Review Articles

Asymptotic properties of Kaplan–Meier estimator and hazard estimator for censored survival time with LENQD data

Yongming Li ,

School of Mathematics and Computer Science, Shangrao Normal University, Shangrao, People's Republic of China

Weicai Pang ,

School of Mathematics and Statistics, Nanning Normal University, Nanning, People's Republic of China

Ziqing Feng ,

College of Sciences, Nanchang University, Nangchang, People's Republic of China

Naiyi Li

School of Mathematics and Computer, Guangdong Ocean University, Zhanjiang, People's Republic of China

Pages | Received 22 Dec. 2021, Accepted 25 Dec. 2023, Published online: 16 Jan. 2024,
  • Abstract
  • Full Article
  • References
  • Citations

In this paper, we consider the estimators of distribution function and hazard rate for censored survival time. First, some properties and inequalities are established for linearly extended negative quadrant-dependent sequence as auxiliary results. Then by applying the properties and inequalities, we investigate the strong consistency and strong representation for the Kaplan–Meier estimator and hazard rate estimator with censored linearly extended negative quadrant-dependent data. Under some mild conditions, we derive that the rates of strong consistency are near O(n−1/2log1/2n)and also obtain the strong representations with the remainder of order O(n−1/2log1/2n). The results established here extend and generalize the corresponding ones in recent literature.


  • Ahmed, I., & Flandre, P. (2020). Weighted Kaplan–Meier estimators motivating to estimate HIV-1 RNA reduction censored by a limit of detection. Statistics in Medicine39(7), 968–983.
  • Anevski, D. (2017). Functional central limit theorems for the Nelson–Aalen and Kaplan–Meier estimators for dependent stationary data. Statistics and Probability Letter124(1), 83–91.
  • Block, H. W., Savits, T. H., & Shaked, M. (1982). Some concepts of negative dependence. The Annals of Probability10(3), 765–772.
  • Hu, X. P., & Jiang, R. (2018). Uniformly asymptotic normality of sample quantiles estimator for linearly negative quadrant dependent samples. Journal of Inequalities and Applications2018(196), 1–12.
  • Hu, X.P., & Wang, J. Y. (2020). A Berry–Esseen bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence. Aims Mathematics5(6), 6985–6995.
  • Joag-Dev, K., & Proschan, F. (1983). Negative association of random variables with applications. Annals of Statistics11(1), 286–295.
  • Kaplan, E. M., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association53(2), 457–481.
  • Ko, M. H., Ryu, D. H., & Kim, T. S. (2007). Limiting behaviors of weighted sums for linearly negative quadrant dependent random variables. Journal of the American Statistical Association11(2), 511–522.
  • Lehmann, E. (1966). Some concepts of dependence. Annals of Mathematical Statistics37(5), 1137–1153.
  • Li, Y. M., Guo, J. H., & Li, N. Y. (2012). Some inequalities for a LNQD sequence with applications. Journal of Inequalities and Applications2012(216), 1–9.
  • Li, Y. M., Pang, W. C., Feng, Z. Q., & Li, N. Y. (2023). On the linearly extended negative quadrant dependent random variables and its inequalities. Communication Statistics-Theory and Method52(24), 8696–8711.
  • Li, Y. M., & Zhou, Y. (2019). Asymptotic properties of the Kaplan–Meier estimator and hazard rate estimator for right censored and widely orthant dependent data. Acta Mathematicae Applicatae Sinica42(1), 71–84.
  • Li, Y. M., & Zhou, Y. (2020). The Kaplan–Meier estimator and hazard estimator for censored END survival time observations. Communication Statistics-Theory and Method49(11), 2690–2702.
  • Liang, H. Y., & Una-Alvarez, J. (2009). A Berry–Esseen type bound in kernel density estimation for strong mixing censored samples. Journal of Multivariate Analysis100(6), 1219–1231.
  • Liebscher, E. (2002). Kernel density and hazard rate estimation for censored data under α-mixing condition. Annals of the Institute of Statistical Mathematics54(1), 19–28.
  • Liu, L. (2009). Precise large deviations for dependent random variables with heavy tails. Statistics and Probability Letter79(9), 1290–1298.
  • 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 Medicine2020(4), 1–11.
  • Newman, C. M. (1984). Asymptotic independent and limit theorems for positively and negatively dependent random variables. In Inequalities in statistics and probability (Lincoln, Neb., 1982), IMS Lecture Notes-Monograph Series (Vol. 5, pp. 127–140). Inst Math Statist.
  • Shen, A. T., & Wang, X. J. (2016). Kaplan–Meier estimator and hazard estimator for censored negatively superadditive dependent data. Statistics50(2), 377–388.
  • Shen, A. T., & Zhu, H. Y. (2015). Complete convergence for weighted sums of LNQD random variables. Stochastics87(1), 160–169.
  • Wang, J. F., & Zhang, L. X. (2006). A Berry–Esseen theorem for weakly negatively dependent random variables and its applications. Acta Mathematica Hungarica110(4), 293–308.
  • Wang, X. J., Hu, S. H., Yang, W. Z., & Li, X. Q. (2010). Exponential inequalities and complete convergence for a LNQD sequence. Journal of the Korean Statistical Society39(4), 555–564.
  • Wu, Q. Y., & Chen, P. Y. (2013). Strong representation results of the Kaplan–Meier estimator for censored negatively associated data. Journal of Inequalities and Applications2013(1), 1–9.
  • Wu, Y., Yu, W., & Wang, X. J. (2022). Strong representations of the Kaplan–Meier estimator and hazard estimator with censored widely orthant dependent. Computational Statistics37(1), 383–402.
  • Yang, S. C. (2003). Consistency of nearest neighbor estimator of density function for negative associated samples. Acta Mathematicae Applicatae Sinica26(3), 385–395.

To cite this article: Yongming Li, Weicai Pang, Ziqing Feng & Naiyi Li (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, DOI: 10.1080/24754269.2024.2302754

To link to this article: