# Confidence Intervals by Bootstrapping Approach: A Significance Review

## Authors

• Siti Fairus Mokhtar ᵃSchool Quantitative Sciences, Universiti Utara Malaysia, Kedah, Malaysia; ᵇDepartment of Mathematical Sciences, Faculty of Computer Science and Mathematics,Universiti Teknologi Mara, Malaysia
• Zahayu Md Yusof ᵃSchool of Quantitative Sciences, Universiti Utara Malaysia 06010 Sintok, Kedah, Malaysia; ᶜInstitute of Strategic Industrial Decision Modelling (ISIDM), Universiti Utara Malaysia, Kedah, Malaysia
• Hasimah Sapiri School Quantitative Sciences, Universiti Utara Malaysia, Kedah, Malaysia

## Abstract

A confidence interval is an interval estimate of a parameter of a population calculated from a sample drawn from the population. Bootstrapping method, which involves producing several new data sets that are resampled from the original data in order to estimate parameter for each newly created data set, allowing an empirical distribution for the parameter to be estimated. Since certain statistics are harder to estimate, confidence intervals are rarely employed. Several statistics might necessitate multi-step formulas assuming that are impractical for calculating confidence intervals. This paper reviews research on the concept of bootstrapping and bootstrap confidence interval. The current narrative analysis was developed to answer the main research question: (1) What is the concept of the bootstrap method and bootstrap confidence interval? (2) What are the methods of bootstrapping to obtain confidence interval? This study has found general bootstrap method idea, various techniques of bootstrap methods, its advantages and disadvantages, and its limitations. There are normal interval method, percentile bootstrap method, basic method, first-order normal approximation method, bias-corrected bootstrap, accelerated bias-corrected bootstrap and bootstrap-t method. This study concludes that the advantages of using bootstrap CI is that it does not require any assumptions about the shape of distribution and universality of the approach. Bootstrapping is a computer-intensive statistical technique that relies significantly on modern high-speed digital computers to do massive computations.

## References

Abu-shawiesh, M. O. A., & Saeed, N. (2022). Estimating the population mean under contaminated normal distribution based on deciles.

Abu-Shawiesh, M. O. A., Sinsomboonthong, J., & Kibria, B. M. G. (2022). A modified robust confidence interval for the population mean of distribution based on deciles. Statistics in Transition New Series, 23(1), 109-128. https://doi.org/10.21307/stattrans-2022-007.

Banjanovic, E. S., & Osborne, J. W. (2016). Confidence intervals for effect sizes: Applying bootstrap resampling. Practical Assessment, Research and Evaluation, 21(5).

Barker, N. (2005). A practical introduction to the bootstrap using the SAS system. Proceedings of the Pharmaceutical Users Software Exchange Conference, Paper PK02. http://www.lexjansen.com/phuse/2005/pk/pk02.pdf.

Berrar, D. (2019). Introduction to the non-parametric bootstrap. April, 0-15. https://doi.org/10.1016/B978-0-12-809633-8

Bickel, D. R., & Frühwirth, R. (2006). On a fast, robust estimator of the mode: Comparisons to other robust estimators with applications. Computational Statistics and Data Analysis, 50(12), 3500-3530. https://doi.org/10.1016/j.csda.2005.07.011.

Boos, D. D., & Hughes-Oliver, J. M. (2000). How large does n have to be for z and t intervals? American Statistician, 54(2), 121-128. https://doi.org/10.1080/00031305.2000.10474524.

Chen, D., & Fritz, M. S. (2021). comparing alternative corrections for bias in the bias-corrected bootstrap test of mediation. evaluation and the health professions, 44(4), 416-427. https://doi.org/10.1177/01632787211024356.

Chou, C. Y., Lin, Y. C., Chang, C. L., & Chen, C. H. (2006). On the bootstrap confidence intervals of the process incapability index Cpp. Reliability Engineering and System Safety, 91(4), 452-459. https://doi.org/10.1016/j.ress.2005.03.004.

Cumming, G. (2007). Inference by eye: Pictures of confidence intervals and thinking about levels of confidence. Teaching Statistics, 29(3), 89-93. https://doi.org/10.1111/j.1467-9639.2007.00267.x.

Das, S. K. (2019). Confidence interval is more informative than P-Value in research. International Journal of Engineering Applied Sciences and Technology, 04(06), 278-282. https://doi.org/10.33564/ijeast.2019.v04i06.045

David, H. A. (1998). Early sample measures of variability. Statistical Science, 13(4), 368-377. https://doi.org/10.1214/ss/1028905831.

DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189-228.

