Bridging Markov Chain Monte Carlo Techniques and Tierney–Kadane Approximations for Progressively Censored Garhy Reliability Models: Simulation Insights and a Medical Application
- 1Department of Mathematics, College of Science, University of Ha’il, Hail 55476, Saudi Arabia
- 2Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
- 3Faculty of Computers and Information Systems, Egyptian Chinese University, Nasr City 4442126, Egypt
Abstract
This paper investigates the estimation of the stress–strength reliability parameter R=P(Y<X) when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for R is explicitly derived, enabling effective and precise calculation without numerical integration. The Garhy distribution, a flexible one-parameter lifetime model with an increasing hazard function, is confirmed by full-scale goodness-of-fit diagnostics. A Bayesian estimation model is trained on non-informative priors (normal and extended Jeffreys priors) under squared error loss. The posterior expectations are analytically intractable; we adopt two complementary methods of computation: (i) Markov Chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm and (ii) the Tierney–Kadane (TK) approximation, which provides extremely precise analytical estimates with significantly reduced computational burden. Monte Carlo simulations are large-scale and compare the proposed estimators under different censoring schemes, sample sizes, and parameter configurations in terms of bias and mean squared error (MSE). The methodology is further applied to a real medical dataset comprising kidney dialysis patient survival times, demonstrating its practical relevance in clinical reliability assessment. Results consistently indicate that Bayesian methods, particularly with the extended Jeffreys prior, outperform classical MLEs in terms of stability and accuracy, especially under heavy censoring. Moreover, the TK approximation yields estimates virtually identical to MCMC while requiring only a fraction of the computational effort. We further extend the TK framework to approximate the posterior variance of R and the expected log-likelihood, providing a fully analytical alternative to MCMC for comprehensive Bayesian inference.
Related
- A family of spectral conjugate gradient algorithms derived by least-squares approximations based on a modified quasi--Newton update with application to a revised robust binary classification model
- Nonparametric Estimation of Negative Cumulative Extropy for Linear Consecutive Systems: Reliability Properties and Image-Processing Application
- Simulation and Application of the Transmuted Unit Burr XII Distribution in Real-World Data Analysis
- Analytical and Bootstrap Confidence Intervals for Double Machine Learning: Simulation Studies and an Application to Rural-Urban Differences in Obesity Prevalence
- Mediation and moderation in distance-based regression with triangular fuzzy data: simulation evidence and an application to academic performance
BibTeX
@article{alenezy2026bridging,
title = {{Bridging Markov Chain Monte Carlo Techniques and Tierney–Kadane Approximations for Progressively Censored Garhy Reliability Models: Simulation Insights and a Medical Application}},
author = {Abdullah H. Alenezy and Anis Ben Ghorbal and Khudhayr A. Rashedi and Ghareeb A. Marei},
journal = {Mathematics},
year = {2026},
month = may,
doi = {10.3390/math14101777},
url = {https://doi.org/10.3390/math14101777},
note = {CC BY},
}
RIS
TY - JOUR
TI - Bridging Markov Chain Monte Carlo Techniques and Tierney–Kadane Approximations for Progressively Censored Garhy Reliability Models: Simulation Insights and a Medical Application
AU - Abdullah H. Alenezy
AU - Anis Ben Ghorbal
AU - Khudhayr A. Rashedi
AU - Ghareeb A. Marei
PY - 2026
DA - 2026/05/21
JO - Mathematics
DO - 10.3390/math14101777
UR - https://doi.org/10.3390/math14101777
AB - This paper investigates the estimation of the stress–strength reliability parameter R=P(Y<X) when both stress and strength follow independent Garhy distributions under progressive Type-II censoring schemes. A closed-form expression for R is explicitly derived, enabling effective and precise calculation without numerical integration. The Garhy distribution, a flexible one-parameter lifetime model with an increasing hazard function, is confirmed by full-scale goodness-of-fit diagnostics. A Bayesian estimation model is trained on non-informative priors (normal and extended Jeffreys priors) under squared error loss. The posterior expectations are analytically intractable; we adopt two complementary methods of computation: (i) Markov Chain Monte Carlo (MCMC) using the Metropolis–Hastings algorithm and (ii) the Tierney–Kadane (TK) approximation, which provides extremely precise analytical estimates with significantly reduced computational burden. Monte Carlo simulations are large-scale and compare the proposed estimators under different censoring schemes, sample sizes, and parameter configurations in terms of bias and mean squared error (MSE). The methodology is further applied to a real medical dataset comprising kidney dialysis patient survival times, demonstrating its practical relevance in clinical reliability assessment. Results consistently indicate that Bayesian methods, particularly with the extended Jeffreys prior, outperform classical MLEs in terms of stability and accuracy, especially under heavy censoring. Moreover, the TK approximation yields estimates virtually identical to MCMC while requiring only a fraction of the computational effort. We further extend the TK framework to approximate the posterior variance of R and the expected log-likelihood, providing a fully analytical alternative to MCMC for comprehensive Bayesian inference.
SN - 2227-7390
ER -