A novel odd Rayleigh-exponential distribution (OR-ED) and its application to lifetime datasets

Abstract

A novel two-parameter Odd Rayleigh-Exponential Distribution (OR-ED), derived by embedding the Exponential Distribution within the Odd Rayleigh-G (OR-G) family of distributions, was proposed in this study. The aim is to introduce a more flexible and robust model capable of accurately capturing the complex hazard rate behaviours and tail properties often observed in real lifetime and reliability datasets. The proposed model is mathematically tractable and exhibits a variety of shapes, including right-skewed and heavy-tailed forms. Key statistical properties such as the quantile function, raw moments, entropy, order statistics, and moment generating functions (MGF) were derived. The Maximum Likelihood Estimation (MLE) method was employed for parameter estimation, and simulation studies demonstrate that the estimators are consistent, with decreasing bias and RMSE, as sample size increases. The model’s suitability and superiority are assessed using three real-life datasets: kidney infection frailty data, COVID-19 mortality rates in Mexico, and failure times of lifting engines in giant machines. Model comparison with six established distributions was based on log-likelihood (LL), information criteria (AIC, BIC, CAIC, HQIC), and goodness-of-fit tests (Anderson-Darling, Cramer von-Mises, Kolmogorov-Smirnov). The proposed model consistently yielded lower information criteria and better fit statistics across all datasets, including the smallest KS distances (KS-D) and highest p-values, in comparison to competing models, aligning with empirical distributions. The OR-ED emerged as a statistically sound, flexible model with superior tail behaviour and hazard rate adaptability, offering significant improvement over other competing distributions in modelling complex lifetime data.