The Accuracy of Approximate Intervals for a Binomial Parameter

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Abstract
This article considers the uniform convergence of the coverage probabilities of some approximate confidence intervals for the binomial parameter p, induced by central limit arguments. A uniform upper bound on the coverage probabilities of any such interval obtained by transformation of the sample proportion [pcirc] n is derived. The coverage probability of the interval induced by the arcsine transformation turns out to be very close to the upper bound. Replacing [pcirc] n by the Bayes estimate (X + β)/(n + 2β) for some β, however, gives even better uniform asymptotic properties; the choice β = z α 2/2 is recommended, where z α is the (1 − α)th quantile of the standard normal distribution. With such β, numerical results show that this interval is satisfactory, even in the uniform sense, since it possesses uniform asymptotic coefficients that are very close to the nominal ones. Furthermore, the best k for β of the form β = kz α 2 such that the uniform asymptotic coefficient is as close to the nominal one as possible is found for 1 − 2α = .90, .95, and .98. Simulation results for the most commonly used nominal confidence coefficients are consistent with the limiting results.

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