Modeling Lifetime Data with the New Marshall-Olkin Weibull-Rayleigh Distribution and Its Bivariate Form

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.