Robust Confidence Intervals for the Bernoulli Parameter

Abstract

Despite the simplicity of the Bernoulli process, developing good confidence interval procedures for its parameter—the probability of success p—is deceptively difficult. The binary data yield a discrete number of successes from a discrete number of trials, n. This discreteness results in actual coverage probabilities that oscillate with the n for fixed values of p (and with p for fixed n). Moreover, this oscillation necessitates a large sample size to guarantee a good coverage probability when p is close to 0 or 1.

It is well known that the Wilson procedure is superior to many existing procedures because it is less sensitive to p than any other procedures, therefore it is less costly. The procedures proposed in this article work as well as the Wilson procedure when 0.1 ≤p ≤ 0.9, and are even less sensitive (i.e., more robust) than the Wilson procedure when p is close to 0 or 1. Specifically, when the nominal coverage probability is 0.95, the Wilson procedure requires a sample size 1, 021 to guarantee that the coverage probabilities stay above 0.92 for any 0.001 ≤ min {p, 1 −p} <0.01. By contrast, our procedures guarantee the same coverage probabilities but only need a sample size 177 without increasing either the expected interval width or the standard deviation of the interval width.

Keywords:

• Bernoulli parameter
• Coverage probability
• Expected interval width

Mathematics Subject Classification:

• Primary 62F25
• Secondary 65C05

Acknowledgments

This article is based upon work supported by the National Science Council under Grant No. NSC-97-2221-E-007-100-MY3. We thank professors Bruce Schmeiser, Dennis Engi, David Goldsman, and Raghu Pasupathy for their perceptive comments.

Notes

The last column indicates the row numbers.