study that the C–P interval with a given average coverage probability \(1-\gamma \) typically has a similar or larger ICP and a smaller average expected length than the corresponding A–C interval, and its ICP approaches to \(1-\gamma \) when the sample size goes large. All mathematical proofs and R-codes for computation in the paper are given in Supplementary Materials.