## The Annals of Statistics

### Confidence Intervals for a binomial proportion and asymptotic expansions

#### Abstract

We address the classic problem of interval estimation of a binomial proportion. The Wald interval $\hat{p}\pm z_{\alpha/2} n^{-1/2} (\hat{p} (1 - \hat{p}))^{1/2}$ 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), the likelihood-ratio-test interval, a Jeffreys prior Bayesian interval and an interval suggested by Agresti and Coull. 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 (Statist. Sci. (2001) 16 101–133).

#### Article information

Source
Ann. Statist., Volume 30, Number 1 (2002), 160-201.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aos/1015362189

Digital Object Identifier
doi:10.1214/aos/1015362189

Mathematical Reviews number (MathSciNet)
MR1892660

Zentralblatt MATH identifier
1012.62026

#### Citation

Brown, Lawrence D.; Cai, T. Tony; DasGupta, Anirban. Confidence Intervals for a binomial proportion and asymptotic expansions. Ann. Statist. 30 (2002), no. 1, 160--201. doi:10.1214/aos/1015362189. https://projecteuclid.org/euclid.aos/1015362189

#### References

• AGRESTI, A. and COULL, B. A. (1998). Approximate is better than "exact" for interval estimation of binomial proportions. Amer. Statist. 52 119-126.
• BARNDORFF-NIELSEN, O. E. and COX, D. R. (1989). Asymptotic Techniques for Use in Statistics. Chapman and Hall, London.
• BERGER, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
• BHATTACHARYA, R. N. and RANGA RAO, R. (1976). Normal Approximation and Asymptotic Expansions. Wiley, New York.
• BLYTH, C. R. and STILL, H. A. (1983). Binomial confidence intervals. J. Amer. Statist. Assoc. 78 108-116.
• BREIMAN, L. (1992). Probability. SIAM, Philadelphia.
• BROWN, L. D., CAI, T. and DASGUPTA, A. (2000). Interval estimation in exponential families. Technical report, Dept. Statistics, Univ. Pennsylvania. Available at www-stat.wharton. upenn.edu/ tcai/.
• BROWN, L. D., CAI, T. and DASGUPTA, A. (2001). Interval estimation for a binomial proportion (with discussion). Statist. Sci. 16 101-133.
• BROWN, L. D., CASELLA, G. and HWANG, J. T. G. (1995). Optimal confidence sets, bioequivalence, and the limacon of Pascal. J. Amer. Statist. Assoc. 90 880-889.
• CASELLA, G., HWANG, J. T. G. and ROBERT, C. P. (1994). Loss functions for set estimation. In Statistical Decision Theory and Related Topics V (S. S. Gupta and J. Berger, eds.) 237- 251. Academic Press, New York.
• CLOPPER C. J. and PEARSON, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26 404-13.
• CROW, E. L. (1956). Confidence intervals for a proportion. Biometrika 43 423-35.
• ESSEEN, C. G. (1945). Fourier analysis of distribution functions: a mathematical study of the Laplace-Gaussian law. Acta Math. 77 1-125.
• GHOSH, B. K. (1979). A comparison of some approximate confidence intervals for the binomial parameter J. Amer. Stat. Assoc. 74 894-900.
• GHOSH, J. K. (1994). Higher Order Asymptotics. IMS, Hayward, CA.
• HALL, P. (1982). Improving the normal approximation when constructing one-sided confidence intervals for binomial or Poisson parameters. Biometrika 69 647-52.
• HALL, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
• JOHNSON, N. L., KOTZ, S. and BALAKRISHNAN, N. (1995). Continuous Univariate Distributions 2, 2nd ed. Wiley, New York.
• JOHNSON, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Statist. 41 851-64.
• PIERCE, D. A. and PETERS, D. (1992). Practical use of higher order asymptotics for multiparameter exponential families. J. Roy. Statist. Soc. Ser. B 54 701-725.
• RAO, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York.
• SANTNER, T. J. (1998). Teaching large-sample binomial confidence intervals. Teaching Statistics 20 20-23.
• SCHADER, M. and SCHMID, F. (1990). Charting small sample characteristics of asymptotic confidence intervals for the binomial parameter p. Statist. Papers 31 251-264.
• STERNE, T. E. (1954). Some remarks on confidence or fiducial limits. Biometrika 41 275-278.
• WASSERMAN, L. (1991). An inferential interpretation of default priors. Technical report, Dept. Statistics, Carnegie Mellon Univ.
• WILSON, E. B. (1927). Probable inference, the law of succession, and statistical inference. J. Amer. Statist. Assoc. 22 209-212.
• WOODROOFE, M. (1986). Very weak expansions for sequential confidence levels. Ann. Statist. 14 1049-1067.
• PHILADELPHIA, PENNSYLVANIA 19104 E-MAIL: lbrown@stat.wharton.upenn.edu tcai@stat.wharton.upenn.edu DEPARTMENT OF STATISTICS PURDUE UNIVERSITY
• WEST LAFAYETTE, INDIANA 47907 E-MAIL: dasgupta@stat.purdue.edu