This paper illustrated how nonparametric bootstrap methods for double-censored data can be used to conduct some hypothesis tests, such as quartiles' hypothesis tests. Through simulation studies, the smoothed bootstrap (SB) method performed better results than Efron's method in most scenarios, particularly for small datasets. The SB method provided smaller discrepancies between the actual and nominal error rates.
Citation: Asamh Saleh M. Al Luhayb. Nonparametric bootstrap methods for hypothesis testing in the event of double-censored data[J]. AIMS Mathematics, 2024, 9(2): 4649-4664. doi: 10.3934/math.2024224
This paper illustrated how nonparametric bootstrap methods for double-censored data can be used to conduct some hypothesis tests, such as quartiles' hypothesis tests. Through simulation studies, the smoothed bootstrap (SB) method performed better results than Efron's method in most scenarios, particularly for small datasets. The SB method provided smaller discrepancies between the actual and nominal error rates.
[1] | B. Efron, Bootstrap methods: another look at the jackknife, Ann. Statist., 7 (1979), 1–26. https://doi.org/10.1214/aos/1176344552 doi: 10.1214/aos/1176344552 |
[2] | B. Efron, R. J. Tibshirani, An introduction to the bootstrap, Chapman and Hall, 1993. |
[3] | A. C. Davison, D. V. Hinkley, Bootstrap methods and their application, Cambridge University Press, 1997. https://doi.org/10.1017/CBO9780511802843 |
[4] | D. Berrar, Introduction to the non-parametric bootstrap, Encycl. Bioinf. Comput. Biol., 1 (2019), 766–773. https://doi.org/10.1016/B978-0-12-809633-8.20350-6 doi: 10.1016/B978-0-12-809633-8.20350-6 |
[5] | D. L. Banks, Histospline smoothing the bayesian bootstrap, Biometrika, 75 (1988), 673–684. https://doi.org/10.2307/2336308 doi: 10.2307/2336308 |
[6] | F. P. A. Coolen, S. B. Himd, Nonparametric predictive inference bootstrap with application to reproducibility of the two-sample Kolmogorov-Smirnov test, J. Stat. Theory Pract., 14 (2020), 26. https://doi.org/10.1007/s42519-020-00097-5 doi: 10.1007/s42519-020-00097-5 |
[7] | B. Efron, Censored data and the bootstrap, J. Amer. Stat. Assoc., 76 (1981), 312–319. https://doi.org/10.2307/2287832 doi: 10.2307/2287832 |
[8] | B. Efron, R. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statist. Sci., 1 (1986), 54–77. https://doi.org/10.1214/ss/1177013815 doi: 10.1214/ss/1177013815 |
[9] | A. S. M. Al Luhayb, F. P. A. Coolen, T. Coolen-Maturi, Generalizing banks' smoothed bootstrap method for right-censored data, Proceedings of the 29th European Safety and Reliability Conference, Hannover, Germany, 2019,894–901. https://doi.org/10.3850/978-981-11-2724-3_0177-cd |
[10] | A. S. M. Al Luhayb, T. Coolen-Maturi, F. P. A. Coolen, Smoothed bootstrap for survival function inference, 2019 International Conference on Information and Digital Technologies (IDT), Zilina, Slovakia, 2019,296–303. https://doi.org/10.1109/DT.2019.8813347 |
[11] | A. S. M. Al Luhayb, F. P. A. Coolen, T. Coolen-Maturi, Smoothed bootstrap for right-censored data, Commun. Stat.-Theory Methods, 2023, 1–25. https://doi.org/10.1080/03610926.2023.2171708 doi: 10.1080/03610926.2023.2171708 |
[12] | F. P. A. Coolen, K. J. Yan, Nonparametric predictive inference with right-censored data, J. Stat. Plan. Infer., 126 (2004), 25–54. https://doi.org/10.1016/j.jspi.2003.07.004 doi: 10.1016/j.jspi.2003.07.004 |
[13] | A. S. M. Al Luhayb, Smoothed bootstrap methods for right-censored data and bivariate data, Ph.D. Thesis, Durham University, 2021. |
