Hamiltonian Monte Carlo–based inference for the flexible exponential power–Weibull distribution with applications in reliability analysis

M. G. M. Ghazal; M. G. M. Ghazal

doi:10.3934/math.2026481

AIMS Mathematics

2026, Volume 11, Issue 4: 11659-11705. doi: 10.3934/math.2026481

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Hamiltonian Monte Carlo–based inference for the flexible exponential power–Weibull distribution with applications in reliability analysis

M. G. M. Ghazal ^{1,2
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1.
Department of Mathematics, Faculty of Science, Minia University, Minia 61519, Egypt
2.
University of Technology and Applied Sciences, Rustaq College of Education, Sultanate of Oman

Received: 17 February 2026 Revised: 07 April 2026 Accepted: 20 April 2026 Published: 27 April 2026
MSC : 60E05, 62E15, 62E20, 62F10, 62F15

In this study, I introduce a new four-parameter lifetime model, termed the flexible exponential power–Weibull (FEPW) distribution, developed by combining the exponential power and Weibull distributions to enhance modeling capability for a wide range of reliability data. The proposed distribution exhibited considerable flexibility in capturing increasing, decreasing, bathtub-shaped, and J-shaped hazard rate behaviors, making it suitable for complex engineering systems. The hazard rate function of the FEPW distribution was derived, and its principal structural properties were rigorously established. Parameter estimation for the FEPW distribution was addressed through maximum likelihood estimation and Bayesian inference. For the Bayesian framework, I implemented Hamiltonian Monte Carlo with the No-U-Turn Sampler (HMC–NUTS) to achieve efficient posterior exploration and stable parameter uncertainty quantification. A detailed simulation study was conducted to evaluate estimator performance under various parameter settings and sample sizes, assessing bias and mean squared error. The practical utility of the proposed model was demonstrated through applications to real-world reliability datasets, where it consistently outperformed several competing lifetime models in terms of goodness-of-fit and information criteria. The results underscored the FEPW distribution as a flexible and useful model for modeling complex failure-time data in reliability engineering and related fields.
Citation: M. G. M. Ghazal. Hamiltonian Monte Carlo–based inference for the flexible exponential power–Weibull distribution with applications in reliability analysis[J]. AIMS Mathematics, 2026, 11(4): 11659-11705. doi: 10.3934/math.2026481

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Abstract

In this study, I introduce a new four-parameter lifetime model, termed the flexible exponential power–Weibull (FEPW) distribution, developed by combining the exponential power and Weibull distributions to enhance modeling capability for a wide range of reliability data. The proposed distribution exhibited considerable flexibility in capturing increasing, decreasing, bathtub-shaped, and J-shaped hazard rate behaviors, making it suitable for complex engineering systems. The hazard rate function of the FEPW distribution was derived, and its principal structural properties were rigorously established. Parameter estimation for the FEPW distribution was addressed through maximum likelihood estimation and Bayesian inference. For the Bayesian framework, I implemented Hamiltonian Monte Carlo with the No-U-Turn Sampler (HMC–NUTS) to achieve efficient posterior exploration and stable parameter uncertainty quantification. A detailed simulation study was conducted to evaluate estimator performance under various parameter settings and sample sizes, assessing bias and mean squared error. The practical utility of the proposed model was demonstrated through applications to real-world reliability datasets, where it consistently outperformed several competing lifetime models in terms of goodness-of-fit and information criteria. The results underscored the FEPW distribution as a flexible and useful model for modeling complex failure-time data in reliability engineering and related fields.

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