Statistical Inference for the Rayleigh–Logarithmic Distributions Under Progressive Type II Censoring: Likelihood Structure and Modeling Flexibility

In reliability and survival studies, lifetime data are frequently subject to progressive Type-II censoring, leading to incomplete failure-time information and challenging statistical inference problems. In this study, statistical inference for the Rayleigh–Logarithmic (RL) distribution is developed under progressive Type-II censoring. The RL distribution provides a flexible lifetime model by combining a Rayleigh lifetime component with a logarithmically distributed number of latent failure causes. A competing-risk interpretation of the model is presented, and parameter estimation is carried out using both maximum likelihood estimation (MLE) and maximum product spacing (MPS) methods. The performance of the proposed inference procedures is investigated through extensive Monte Carlo simulations under different parameter settings and censoring schemes. The results indicate that both MLE and MPS provide reliable estimates, with estimation accuracy improving as the sample size increases. The methodology is further illustrated using simulated and real lifetime data sets and compared with classical lifetime distributions. The findings show that the RL distribution offers a flexible and effective framework for modeling progressively censored lifetime data, particularly in the presence of heterogeneous and latent failure mechanisms.