Confidence Intervals for a Binomial Proportion and Asymptotic Expansions
Lawrence Brown, Tony Cai, and Anirban DasGupta
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Abstract:
We address the classic problem of interval estimation of a binomial
proportion. The Wald interval 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 (1927)) the likelihood-ratio-test interval, a Jeffreys prior Bayesian interval and an interval suggested in Agresti and Coull (1998). 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 (1999).
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pdf file.
- Other related papers:
Brown, L.D., Cai, T. & DasGupta, A. (2001).
Interval estimation for a binomial proportion (with discussion).
Statistical Science 16, 101-133.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.