Transmuted Generalized Weibull Lindley (TGWL) distribution: Bayesian inference and Bayesian neural network approaches for lifetime data modeling

<h2>Abstract</h2> Due to the complexity of lifetime data, inability, and lack of robustness in the primarily used classical approaches, developing a new approach is necessary to address the challenges observed in the real world effectively. Focusing on addressing the wide range of behavior in lifetime data, we introduce a new distribution termed Transmuted Generalized Weibull Lindley (TGWL) specifically for lifetime datasets. This distribution utilizes the flexibility of the Generalized Weibull Lindley (GWL) and the transmuted family of distribution to offer a more adaptable model capable of capturing complexity observed in real-world problems. We derive and discuss the significance of the fundamental statistical properties of TGWL distribution in modeling lifetime data. Due to the limitation of the classical approach for parameter estimation in complex models, we leverage Bayesian and machine learning techniques to obtain the most precise and robust estimates. We presented frequentist, Bayesian, and machine-learning approaches for parameter estimation. Through extensive simulation, we demonstrate that classical techniques such as Maximum Likelihood Estimation (MLE) struggle to address uncertainty in model parameters in complex lifetime models such as TGWL, while Bayesian Inference and Bayesian Neural Network (BNN) achieve outstanding performance both in terms of accuracy and robustness. We finally validate the performance of our model by using two real datasets. Compared with closely related distributions, the TGWL shows great accuracy in modeling complex lifetime data.