(untitled) (Contemporary Mathematics, 2025)
Lifetime distributions are indispensable in various scientific applications, particularly for analyzing bivariate measurements in reliability, engineering, and biomedical sciences. This paper introduces novel univariate and bivariate Marshall-Olkin Weibull models. Specifically, the Marshall-Olkin Weibull-Rayleigh distribution is proposed, offering enhanced flexibility to accommodate diverse univariate hazard rate shapes, including increasing, decreasing, and bathtub curves. Its novel bivariate form, the bivariate Marshall-Olkin Weibull-Rayleigh distribution, is derived using the Farlie-Gumbel-Morgenstern (FGM) copula function, designed to effectively capture various dependence structures in paired lifetime data. For parameter estimation, a comprehensive investigation is conducted comparing maximum likelihood estimation and Two-stage estimation to establish robust and efficient strategies. Furthermore, an accurate methodology for assessing copula goodness-of-fit is applied, detailing the empirical process approach, the use of pseudo-observations, and the Cramer-von Mises test statistic. Theoretical results are numerically examined through simulation. Finally, the superior performance and practical utility of the proposed bivariate Marshall-Olkin Weibull-Rayleigh model are demonstrated through a thorough analysis of real-world datasets and a comparative study against other established distributions, showcasing its significant advantages in reliability engineering applications.