On the Marshall–Olkin Exponentiated Rayleigh Distribution: Properties, Estimation, and Applications

This study introduces a new Marshall–Olkin Exponentiated Rayleigh (MO-ER) distribution for modelling lifetime data. The MO-ER distribution is derived by taking the exponentiated Rayleigh basis through the Marshall–Olkin transformation to obtain a flexible three-parameter distribution. Some statistical properties of the distribution are obtained, including its probability density, cumulative distribution, and survival and hazard functions. The identifiability of the model is formally established. A Monte Carlo simulation study is conducted to examine the performance of the proposed distribution across different sample sizes, and a Bayesian estimation approach is developed to address the instability of the maximum likelihood estimator of the tilt parameter in small samples. The results indicate that there is consistency in the behaviour of the MO-ER distribution with varying sample sizes and the accuracy increases with sample size. The performance of the distribution is also studied with the help of two real datasets, where the MO-ER distribution exhibits a better fit than a number of competing distributions based on information criteria, a likelihood ratio test against the nested exponentiated Rayleigh baseline, and Kolmogorov–Smirnov statistics. The MO-ER distribution in general is a flexible and practical alternative to modelling lifetime data in reliability and survival analysis.