• Abstract: We revisit the problem of interval estimation of a binomial proportion. The erratic behavior of the coverage probability of the standard Wald confidence interval has previously been remarked on in the literature (Blyth & Still (1983), Agresti & Coull (1998), Santner (1998), and others). We begin by showing that the chaotic coverage properties of the Wald interval are far more persistent than is appreciated. Furthermore, common textbook prescriptions regarding its safety are misleading and defective in several respects and cannot be trusted.


    This leads us to consideration of alternative intervals. A number of natural alternatives are presented, each with its motivation and context. Each interval is examined as regards its coverage probability and its length. Based on this analysis, we recommend the Wilson interval (Wilson (1927)) or the equal tailed Jeffreys prior interval for small n, and the interval suggested in Agresti and Coull (1998) for larger n. We also provide an additional frequentist justification for use of the Jeffreys interval.


  • Paper with discussion: pdf file.


  • Other related papers:

    Brown, L.D., Cai, T. & DasGupta, A. (2002).
    Confidence intervals for a binomial proportion and asymptotic expansions.
    The Annals of Statistics 30, 160-201.

    Brown, L.D., Cai, T. & DasGupta, A. (2003).
    Interval estimation in exponential families.
    Statistica Sinica 13 , 19-49.

    Cai, T. (2005).
    One-sided confidence intervals in discrete distributions.
    J. Statistical Planning and Inference 131, 63-88.


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