Desharnais, B., Camirand-Lemyre, F., Mireault, P., & Skinner, C. D. (2015). Determination of confidence intervals in non-normal data: Application of the bootstrap to cocaine concentration in femoral blood. Journal of Analytical Toxicology, 39(2), 113-117. https://doi.org/10.1093/jat/bku127.

Doğan, C. D. (2017). Applying bootstrap resampling to compute confidence intervals for various statistics with R. Egitim Arastirmalari - Eurasian Journal of Educational Research, 2017(68), 1-18. https://doi.org/10.14689/ejer.2017.68.1.

Dogan, G. (2004). Confidence interval estimation in system dynamics models: bootstrapping vs. likelihood ratio method. Proceedings of the 22nd International Conference of the System Dynamics Society, 1-38.

Efron, B. (1979). Bootstrap method_another look at Jackknife. Breakthroughs in Statistics, 569-593.

Efron, B. (1981). Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods. Biometrika, 68(3), 589-599. https://doi.org/10.1093/biomet/68.3.589.

Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397), 171-185.

Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. https://doi.org/10.1201/9780429246593.

Endo, T., Watanabe, T., & Yamamoto, A. (2015). Confidence interval estimation by bootstrap method for uncertainty quantification using random sampling method. Journal of Nuclear Science and Technology, 52(7-8), 993-999. https://doi.org/10.1080/00223131.2015.1034216.

Flowers-Cano, R. S., Ortiz-Gómez, R., León-Jiménez, J. E., Rivera, R. L., & Cruz, L. A. P. (2018). Comparison of bootstrap confidence intervals using Monte Carlo simulations. Water (Switzerland), 10(2), 1-21. https://doi.org/10.3390/w10020166.

Good, P. (2005). Multivariate analysis. Springer.

Hanna, J. P., Stone, P., & Niekum, S. (2017). Bootstrapping with models: confidence intervals for off-policy evaluation. Proceedings of the International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS, 1(May), 538-546.

Haukoos, J. S., & Lewis, R. J. (2005). Advanced statistics: Bootstrapping confidence intervals for statistics with “difficult” distributions. Academic Emergency Medicine, 12(4), 360-365. https://doi.org/10.1197/j.aem.2004.11.018.

Hedges, S. B. (1992). The number of replications needed for accurate estimation of the bootstrap P value in phylogenetic studies. Molecular Biology and Evolution, 9(2), 366-369. https://doi.org/10.1093/oxfordjournals.molbev.a040725.

Hongyi, Li, G. S., & Maddala. (2007). Bootstrapping time series. Econometric Review, 15(2), 115-158.

Horowitz, J. L. (2019). Bootstrap methods in econometrics. Annual Review of Economics, 11. https://doi.org/10.1146/annurev-economics-080218-025651.

Hoyle, S. D., & Cameron, D. S. (2003). Confidence intervals on catch estimates from a recreational fishing survey: A comparison of bootstrap methods. Fisheries Management and Ecology, 10(2), 97-108. https://doi.org/10.1046/j.1365-2400.2003.00321.x.

Jung, K., Lee, J., Gupta, V., & Cho, G. (2019). Comparison of bootstrap confidence interval methods for GSCA using a Monte Carlo simulation. Frontiers in Psychology, 10(October), 24-25. https://doi.org/10.3389/fpsyg.2019.02215.

Kennedy, K., & Schumacher, P. (1993). The introduction of bootstrapping techniques for finding confidence intervals using simulation. Journal of Computers in Mathematics and Science Teaching, 12(1), 83-92. http://www.eric.ed.gov/ERICWebPortal/detail?accno=EJ467667.

Klaudia, J., & Łukasz, S. (2020). The use of the bootstrap method for the assessment of investment effectiveness and risk – the case of confidence intervals estimation for the Sharpe ratio and TailVaR. Journal of Banking and Financial Economics, 1/2020(13), 40-50. https://doi.org/10.7172/2353-6845.jbfe.2020.1.3.

LaFontaine, D. (2021). The history of bootstrapping: tracing the development of resampling with replacement. Mathematics Enthusiast, 18(1-2), 78-99. https://doi.org/10.54870/1551-3440.1515.

Mahmudah, U., Surono, S., Prasetyo, P. W., & Haryati, A. E. (2023). Forecasting educated unemployed people in indonesia using the bootstrap technique. 12(July 2022), 171-182. https://doi.org/10.22103/jmmr.2022.19368.1239.

Mesabbah, M., Rashwan, W., & Arisha, A. (2015). An empirical estimation of statistical inferences for system dynamics model parameters. Proceedings - Winter Simulation Conference, 2015-Janua, 686–697. https://doi.org/10.1109/WSC.2014.7019932

Onyesom, C., & S.I., A. (2021). Bootstrapping - An introduction and its applications in statistics. 9(3), 22-28.

