Estimation for partially observed left truncation and right censored competing risks data from a generalized inverted exponential distribution with illustrations

ABSTRACT

In this paper, studies of competing risks model are considered when the observations are left-truncated and right-censored data. When the failure times of the competing risks are distributed by a generalized inverted exponential model with same scale but different shape parameters with partially observed failure causes, statistical inference for the unknown model parameters is discussed from classical and Bayesian approaches, respectively. Maximum likelihood estimators of the unknown parameters, along with associated existence and uniqueness, are established, and the asymptotic likelihood theory is also used to construct the confidence interval via the observed Fisher information matrix. Moreover, Bayesian estimates and the corresponding highest posterior density credible intervals are also obtained based a flexible Gamma-Beta prior, and a Gibbs sampling technique is constructed to compute associated estimates. Further, under a general practical assumption with order-restriction parameter case, classical and Bayesian estimations are also established under order restriction situations, respectively. Extensive Monte-Carlo simulations are carried out to investigate the performances of our results and two real-life examples are analyzed to show the applicability of the proposed methods.

KEYWORDS:

• Left-truncated and right-censored data
• competing risks model
• order restriction
• generalized inverted exponential model
• maximum likelihood estimation
• posterior analysis

Acknowledgements

The authors would like to thank the editor and the referees for their insightful comments that have led to a substantial improvement to an earlier version of the paper. This work of Liang Wang was supported by the National Natural Science Foundation of China (No. 12061091), the Yunnan Fundamental Research Projects (No. 202101AT070103), the Doctoral Research Foundation of Yunnan Normal University (No. 00800205020503129) and Yunnan Key Laboratory of Modern Analytical Mathematics and Applications. The work of Chunfang Zhang is funded by the National Natural Science Foundation of China (No. 12101476) and the Natural Science Basic Research Program of Shaanxi Province (No. 2020JQ-285).

Disclosure statement

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

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [12101476]; National Natural Science Foundation of China [12061091]; Natural Science Basic Research Program of Shaanxi Province [2020JQ-285]; Yunnan Fundamental Research Projects [202101AT070103]; Doctoral Research Foundation of Yunnan Normal University [00800205020503129]

Notes on contributors

Liang Wang

Liang Wang received the Ph.D. degree in Applied Mathematics from Northwestern Polytechnical University, Xi'an, China, in 2012. He is currently a full Professor with the School of Mathematics, Yunnan Normal University, Kungming, China. His research interests include applied probabilityand statistics, reliability analysis and life testing.

Chunfang Zhang

Chunfang Zhang received the Ph.D. degree in Mathematics from Northwestern Polytechnical University, Xi'an, China, in 2017. She is currently a lecturer with the School of Mathematics and Statistics, Xidian University, Xi'an, China. Her research interests include multivariate statistics, Bayesian statistics, and reliability analysis.

Shuo-Jye Wu

Shuo-Jye Wu is a professor in the Department of Statistics at Tamkang University. He received his PhD in statistics from the University of Wisconsin-Madison. His professional interests are in the development and application of statistical methodology for problems in reliability.

Sanku Dey

Sanku Dey is currently working as Associate Professor in the Department of Statistics, St. Anthony's College, Shillong, Meghalaya, India. He did his M.Sc. in Statistics in the year of 1991 from Gauhati University, Guwahati, India and Ph.D. in Statistics (reliability theory) in the year 1998 from the same university. He has published 270 research papers in journals of repute. He is also an Academic Editor, Associate Editor and Section Editor of renowned international journals. He has a good number of contributions in almost all fields of Statistics viz, distribution theory, reliability theory, Bayesian inference, Record Statistics, Statistical quality control, order statistics, lifetime performance index based on classical and Bayesian approach as well as different types of censoring schemes etc.

Yuhlong Lio

Yuhlong Lio is a Professor in the Department of Mathematical Sciences at the University of South Dakota. His research interests include reliability, smooth estimation, survival analysis, and statistical process control.He received his B.S. in Mathematics from National Cheng-Kung University, his M.S. in Mathematics from National Central University, and his Ph.D from the University of South Carolina in 1987.