On progressive first-failure reliability analysis: Classical and Bayesian approaches

Hisham M. Almongy; Ehab M. Almetwally; Eslam Hussam; Mahmoud H. Abu-Moussa; T. S. Taher; Ali M. Sharawy; Hisham M. Almongy; Ehab M. Almetwally; Eslam Hussam; Mahmoud H. Abu-Moussa; T. S. Taher; Ali M. Sharawy

doi:10.3934/math.2026554

AIMS Mathematics

2026, Volume 11, Issue 5: 13449-13484. doi: 10.3934/math.2026554

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On progressive first-failure reliability analysis: Classical and Bayesian approaches

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
2.
Department of Accounting, College of Business Administration, Prince Sattam bin Abdulaziz University, Hawtat Bani Tamim, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Capital University, Egypt
4.
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
5.
Faculty of Education and Arts, Sohar University, Sohar, Oman
6.
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt
7.
Faculty of Engineering, Egyptian Russian University, Cairo, Egypt

Received: 12 February 2026 Revised: 28 April 2026 Accepted: 07 May 2026 Published: 14 May 2026
MSC : 62F15, 62N05

The progressive first-failure censoring (PF-FC) plan is widely used in reliability settings where test items are arranged into groups of size $ k $, and only the earliest failure in each group is observed. In this study, statistical inference for the Gompertz–Lindley distribution (GLD) under PF-FC was considered, with emphasis on estimating the model parameters together with the reliability and hazard rate functions (HRFs). Classical inference was performed via the maximum likelihood method (MLE), and confidence intervals (CIs) were formed using the large-sample behavior of the estimators. A Bayesian framework was also constructed using independent gamma priors and non-informative priors (NIPs) under loss structures. Markov Chain Monte Carlo (MCMC) algorithms were used to generate Bayesian estimates (BEs) and credible intervals (CRIs). Reliability measures are examined from both classical and BE. To evaluate the proposed procedures, a MCMC simulation study was carried out to examine their precision and robustness. The practical relevance of the developed methodology was illustrated using a real lifetime dataset.
- MLEs,
- progressive first-failure censoring,
- reliability inference,
- Gompertz-Lindley distribution,
- Bayesian estimation
Citation: Hisham M. Almongy, Ehab M. Almetwally, Eslam Hussam, Mahmoud H. Abu-Moussa, T. S. Taher, Ali M. Sharawy. On progressive first-failure reliability analysis: Classical and Bayesian approaches[J]. AIMS Mathematics, 2026, 11(5): 13449-13484. doi: 10.3934/math.2026554

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Abstract

The progressive first-failure censoring (PF-FC) plan is widely used in reliability settings where test items are arranged into groups of size $ k $, and only the earliest failure in each group is observed. In this study, statistical inference for the Gompertz–Lindley distribution (GLD) under PF-FC was considered, with emphasis on estimating the model parameters together with the reliability and hazard rate functions (HRFs). Classical inference was performed via the maximum likelihood method (MLE), and confidence intervals (CIs) were formed using the large-sample behavior of the estimators. A Bayesian framework was also constructed using independent gamma priors and non-informative priors (NIPs) under loss structures. Markov Chain Monte Carlo (MCMC) algorithms were used to generate Bayesian estimates (BEs) and credible intervals (CRIs). Reliability measures are examined from both classical and BE. To evaluate the proposed procedures, a MCMC simulation study was carried out to examine their precision and robustness. The practical relevance of the developed methodology was illustrated using a real lifetime dataset.

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