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, <CitationRef CitationID="CR2">1998</CitationRef>) and the Clopper–Pearson (C–P) interval (Biometrika 26:404–413, <CitationRef CitationID="CR5">1934</CitationRef>). 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$1-\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo>-</mo> <mi mathvariant="italic">α</mi> </mrow> </math> </EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$1-\gamma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo>-</mo> <mi mathvariant="italic">γ</mi> </mrow> </math> </EquationSource> </InlineEquation> typically has a similar or larger ICP and a smaller average expected length than the corresponding A–C interval, and its ICP approaches to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$1-\gamma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>1</mn> <mo>-</mo> <mi mathvariant="italic">γ</mi> </mrow> </math> </EquationSource> </InlineEquation> when the sample size goes large. All mathematical proofs and R-codes for computation in the paper are given in Supplementary Materials. Copyright Springer-Verlag Berlin Heidelberg 2014