This paper investigates the asymptotic properties of the Kaplan–Meier and hazard estimators for censored survival time data. We conduct this analysis under the assumption of m-widely acceptable (m-WA) dependence, a generalized form of weak correlation. Using the Fuk–Nagaev inequality, we establish strong consistency and strong representation results for these estimators. Our findings show that the rate of strong consistency is near 𝑂⁡(√𝑔⁡(𝑛)⁢log⁡𝑛/𝑛) and the remainder term in the strong representation is of the same order. These results generalize and extend existing work for other types of dependent data, such as linearly extended negative quadrant-dependent (LENQD) and extended negative dependent (END) sequences, thereby broadening the theoretical foundation for these widely used statistical tools.

References

• Aalen, O. O (1978). Nonparametric inference for a family of counting processes. The Annals of Statistics, 6(4), 701–726. https://dx.doi.org/10.1214/aos/1176344247
• Ahmed, I., & Flandre, P. (2020). Weighted Kaplan–Meier estimators motivating to estimate HIV-1 RNA reduction censored by a limit of detection. Statistics in Medicine, 39(7), 968–983. https://doi.org/10.1002/sim.8455
• Andersen, P. K., Borgan, O., Gill, R. D., & Keiding, N (1993). Statistical Models Based on Counting Processes. Springer-Verlag.
• Anevski, D. (2017). Functional central limit theorems for the Nelson–Aalen and Kaplan–Meier estimators for dependent stationary data. Statistics & Probability Letters, 124(1), 83–91. https://doi.org/10.1016/j.spl.2017.01.005
• Fleming, T. R., & Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley.
• Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457–481. https://doi.org/10.1080/01621459.1958.10501452
• Klein, J. P., & Moeschberger, M. L (2003). Survival Analysis: Techniques for Censored and Truncated Data (Vol. 1230). Springer.
• Li, Y., Pang, W., Feng, Z., & Li, N. (2024). Asymptotic properties of Kaplan–Meier estimator and hazard estimator for censored survival time with LENQD data. Statistical Theory and Related Fields, 8(2), 107–116. https://doi.org/10.1080/24754269.2024.2302754
• Li, Y. M., & Zhou, Y. (2020). The Kaplan–Meier estimator and hazard estimator for censored END survival time observations. Communications in Statistics -- Theory and Methods, 49(11), 2690–2702. https://doi.org/10.1080/03610926.2019.1580737
• Li, Y. M., & Zhou, Y. (2024). Asymptotic properties of Kaplan–Meier estimator and hazard estimator for censored survival time with LENQD data. Statistical Theory and Related Fields, 8(2), 107–116. https://doi.org/10.1080/24754269.2024.2302754
• Liebscher, E. (2002). Kernel density and hazard rate estimation for censored data under α-mixing condition. Annals of the Institute of Statistical Mathematics, 54(1), 19–28. https://doi.org/10.1023/A:1016157519826
• Nelson, W. (1969). Hazard plotting for incomplete failure data. Journal of Quality Technology, 1(1), 27–52. https://doi.org/10.1080/00224065.1969.11980344
• Nematolahi, S., Nazari, S., Shayan, Z., Ayatollahi, S. M. T., & Amanati, A. (2020). Improved Kaplan–Meier estimator in survival analysis based on partially rank-ordered set samples. Computational and Mathematical Methods in Medicine, 2020, 1–11. https://doi.org/10.1155/2020/7827434
• Peng, S. G. (2006). G-expectation, G-Brownian motion and related stochastic calculus of Itô's type G-expectation, G-Brownian motion and related stochastic calculus of Itô's type. In H. Holden, K. H. Karlsen, K. A. Lie, & N. H. Risebro (Eds.), The Abel Symposium 2005, Abel Symposia (pp. 541–567). Springer-Verlag.
• Shen, A. T., & Wang, X. H. (2016). Kaplan–Meier estimator and hazard estimator for censored negatively superadditive dependent data. Statistics, 50(2), 377–388. https://doi.org/10.1080/02331888.2015.1038269
• Wu, Q. Y., & Chen, P. Y. (2013). Strong representation results of the Kaplan–Meier estimator for censored negatively associated data. Journal of Inequalities and Applications, 2013(1), 1–9. https://doi.org/10.1186/1029-242X-2013-1
• Wu, Y., Deng, X., & Wang, X. J. (2023). Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations. Journal of Mathematical Analysis and Applications, 525(1), 127–282. https://doi.org/10.1016/j.jmaa.2023.127282

To cite this article: Ikhlasse Chebbab (28 May 2026): Strong representation of the Kaplan–Meier and hazard estimators for censored data with m-widely acceptable dependence, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2026.2679126
To link to this article: https://doi.org/10.1080/24754269.2026.2679126