Pek, J., Wong, A. C. M., & Wong, O. C. Y. (2017). confidence intervals for the mean of non-normal distribution: transform or not to transform. Open Journal of Statistics, 07(03), 405-421. https://doi.org/10.4236/ojs.2017.73029.

Petty, M. D. (2012). Calculating and using confidence intervals for model validation. Fall Simulation Interoperability Workshop 2012, 2012 Fall SIW, 37-45.

Puth, M. T., Neuhäuser, M., & Ruxton, G. D. (2015). On the variety of methods for calculating confidence intervals by bootstrapping. Journal of Animal Ecology, 84(4), 892-897. https://doi.org/10.1111/1365-2656.12382.

Rahmandad, H., Jalali, M. S., & Ghoddusi, H. (2013). Estimation of uknown parameters in system dynamics models using the method of simulated moments. 1212-1221. https://doi.org/10.1128/AAC.03728-14.

Rousselet, G. A., Pernet, C. R., & Wilcox, R. R. (2021). The percentile bootstrap: a primer with step-by-step instructions in R. Advances in Methods and Practices in Psychological Science, 4(1). https://doi.org/10.1177/2515245920911881.

Saha, D., & Kapilesh, B. (2016). Estimation of concrete characteristic strength from limited data by bootstrap. Journal of Asian Concrete Federation, 2(1), 81. https://doi.org/10.18702/acf.2016.06.2.1.81.

Saito, Y., & Dohi, T. (2018). Parametric bootstrap methods for estimating model parameters of Non-homogeneous gamma process. International Journal of Mathematical, Engineering and Management Sciences, 3(2), 167-176. https://doi.org/10.33889/ijmems.2018.3.2-013.

Shao, J., & Tu, D. (1995). The jackknife and bootstrap. Springer Science & Business Media. http://orton.catie.ac.cr/cgi-bin/wxis.exe/?IsisScript=BOSQUE.xis&method=post&formato=2&cantidad=1&expresion=mfn=004867%5Cnpapers2://publication/uuid/3565770D-2B58-427B-AFBC-BD0A21C7CC8D.

Shi, W., & Golam Kibria, B. M. (2007). On some confidence intervals for estimating the mean of a skewed population. International Journal of Mathematical Education in Science and Technology, 38(3), 412-421. https://doi.org/10.1080/00207390601116086.

Sinsomboonthong, J., Abu-Shawiesh, M. O. A., & Kibria, B. M. G. (2020). Performance of robust confidence intervals for estimating population mean under both non-normality and in presence of outliers. Advances in Science, Technology and Engineering Systems, 5(3), 442-449. https://doi.org/10.25046/aj050355.

Tapia, J. A., Salvador, B., & Rodríguez, J. M. (2020). Data envelopment analysis with estimated output data: Confidence intervals efficiency. Measurement: Journal of the International Measurement Confederation, 152. https://doi.org/10.1016/j.measurement.2019.107364.

Tong, L. I., Saminathan, R., & Chang, C. W. (2016). Uncertainty assessment of non-normal emission estimates using non-parametric bootstrap confidence intervals. Journal of Environmental Informatics, 28(1), 61–70. https://doi.org/10.3808/jei.201500322.

Tong, Lee Ing, Chang, C. W., Jin, S. E., & Saminathan, R. (2012). Quantifying uncertainty of emission estimates in National Greenhouse Gas Inventories using bootstrap confidence intervals. Atmospheric Environment, 56, 80-87. https://doi.org/10.1016/j.atmosenv.2012.03.063.

Wang, F.-K. (2001). Non-normal data. Quality and Reliability Engineering International, 267(259), 257-267. https://doi.org/10.1002/qre400.

Wilcox, R. R. (2021). A note on computing a confidence interval for the mean. Communications in Statistics: Simulation and Computation, 1-3. https://doi.org/10.1080/03610918.2021.2011926.

Yan, L. (2022). Confidence interval estimation of the common mean of several gamma populations. PLoS ONE, 17(6 June), 1-13. https://doi.org/10.1371/journal.pone.0269971.

Zaman, T. (2016). A study based on the application of bootstrap and jackknife methods in simple linear regression analysis. International Journal of Sciences.

Zhou, X. H., & Dinh, P. (2005). Nonparametric confidence intervals for the one- And two-sample problems. Biostatistics, 6(2), 187-200. https://doi.org/10.1093/biostatistics/kxi002.

Zhao, S., Yang, Z., Musa, S. S., Ran, J., Chong, M. K. C., Javanbakht, M., He, D., & Wang, M. H. (2021). Attach importance of the bootstrap t-test against Student’s t-test in clinical epidemiology: A demonstrative comparison using COVID-19 as an example. Epidemiology and Infection. https://doi.org/10.1017/S0950268821001047x.