Kaplan–Meier estimator and hazard estimator for censored negatively superadditive dependent data

Abstract

In this paper, we investigate the strong convergence properties for the Kaplan–Meier estimator and hazard estimator based on censored negatively superadditive dependent data. Under some mild conditions, the strong convergence rate of the Kaplan–Meier estimator and hazard estimator is established. In addition, the strong representation of the Kaplan–Meier estimator and hazard estimator is also obtained with the remainder of order $O(n^{-1/2}{\log }^{1/2}n)$ $a.s.$ Our results established in the paper generalize the corresponding ones for independent random variables and negatively associated random variables.

Keywords:

• Kaplan–Meier estimator
• hazard estimator
• negatively superadditive dependent random variables
• strong convergence rate

Mathematics Subject Classifications:

• 60F15
• 60F05

Acknowledgements

The authors are most grateful to the Editor in Chief, Associate Editor and anonymous referees for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11201001, 11501004), the Natural Science Foundation of Anhui Province (1508085J06, 1308085QA03), the Provincial Natural Science Research Project of Anhui Colleges (KJ2015A018), the Open Project of School of Mathematical Sciences, Anhui University (ADSY201503) and the Quality Improvement Projects for Undergraduate Education of Anhui University (ZLTS2015035).