first_pagesettingsOrder Article Reprints Open AccessArticle Optimum Progressive Data Analysis and Bayesian Inference for Unified Progressive Hybrid INH Censoring with Applications to Diamonds and Gold by Heba S. Mohammed 1,*,Osama E. Abo-Kasem 2 andAhmed Elshahhat 3ORCID 1 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia 2 Department of Statistics, Faculty of Commerce, Zagazig University, Zagazig 44519, Egypt 3 Faculty of Technology and Development, Zagazig University, Zagazig 44519, Egypt * Author to whom correspondence should be addressed. Axioms 2025, 14(8), 559; https://doi.org/10.3390/axioms14080559 Submission received: 23 June 2025 / Revised: 15 July 2025 / Accepted: 21 July 2025 / Published: 23 July 2025 (This article belongs to the Special Issue Applications of Bayesian Methods in Statistical Analysis) Downloadkeyboard_arrow_down Browse Figures Versions Notes Abstract A novel unified progressive hybrid censoring is introduced to combine both progressive and hybrid censoring plans to allow flexible test termination either after a prespecified number of failures or at a fixed time. This work develops both frequentist and Bayesian inferential procedures for estimating the parameters, reliability, and hazard rates of the inverted Nadarajah–Haghighi lifespan model when a sample is produced from such a censoring plan. Maximum likelihood estimators are obtained through the Newton–Raphson iterative technique. The delta method, based on the Fisher information matrix, is utilized to build the asymptotic confidence intervals for each unknown quantity. In the Bayesian methodology, Markov chain Monte Carlo techniques with independent gamma priors are implemented to generate posterior summaries and credible intervals, addressing computational intractability through the Metropolis—Hastings algorithm. Extensive Monte Carlo simulations compare the efficiency and utility of frequentist and Bayesian estimates across multiple censoring designs, highlighting the superiority of Bayesian inference using informative prior information. Two real-world applications utilizing rare minerals from gold and diamond durability studies are examined to demonstrate the adaptability of the proposed estimators to the analysis of rare events in precious materials science. By applying four different optimality criteria to multiple competing plans, an analysis of various progressive censoring strategies that yield the best performance is conducted. The proposed censoring framework is effectively applied to real-world datasets involving diamonds and gold, demonstrating its practical utility in modeling the reliability and failure behavior of rare and high-value minerals.