Beta-Marshal-Olkin Extended Log-Logistic Distribution: Theory and Applications

Accepted March 2026

Abstract

This paper introduces a new five-parameter lifetime model, the Beta-Marshall-Olkin Extended Log-Logistic (BMOELL) distribution, constructed by compounding the Beta-Generated and Marshall-Olkin-Generated families. The distribution accommodates symmetric, skewed, J-shaped, and bathtub-shaped hazard patterns. We derive its statistical properties including moments, entropy measures, and mean deviations. A key sub-model, the Exponentiated-Marshall-Olkin Extended Log-Logistic (EMOELL) distribution, obtained by setting ψ = 1, admits a closed-form cumulative distribution function and serves as the basis for all estimation and data-analytic work in this paper. Maximum likelihood estimation is developed for complete samples, randomly censored samples, and survival data with a cure fraction. Four real-world applications from medical and engineering domains illustrate the practical utility of the proposed model.