[14] | A. S. M. Al Luhayb, T. Coolen-Maturi, F. P. A. Coolen, Smoothed bootstrap methods for bivariate data, J. Stat. Theory Pract., 17 (2023), 37. https://doi.org/10.1007/s42519-023-00334-7 doi: 10.1007/s42519-023-00334-7 |
[15] | J. L. Rasmussen, Estimating correlation coefficients: bootstrap and parametric approaches, Psychol. Bull., 101 (1987), 136–139. https://doi.org/10.1037/0033-2909.101.1.136 doi: 10.1037/0033-2909.101.1.136 |
[16] | M. J. Strube, Bootstrap Type Ⅰ error rates for the correlation coefficient: an examination of alternate procedures, Psychol. Bull., 104 (1988), 290–292. https://doi.org/10.1037/0033-2909.104.2.290 doi: 10.1037/0033-2909.104.2.290 |
[17] | H. J. Vaman, P. Tattar, Survival analysis, 1 Ed., New York: Chapman and Hall/CRC, 2022. https://doi.org/10.1201/9781003306979 |
[18] | M. Dolker, S. Halperin, D. R. Divgi, Problems with bootstrapping pearson correlations in very small bivariate samples, Psychometrika, 47 (1982), 529–530. https://doi.org/10.1007/BF02293714 doi: 10.1007/BF02293714 |
[19] | J. G. MacKinnon, Bootstrap hypothesis testing, In: D. A. Belsley, E. J. Kontoghiorghes, Handbook of computational econometrics, 2009,183–213. https://doi.org/10.1002/9780470748916.ch6 |
[20] | T. Hesterberg, Bootstrap, Wiley Int. Rev.: Comput. Stat., 3 (2011), 497–526. https://doi.org/10.1002/wics.182 doi: 10.1002/wics.182 |
[21] | T. Coolen-Maturi, F. P. A. Coolen, N. Muhammad, Predictive inference for bivariate data: combining nonparametric predictive inference for marginals with an estimated copula, J. Stat. Theory Pract., 10 (2016), 515–538. https://doi.org/10.1080/15598608.2016.1184112 doi: 10.1080/15598608.2016.1184112 |
[22] | N. Muhammad, Predictive inference with copulas for bivariate data, Ph.D. Thesis, Durham University, UK, 2016. |
[23] | N. Muhammad, F. P. A. Coolen, T. Coolen-Maturi, Predictive inference for bivariate data with nonparametric copula, AIP Conf. Proc., 1750 (2016), 060004. https://doi.org/10.1063/1.4954609 doi: 10.1063/1.4954609 |
[24] | J. P. Klein, M. L. Moeschberger, Survival analysis: techniques for censored and truncated data, New York: Springer, 2003. https://doi.org/10.1007/b97377 |
[25] | B. M. Hill, Posterior distribution of percentiles: Bayes' theorem for sampling from a population, J. Amer. Stat. Assoc., 63 (1968), 677–691. https://doi.org/10.1080/01621459.1968.11009286 doi: 10.1080/01621459.1968.11009286 |
[26] | B. M. Hill, De finetti's theorem, induction, and $A_{(n)}$ or bayesian nonparametric predictive inference (with discussion), In: J. M. Bernardo, M. H. DeGroot, D. V. Lindley, A. F. M. Smith, Bayesian statistics, Oxford University Press, 3 (1988), 211–241. |
[27] | A. S. M. Al Luhayb, Nonparametric statistical method for prediction in case of data including double-censored observations, Pak. J. Statist., 39 (2023), 485–500. |
[28] | L. M. Berliner, B. M. Hill, Bayesian nonparametric survival analysis, J. Amer. Stat. Assoc., 83 (1988), 772–779. https://doi.org/10.1080/01621459.1988.10478660 doi: 10.1080/01621459.1988.10478660 |
[29] | E. L. Kaplan, P. Meier, Nonparametric estimation from incomplete observations, J. Amer. Stat. Assoc., 53 (1958), 457–481. https://doi.org/10.1080/01621459.1958.10501452 doi: 10.1080/01621459.1958.10501452 |
[30] | F. Wan, Simulating survival data with predefined censoring rates for proportional hazards models, Stat. Med., 36 (2017), 838–854. https://doi.org/10.1002/sim.7178 doi: 10.1002/sim.7178 |