A Three-Parameter Record-Based Transmuted Rayleigh Distribution (Order 3): Theory and Real-Data Applications

This paper introduces the record-based transmuted Rayleigh distribution of order 3 (rbt-R), a three-parameter extension of the classical Rayleigh model designed to address data characterized by high skewness and heavy tails. While traditional generalizations of the Rayleigh distribution enhance model flexibility, they often lack sufficient adaptability to capture the complexity of empirical distributions encountered in applied statistics. The rbt-R model incorporates two additional shape parameters, a and b, enabling it to represent a wider range of distributional shapes. Parameter estimation for the rbt-R model is performed using the maximum likelihood method. Simulation studies are conducted to evaluate the asymptotic properties of the estimators, including bias and mean squared error. The performance of the rbt-R model is assessed through empirical applications to four datasets: nicotine yields and carbon monoxide emissions from cigarette data, as well as breaking stress measurements from carbon-fiber materials. Model fit is evaluated using standard goodness-of-fit criteria, including AIC, AICc, BIC, and the Kolmogorov–Smirnov statistic. In all cases, the rbt-R model demonstrates a superior fit compared to existing Rayleigh-based models, indicating its effectiveness in modeling highly skewed and heavy-tailed data.