Abstract
The problems of constructing confidence intervals (CIs) for a proportion, prediction intervals (PIs) for a future sample size in a negative binomial sampling to observe a specified number of successes and tolerance intervals (TIs) for negative binomial distributions are considered. For interval estimating the success probability, we propose CIs based on the fiducial approach and the score method, evaluate them and compare them with available CIs with respect to coverage probability and precision. We propose PIs based on the fiducial approach and joint sampling approach, and compare them with the exact and other approximate PIs. We also propose TIs on the basis of our new CIs and evaluate them with respect to coverage probability and expected width. All three statistical intervals are illustrated using two examples with real data.
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References
Agresti A, Coull BA (1988) Approximate is better than “exact" for interval estimation of binomial proportion. Am Stat 52:119–125
Brown PJ (1982) Multivariate calibration. J R Stat Soc B 44:287–321
Brown LD, Cai T, Das Gupta A (2001) Interval estimation for a binomial proportion (with discussion). Stat Sci 16:101–133
Cai TT, Wang H (2009) Tolerance intervals for discrete distributions in exponential families. Stat Sin 19:905–923
Casella G, Berger RL (2001) Statistical inference, 2nd edn. Duxbury, Belmont
Clemans KG (1959) Confidence limits in the case of the geometric distribution. Biometrika 46:260–264
Dunsmore IR (1976) A note on Faulkenberry’s method of obtaining prediction intervals. J Am Stat Assoc 71:193–194
Fisher RA (1935) The fiducial argument in statistical inference. Ann Eugen 6(4):391–398
George VT, Elston RC (1993) Confidence limits based on the first occurrence of an event. Stat Med 11:685–690
Hahn GJ, Chandra R (1981) Tolerance intervals for Poisson and binomial random variables. J Qual Technol 13:100–110
Hahn GJ, Meeker WQ (1991) Statistical intervals. Wiley, Hoboken
Haldane JBS (1945) On a method of estimating frequencies. Biometrika 33:222–225
Hannig J (2009) On generalized fiducial inference. Stat Sin 19:491–544
Hilbe JM (2011) Negative binomial regression, 2nd edn. University Press, Cambridge
Johnson NL, Kemp AW, Kotz S (2005) Univariate discrete distributions, 3rd edn. Wiley, Hoboken
Kikuchi DA (1987) Inverse sampling in case control studies involving a rare exposure. Biom J 29:243–246
Knüsel L (1994) The prediction problem as the dual form of the two-sample problem with applications to the Poisson and the binomial distribution. Am Stat 48:214–219
Krishnamoorthy K, Lee M (2010) Inference for functions of parameters in discrete distributions based on fiducial approach: binomial and Poisson cases. J Stat Plan Inference 140:1182–1192
Krishnamoorthy K, Mathew T (2009) Statistical tolerance regions: theory, applications, and computation. Wiley, Hoboken
Krishnamoorthy K, Peng J (2011) Improved closed-form prediction intervals for binomial and Poisson distributions. J Stat Plan Inference 141:1709–1718
Krishnamoorthy K, Xia Y, Xie F (2011) A simple approximate procedure for constructing binomial and Poisson tolerance intervals. Commun Stat 40:2443–2458
Lui KJ (1995) Confidence limits for the population prevalence rate based on the negative binomial distribution. Stat Med 14:1471–1477
Madden LV, Hughes G, Munkvold GP (1996) Plant disease incidence: inverse sampling, sequential sampling, and confidence intervals when observed mean incidence is zero. Crop Prot 15:621–632
Mathew T, Young D (2013) Fiducial-based tolerance intervals for some discrete distributions. Comput Stat Data Anal 61:38–49
Patil GP (1960) On the evaluation of the negative binomial distribution with examples. Technometrics 2:501–505
Shilane D, Evans SN, Hubbard AE (2010) Confidence intervals for negative binomial random variables of high dispersion. Int J Biostat Article10 6:1–10
Thatcher AR (1964) Relationships between Bayesian and confidence limits for prediction. J Roy Stat Soc B 26:176–192
Thulin M, Zwanzig S (2017) Exact confidence intervals and hypothesis tests for parameters of discrete distributions. Bernoulli 23:479–502
Tian M, Tang ML, Ng HKT, Chan PS (2009) A comparative study of confidence intervals for negative binomial proportions. J Stat Comput Simul 79:241–249
Wald A (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Am Math Soc 54:426–482
Wang H, Tsung F (2009) Tolerance intervals with improved coverage probabilities for binomial and Poisson variables. Technometrics 51:25–33
Wang CM, Hannig J, Iyer HK (2012) Fiducial prediction intervals. J Stat Plan Inference 142:1980–1990
Young D (2014) A procedure for approximate negative binomial tolerance intervals. J Stat Comput Simul 84:438–450
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The authors are grateful to three reviewers for providing valuable comments, suggestions and references relevant to this article.
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Appendix
Appendix
R code to compute HPM-PI
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Dang, BA., Krishnamoorthy, K. Confidence intervals, prediction intervals and tolerance intervals for negative binomial distributions. Stat Papers 63, 795–820 (2022). https://doi.org/10.1007/s00362-021-01255-y
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DOI: https://doi.org/10.1007/s00362-021-01255-y