Confidence Intervals for a binomial proportion and asymptotic expansions

Abstract

We address the classic problem of interval estimation of a binomial proportion. The Wald interval $\hat{p}\pm z_{\alpha/2} n^{-1/2} (\hat{p} (1 - \hat{p}))^{1/2}$ is currently in near universal use. We first show that the coverage properties of the Wald interval are persistently poor and defy virtually all conventional wisdom. We then proceed to a theoretical comparison of the standard interval and four additional alternative intervals by asymptotic expansions of their coverage probabilities and expected lengths.

The four additional interval methods we study in detail are the score-test interval (Wilson), the likelihood-ratio-test interval, a Jeffreys prior Bayesian interval and an interval suggested by Agresti and Coull. The asymptotic expansions for coverage show that the first three of these alternative methods have coverages that fluctuate about the nominal value, while the Agresti–Coull interval has a somewhat larger and more nearly conservative coverage function. For the five interval methods we also investigate asymptotically their average coverage relative to distributions for $p$ supported within (0 1) . In terms of expected length, asymptotic expansions show that the Agresti–Coull interval is always the longest of these. The remaining three are rather comparable and are shorter than the Wald interval except for $p$ near 0 or 1.

These analytical calculations support and complement the findings and the recommendations in Brown, Cai and DasGupta (Statist. Sci. (2001) 16 101–133).