Bootstrap Confidence Intervals for the Parameter of the Poisson-Garima Distribution:  A Case Study of the Number of Thunderstorms in the USA

Wararit Panichkitkosolkul

Authors

Wararit Panichkitkosolkul Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University

Keywords:

interval estimation, Poisson distribution, parameter, bootstrap method, simulation

Abstract

Recently, the Poisson-Garima distribution has been proposed for studying count data, which is of primary interest in several fields, such as biological science, medical science, demography, ecology and technology. However, estimating the confidence interval for its parameter has not yet been examined. In this study, confidence interval estimation based on the percentile, basic, and biased-corrected and accelerated bootstrap methods was examined in terms of their coverage probabilities and average lengths via Monte Carlo simulation. The results indicate that attaining the nominal confidence level using the bootstrap methods was not possible for small sample sizes regardless of the other settings. Moreover, when the sample size was large, the performances of the methods were not substantially different. Overall, the bias-corrected and accelerated bootstrap methods outperformed the others for all of the cases studied. Lastly, the efficacies of the bootstrap methods were illustrated by applying them to the number of thunderstorms in the USA, the results of which match those from the simulation study.

References

Andrew, F.S., and Michael, R.W. (2022). Practical business statistics. 8th Ed. San Diego: Academic Press.

Canty, A., and Ripley, B. (2022). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-28.1.

Chernick, M.R., and LaBudde, R.A. (2011). An introduction to bootstrap methods to R. Singapore: John Wiley & Sons.

David, F., and Johnson, N. (1952). The Truncated Poisson. Biometrics, 8(4), 275–285.

Davison, A.C., and Hinkley, D.V. (1997). Bootstrap methods and their application. 1st Ed. Cambridge: Cambridge University Press.

Efron, B. (1982). The Jackknife, the Bootstrap, and Other Resampling Plans, in CBMS-NSF regional conference series in applied mathematics, Philadelphia: SIAM.

Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(297), 171–185.

Efron, B., and Tibshirani, R.J. (1993). An introduction to the bootstrap. 1st Ed. New York: Chapman and Hall.

Falls, L.W., Williford, W.O., and Carter, M.C. (1971). Probability distributions for thunderstorm activity at Cape Kennedy, Florida. Journal of Applied Meteorology, 10(1), 97–104.

Florida Division of Emergency Management. (1 March 2023). Thunderstorms. https://www.floridadisaster.org/

hazards/thunderstorms/.

Ghitany, M.E., Al-Mutairi, D.K., and Nadarajah, S. (2008). Zero-truncated Poisson-Lindley distribution and its application. Mathematics and Computers in Simulation, 79(3), 279–287.

Henningsen, A., and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3), 443–458.

Hussain, T. (2020). A zero truncated discrete distribution: Theory and applications to count data. Pakistan Journal of Statistics and Operation Research, 16(1), 167–190.

Ihaka, R., and Gentleman, R. (1996). R: A Language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5(3), 299–314.

Lindley, D.V. (1958). Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society: Series B, 20(1), 102–107.

Meeker, W.Q., Hahn, G.J., and Escobar, L.A. (2017). Statistical intervals: A guide for practitioners and researchers. 2nd Ed. New Jersey: John Wiley and Sons.

National Weather Service. (1 March 2023). Introduction to Thunderstorms. https://www.weather.gov/jetstream/

tstorms_intro

Ong, S.-H., Low, Y.-C., and Toh, K.-K. (2021). Recent developments in mixed Poisson distributions. ASM Science Journal, https://doi.org/10.32802/asmscj.2020.464.

Sankaran, M. (1970). The discrete Poisson-Lindley distribution. Biometrics, 26(1), 145–149.

Shanker, R. (2016). Garima distribution and its application to model behavioral science data. Biometrics and Biostatistics International Journal, 4(7), 275–281.

Shanker, R. (2017). The discrete Poisson-Garima distribution. Biometrics and Biostatistics International Journal, 5(2), 48–53.

Siegel, A.F. (2017). Practical business statistics. 7th Ed. London: Academic Press.

Tan, S.H., and Tan, S.B. (2010). The correct interpretation of confidence intervals. Proceedings of Singapore Healthcare, 19(3), 276–278.

Tharshan, R., and Wijekoon, P. (2022). A new mixed Poisson distribution for over-dispersed count data: Theory and applications. Reliability: Theory & Applications, 17(1), 33–51.

Turhan, N.S. (2020). Karl Pearson’s chi-square tests. Educational Research Review, 15(9), 575–580.

Ukoumunne, O.C., Davison, A.C., Gulliford, M.C., Chinn, S. (2003). Non-parametric bootstrap confidence intervals for the intraclass correlation coefficient. Statistics in Medicine, 22(24), 3805–3821.

Wikipedia. (1 March 2023). Thunderstorm. https://en.wikipedia.org/wiki/Thunderstorm.

Wood, M. (2004). Statistical inference using bootstrap confidence intervals. Significance, 1(4), 180–182.