The Four-Parameter Odd Generalized Rayleigh Lomax Distribution: Theory, Simulation, and Applications

The fundamental problem with current Rayleigh-Lomax-based distributions lies in their limited flexibility to model both symmetry and tail weight simultaneously. Therefore, this study aims to introduce the OGRLx anomalous general distribution as an innovative mathematical framework that addresses these shortcomings by providing precise control over the distribution’s shape and risk ratios. We derived the basic statistical properties of the model, and used six different estimation methods that proved their efficiency through an intensive simulation study, with the Maximum Likelihood Estimator showing the best performance in terms of bias criteria and root mean square error. The practical value of the model is evident in its superior ability to fit data with high skewness and variable risks; experimental results using economic and medical data (bladder cancer) have proven the OGRLx distribution to be significantly superior to nine competing models. It achieved the lowest values for information standards Akaike Information Criteria, Consistent AIC, Bayesian Information Criteria, Hanan and Quinn Information Criteria, Anderson–Darling, Cramer–von Mises, Kolmogorov–Smirnov, and the highest p-value tests, making it a more accurate statistical tool for reliability analysis and medical studies compared to traditional extensions. Finally, it should be noted that all analyses, programming, and statistical operations in this study were performed using the R statistical software.