A New One-Parameter Model by Extending Maxwell–Boltzmann Theory to Discrete Lifetime Modeling

Ahmed Elshahhat, Hoda Rezk and Refah Alotaibi ()
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Ahmed Elshahhat: Faculty of Technology and Development, Zagazig University, Zagazig 44519, Egypt
Hoda Rezk: Department of Statistics, Al-Azhar University, Cairo 11884, Egypt
Refah Alotaibi: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Mathematics, 2025, vol. 13, issue 17, 1-27

Abstract: The Maxwell–Boltzmann (MB) distribution is fundamental in statistical physics, providing an exact description of particle speed or energy distributions. In this study, a discrete formulation derived via the survival function discretization technique extends the MB model’s theoretical strengths to realistically handle lifetime and reliability data recorded in integer form, enabling accurate modeling under inherently discrete or censored observation schemes. The proposed discrete MB (DMB) model preserves the continuous MB’s flexibility in capturing diverse hazard rate shapes, while directly addressing the discrete and often censored nature of real-world lifetime and reliability data. Its formulation accommodates right-skewed, left-skewed, and symmetric probability mass functions with an inherently increasing hazard rate, enabling robust modeling of negatively skewed and monotonic-failure processes where competing discrete models underperform. We establish a comprehensive suite of distributional properties, including closed-form expressions for the probability mass, cumulative distribution, hazard functions, quantiles, raw moments, dispersion indices, and order statistics. For parameter estimation under Type-II censoring, we develop maximum likelihood, Bayesian, and bootstrap-based approaches and propose six distinct interval estimation methods encompassing frequentist, resampling, and Bayesian paradigms. Extensive Monte Carlo simulations systematically compare estimator performance across varying sample sizes, censoring levels, and prior structures, revealing the superiority of Bayesian–MCMC estimators with highest posterior density intervals in small- to moderate-sample regimes. Two genuine datasets—spanning engineering reliability and clinical survival contexts—demonstrate the DMB model’s superior goodness-of-fit and predictive accuracy over eleven competing discrete lifetime models.

Keywords: discrete Maxwell–Boltzmann; survival discretization; type-II censoring; Bayesian with Markov iterative; failure rate; moments; bootstrapping; genuine data modeling (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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