Electronic Journal of Statistics

The cost of using exact confidence intervals for a binomial proportion

Måns Thulin

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When computing a confidence interval for a binomial proportion $p$ one must choose between using an exact interval, which has a coverage probability of at least $1-\alpha$ for all values of $p$, and a shorter approximate interval, which may have lower coverage for some $p$ but that on average has coverage equal to $1-\alpha$. We investigate the cost of using the exact one and two-sided Clopper–Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper–Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper–Pearson interval and Bayesian intervals based on noninformative priors.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 817-840.

First available in Project Euclid: 16 June 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F25: Tolerance and confidence regions
Secondary: 62F12: Asymptotic properties of estimators

Asymptotic expansion binomial distribution confidence interval expected length sample size determination proportion


Thulin, Måns. The cost of using exact confidence intervals for a binomial proportion. Electron. J. Statist. 8 (2014), no. 1, 817--840. doi:10.1214/14-EJS909. https://projecteuclid.org/euclid.ejs/1402927499

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