Document Type : Original Paper


1 1 Department of Mathematics and Statistics, Thammasat University, Pathumthani, 12120 Thailand

2 2 Department of Mathematics, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun, 67000 Thailand


In a number of real-world situations, one encounters count data with over-dispersion such that the typical Poisson distribution does not suit the data. In the current situation, it is appropriate to employ a combination of mixed Poisson and Poisson-Sujatha (PS) distributions. The PS distribution has been investigated for count data, which is of primary interest to a number of disciplines, including biology, medicine, demography, and agriculture. However, no research has been conducted regarding generating bootstrap confidence intervals for its parameter. The coverage probabilities and average lengths of bootstrap confidence intervals derived from the percentile, basic, and biased-corrected and accelerated bootstrap methods were compared using Monte Carlo simulation. The results indicated that it was impossible to achieve the nominal confidence level using bootstrap confidence intervals for tiny sample sizes, regardless of the other settings. Furthermore, when the sample size was large, there was not much of a difference in the performance of the several bootstrap confidence intervals. The bias-corrected and accelerated bootstrap confidence interval demonstrated superior performance compared to the other methods in all of the cases examined. Moreover, the effectiveness of the bootstrap confidence intervals was proven through their application to agricultural data sets. The calculations offer significant evidence in favor of the suggested bootstrap confidence intervals.


Main Subjects

  1. Andrew FS, Michael RW. Practical business statistics. 8th ed. San Diego: Academic Press; 2022.
  2. Siegel AF. Practical business statistics. 7th ed. London: Academic Press; 2017.
  3. Hougaard P, Lee M-LT, Whitmore GA. Analysis of overdispersed count data by mixtures of Poisson variables and Poisson processes. Biometrics. 1997;53(4):1225-1238.
  4. Ong S-H, Low Y-C, Toh K-K. Recent developments in mixed Poisson distributions. ASM Sci J. 2021.
  5. McElduff F.C. Models for discrete epidemiological and clinical data [PhD Thesis]. London: University College London; 2012.
  6. Tharshan R, Wijekoon P. A new mixed Poisson distribution for over-dispersed count data: theory and applications. Reliab: Theory Appl. 2022;17(1):33-51.
  7. Shanker R. On Poisson-Sujatha distribution and its applications to model count data from biological sciences. Biom Biostat Int J. 2016;3(4):1-8.
  8. Sankaran M. The discrete Poisson-Lindley distribution. Biometrics. 1970;1926(1):145-149.
  9. Shanker R. Sujatha distribution and its applications. Stat Transit New Series. 2016;17(3):391-410.
  10. Lindley DV. Fiducial distributions and Bayes’ theorem. J R Stat Soc Series B. 1958;20(1):102-107.
  11. Shanker R. Akash distribution and its applications. Int J Prob Stat. 2015;4(3):65-75.
  12. Tan SH, Tan SB. The correct interpretation of confidence intervals. Proc Singapore Healthc. 2010;19(3):276-278.
  13. Wood M. Statistical inference using bootstrap confidence intervals. Signif. 2004;1(4):180-182.
  14. Chernick MR, LaBudde RA. An introduction to bootstrap methods to R. 1st ed. Singapore: John Wiley & Sons; 2011.
  15. Reiser M, Yao L, Wang X, Wilcox J, Gray S. A Comparison of bootstrap confidence intervals for multi-level longitudinal data using Monte-Carlo simulation. In: Chen DG, Chen J. (eds) Monte-Carlo simulation-based statistical modeling. Springer; 2017.
  16. Flowers-Cano RS, Ortiz-Gómez R, León-Jiménez JE, Rivera RL, Cruz LAP. Comparison of bootstrap confidence intervals using Monte Carlo simulations. Water. 2018;10(2).
  17. Mostajeran A, Iranpanah N, Noorossana R. A new bootstrap based algorithm for Hotelling’s T2 multivariate control chart. J Sci I R I. 2016;27(3):269-278.
  18. Henningsen A, Toomet O. maxLik: a package for maximum likelihood estimation in R. Comput Stat. 2011;26(3):443-458.
  19. Ihaka R, Gentleman R. R: a language for data analysis and graphics. J Comput Graph Stat. 1996;5(3):299-314.
  20. Meeker WQ, Hahn GJ, Escobar LA. Statistical intervals: a guide for practitioners and researchers. 2nd ed. New Jersey: John Wiley and Sons; 2017.
  21. Canty A, Ripley B. boot: bootstrap R (S-Plus) functions. R package version 1.3-28.1, 2022.
  22. Efron B. The jackknife, the bootstrap, and other resampling plans, in CBMS-NSF regional conference series in applied mathematics, Philadelphia: SIAM; 1982.
  23. Efron B, Tibshirani RJ. An introduction to the bootstrap. 1st ed. New York: Chapman and Hall; 1993.
  24. Efron B. Better bootstrap confidence intervals. J Am Stat Assoc. 1987;82(297):171-185.
  25. Davison AC, Hinkley DV. Bootstrap methods and their application. 1st ed. Cambridge: Cambridge University Press; 1997.
  26. Ukoumunne OC, Davison AC, Gulliford MC, Chinn S. Non-parametric bootstrap confidence intervals for the intraclass correlation coefficient. Stat Med. 2003;22(24):3805-3821.
  27. Bliss CI, Fisher RA. Fitting the negative binomial distribution to biological data. Biometrics. 1953;9(2):176-200.
  28. Turhan NS. Karl Pearson’s chi-square tests. Educ Res Rev. 2020;15(9):575-580.
  29. McGuire JU, Brindley TA, Bancroft TA. The distribution of european corn borer larvae pyrausta nubilalis (Hbn.), in field corn. Biometrics. 1957;13(1):65-78.
  30. Kostyshak S. bootstrap: functions for the book “An introduction to the bootstrap”. R package version, 2019.6, 2022.
  31. Murphy MV. semEff: automatic calculation of effects for piecewise structural equation models. R package version 0.6.1, 2022.
  32. Kirby KN, Gerlanc D. BootES: an R package for bootstrap confidence intervals on effect sizes. Behav Res Methods. 2013;45(4):905-927.