• Bernoulli
  • Volume 23, Number 1 (2017), 479-502.

Exact confidence intervals and hypothesis tests for parameters of discrete distributions

Måns Thulin and Silvelyn Zwanzig

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We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing $p$-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, including the Clopper–Pearson interval for a binomial proportion and the Garwood interval for a Poisson mean, are optimal.

Article information

Bernoulli, Volume 23, Number 1 (2017), 479-502.

Received: December 2014
Revised: March 2015
First available in Project Euclid: 27 September 2016

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binomial distribution confidence interval expected length fiducial interval hypothesis test Poisson distribution


Thulin, Måns; Zwanzig, Silvelyn. Exact confidence intervals and hypothesis tests for parameters of discrete distributions. Bernoulli 23 (2017), no. 1, 479--502. doi:10.3150/15-BEJ750.

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  • [1] Agresti, A. (2003). Dealing with discreteness: Making “exact” confidence intervals for proportions, differences of proportions, and odds ratios more exact. Stat. Methods Med. Res. 12 3–21.
  • [2] Agresti, A. and Min, Y. (2001). On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57 963–971.
  • [3] Birnbaum, A. (1961). Confidence curves: An omnibus technique for estimation and testing statistical hypotheses. J. Amer. Statist. Assoc. 56 246–249.
  • [4] Blaker, H. (2000). Confidence curves and improved exact confidence intervals for discrete distributions. Canad. J. Statist. 28 783–798.
  • [5] Blyth, C.R. and Still, H.A. (1983). Binomial confidence intervals. J. Amer. Statist. Assoc. 78 108–116.
  • [6] Bol’šev, L.N. (1965). On the construction of confidence limits. Teor. Verojatnost. i Primenen. 10 187–192.
  • [7] Brown, L.D., Cai, T.T. and DasGupta, A. (2001). Interval estimation for a binomial proportion. Statist. Sci. 16 101–133.
  • [8] Byrne, J. and Kabaila, P. (2005). Comparison of Poisson confidence intervals. Comm. Statist. Theory Methods 34 545–556.
  • [9] Casella, G. (1986). Refining binomial confidence intervals. Canad. J. Statist. 14 113–129.
  • [10] Casella, G. and Robert, C. (1989). Refining Poisson confidence intervals. Canad. J. Statist. 17 45–57.
  • [11] Clopper, C.J. and Pearson, E.S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26 404–413.
  • [12] Cohen, A. and Strawderman, W.E. (1973). Admissibility implications for different criteria in confidence estimation. Ann. Statist. 1 363–366.
  • [13] Crow, E.L. (1956). Confidence intervals for a proportion. Biometrika 43 423–435.
  • [14] Crow, E.L. and Gardner, R.S. (1959). Confidence intervals for the expectation of a Poisson variable. Biometrika 46 441–453.
  • [15] Fay, M.P. (2010). Confidence intervals that match Fisher’s exact or Blaker’s exact tests. Biostatistics 11 373–374.
  • [16] Fay, M.P. (2010). Two-sided exact tests and matching confidence intervals for discrete data. R Journal 2 53–58.
  • [17] Fisher, R.A. (1930). Inverse probability. Proc. Camb. Philos. Soc. 26 528–535.
  • [18] Garwood, F. (1936). Fiducial limits for the Poisson distribution. Biometrika 28 437–442.
  • [19] Göb, R. and Lurz, K. (2014). Design and analysis of shortest two-sided confidence intervals for a probability under prior information. Metrika 77 389–413.
  • [20] Hirji, K.F. (2006). Exact Analysis of Discrete Data. Boca Raton, FL: Chapman & Hall/CRC.
  • [21] Kabaila, P. and Byrne, J. (2001). Exact short Poisson confidence intervals. Canad. J. Statist. 29 99–106.
  • [22] Lecoutre, B. and Poitevineau, J. (2014). New results for computing Blaker’s exact confidence interval for one parameter discrete distributions. Comm. Statist. Simulation Comput. DOI:10.1080/03610918.2014.911900.
  • [23] Liese, F. and Miescke, K.-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer Series in Statistics. New York: Springer.
  • [24] Newcombe, R.G. (2011). Measures of location for confidence intervals for proportions. Comm. Statist. Theory Methods 40 1743–1767.
  • [25] Reiczigel, J. (2003). Confidence intervals for the binomial parameter: Some new considerations. Stat. Med. 22 611–621.
  • [26] Schilling, M.F. and Doi, J.A. (2014). A coverage probability approach to finding an optimal binomial confidence procedure. Amer. Statist. 68 133–145.
  • [27] Sommerville, M.C. and Brown, R.S. (2013). Exact likelihood ratio and score confidence intervals for the binomial proportion. Pharmaceutical Statistics 12 120–128.
  • [28] Sterne, T.E. (1954). Some remarks on confidence or fiducial limits. Biometrika 41 275–278.
  • [29] Thulin, M. (2014). On split sample and randomized confidence intervals for binomial proportions. Statist. Probab. Lett. 92 65–71.
  • [30] Thulin, M. (2014). Coverage-adjusted confidence intervals for a binomial proportion. Scand. J. Stat. 41 291–300.
  • [31] Thulin, M. (2014). The cost of using exact confidence intervals for a binomial proportion. Electron. J. Stat. 8 817–840.
  • [32] Vos, P.W. and Hudson, S. (2005). Evaluation criteria for discrete confidence intervals: Beyond coverage and length. Amer. Statist. 59 137–142.
  • [33] Vos, P.W. and Hudson, S. (2008). Problems with binomial two-sided tests and the associated confidence intervals. Aust. N. Z. J. Stat. 50 81–89.
  • [34] Wang, W. (2006). Smallest confidence intervals for one binomial proportion. J. Statist. Plann. Inference 136 4293–4306.
  • [35] Wang, W. (2014). Exact optimal confidence intervals for hypergeometric parameters. J. Amer. Statist. Assoc. To appear. DOI:10.1080/01621459.2014.966191.
  • [36] Wang, Y.H. (2000). Fiducial intervals: What are they? Amer. Statist. 54 105–111.
  • [37] Xie, M.-g. and Singh, K. (2013). Confidence distribution, the frequentist distribution estimator of a parameter: A review. Int. Stat. Rev. 81 3–39.