A New Generalized ZLindley Model: Theory, Inference, and Engineering Reliability Applications

This study presents a new version of the ZLindly (ZL) model that improves modeling flexibility while maintaining ease of analysis, allowing for the simultaneous accommodation of redundant zeros, thick-tailed behavior, and complex failure rate dynamics within a unified probabilistic framework. Marshall–Olkin (MO) theory facilitates this advancement. The MOZL hazard rate can exhibit several patterns, including increasing, decreasing, bathtub, or upside-down bathtub-shaped. These features enable the model to capture diverse reliability phenomena such as early-life failures, random shocks, and wear-out effects. Comprehensive theoretical investigations were conducted and shown to be governed by an interpretable dual-parameter mechanism, where the Marshall–Olkin parameter controls tail behavior and dispersion, while the scale parameter regulates skewness and hazard evolution. A likelihood-based approach was developed under Type-II censoring conditions, and rigorous evidence is provided for the existence and uniqueness. To address inferential uncertainty, both classical asymptotic confidence intervals and log-normal approximations were constructed. Within a Bayesian framework, independent gamma priors were assumed, and posterior inference was performed via an efficient Metropolis–Hastings algorithm. Bayesian point and credible estimators were obtained and compared with their classical counterparts. An extensive simulation study demonstrates that Bayesian estimators, particularly with informative priors, consistently outperform likelihood-based estimators in terms of bias, mean squared error, interval length, and coverage probability, especially for moderate sample sizes and higher censoring levels. Three engineering applications are provided to assess the practical utility of the MOZL model, where it provides superior goodness-of-fit relative to 15 competing models, including MO–Exponential, MO–Gompertz, MO–Nadarajah–Haghighi, MO–Exponentiated Weibull, and Birnbaum–Saunders, among others. Overall, the proposed MOZL distribution emerges as a flexible, interpretable, and computationally efficient lifetime model whose structurally meaningful parameter interactions enhance distributional balance and flexible hazard behavior, thereby contributing to modern symmetry-oriented distribution theory while offering valuable applications in reliability engineering, survival analysis, and applied statistical modeling.