Flexible Bayesian Framework for Analyzing Heterogeneous Lifetime Data Using Weighted Inverse Rayleigh Mixtures

In data analysis, understanding the complexity and variability inherent in lifetime data is critical to various fields of studies, such as reliability and survival analyses, as conventional distributions often fail to effectively capture the heterogeneity of real‐world systems. This study introduces a three‐component mixture of weighted inverse Rayleigh (3‐CMWIR) distributions to model complex lifetime data. The model is designed to capture the heterogeneity and latent subpopulations that are often present in real‐world systems, addressing the gap left by traditional distributions that do not account for this variability. This study proposed the Bayesian formulation to estimate the parameters of the 3‐CMWIR distributions. Bayesian inference was performed under Type‐I censoring using uniform priors (UPs) and gamma priors (GPs), coupled with three loss functions: squared error loss function (SELF), quadratic loss function (QLF), and precautionary loss function (PLF). A comprehensive simulation study is used to assess the performance of Bayes estimators (BEs) and posterior risks (PRs), examining the impact of sample size and test termination time on parameter estimation. The results demonstrate that larger sample sizes and longer test termination times enhance the precision of BE. The GP consistently outperformed the UP, achieving lower PRs, particularly under QLF and PLF for component parameters, SELF and PLF for proportional parameters. Among the loss functions, PLF was identified as the most suitable, providing superior reliability for parameter estimation.