Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Abstract

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F̄=1−F and construct point-wise credible intervals. The resulting Bayes estimator of F̄ extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.

Keywords:

• Dirichlet process
• empirical Bayes
• nonparametric prior
• PL-type estimator
• sum-quota accrual

AMS Subject Classification::

• Primary: 62N01
• Secondary: 62F15

Acknowledgements

The authors are indebted to Professor Jayaram Sethuraman and Professor Timothy Hanson for helpful comments. We also thank the Editor, the Associate Editor, and the two reviewers for providing us valuable comments and suggestions which led to improvements. This work is part of the first author's PhD dissertation.

Funding

Research is partially supported by NSF Grant DMS1106435 and NIH Grants R01 CA154731 and P20 RR17698.