New empirical likelihood inference for the mean residual life with length-biased and right-censored data

ABSTRACT

The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

Keywords:

• Empirical likelihood
• i.i.d. representation
• length-biased data
• mean residual life
• right censoring

Acknowledgments

The authors would like to thank the Editor-in-Chief, Dr. Ricardo Cao, an Associate Editor, and the anonymous reviewer for their useful comments and constructive suggestions, which result in significant improvements in the article. Yichuan Zhao acknowledges the support from the Simons Foundation, the NSF Grant (DMS-2006304) and the NSA Grant (H98230-19-1-0024).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Science Foundation [grant number DMS-2006304] and National Security Agency [grant numberH98230-19-1-0024] and Simons Foundation [grant number 638679].