Open Access

ARTICLE

A Flexible Exponential Log-Logistic Distribution for Modeling Complex Failure Behaviors in Reliability and Engineering Data

Hadeel AlQadi¹, Fatimah M. Alghamdi², Hamada H. Hassan³, Mohamed E. Mead⁴, Ahmed Z. Afify^5,*

1 Department of Mathematics, College of Science, Jazan University, Jazan, 45142, Saudi Arabia
2 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia
3 Higher Institute of Computer Science and Information Technology, El Shorouk Academy, Shorouk, 11837, Egypt
4 Department of Statistics, Mathematics, and Insurance, Zagazig University, Zagazig, 44519, Egypt
5 Department of Statistics, Mathematics, and Insurance, Benha University, Benha, 13511, Benha

* Corresponding Author: Ahmed Z. Afify. Email:

(This article belongs to the Special Issue: Frontiers in Parametric Survival Models: Incorporating Trigonometric Baseline Distributions, Machine Learning, and Beyond)

Computer Modeling in Engineering & Sciences 2025, 144(2), 2029-2061. https://doi.org/10.32604/cmes.2025.069801

Received 01 July 2025; Accepted 04 August 2025; Issue published 31 August 2025

Abstract

Parametric survival models are essential for analyzing time-to-event data in fields such as engineering and biomedicine. While the log-logistic distribution is popular for its simplicity and closed-form expressions, it often lacks the flexibility needed to capture complex hazard patterns. In this article, we propose a novel extension of the classical log-logistic distribution, termed the new exponential log-logistic (NExLL) distribution, designed to provide enhanced flexibility in modeling time-to-event data with complex failure behaviors. The NExLL model incorporates a new exponential generator to expand the shape adaptability of the baseline log-logistic distribution, allowing it to capture a wide range of hazard rate shapes, including increasing, decreasing, J-shaped, reversed J-shaped, modified bathtub, and unimodal forms. A key feature of the NExLL distribution is its formulation as a mixture of log-logistic densities, offering both symmetric and asymmetric patterns suitable for diverse real-world reliability scenarios. We establish several theoretical properties of the model, including closed-form expressions for its probability density function, cumulative distribution function, moments, hazard rate function, and quantiles. Parameter estimation is performed using seven classical estimation techniques, with extensive Monte Carlo simulations used to evaluate and compare their performance under various conditions. The practical utility and flexibility of the proposed model are illustrated using two real-world datasets from reliability and engineering applications, where the NExLL model demonstrates superior fit and predictive performance compared to existing log-logistic-based models. This contribution advances the toolbox of parametric survival models, offering a robust alternative for modeling complex aging and failure patterns in reliability, engineering, and other applied domains.

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