Another Generalised Transmuted Rayleigh Distribution: Theory, Properties, and Inference

Abstract This paper introduces the Another Generalised Transmuted Rayleigh (AGTR) distribution, obtained by embedding the classical Rayleigh law within the Another Generalised Transmuted (AGT) family. The new model inherits the physical interpretability of the Rayleigh distribution while gaining two additional shape parameters that govern skewness and tail weight, enabling it to capture highly skewed and leptokurtic phenomena. We derive closed-form expressions for the probability density, cumulative, survival, hazard, and reverse-hazard functions and establish legitimacy conditions. Further analytical results include quantile and moment functions, R´enyi and Shannon entropies, incomplete moments, Lorenz and Bonferroni inequality curves, and several stochastic orderings. Parameter estimation is addressed via maximum likelihood; the observed-information matrix is provided for inference. A extensive Monte Carlo study, across sample sizes n = 25–1000, demonstrates that the maximum-likelihood estimators are consistent, nearly unbiased, and achieve nominal 95 % coverage with declining RMSE as n increases. Graphical illustrations confirm the AGTR distribution’s ability to reproduce monotone, bathtub, and non-monotone hazard shapes, underscoring its applicability to reliability analysis, survival studies, and risk modelling. The AGTR distribution thus offers a flexible, analytically tractable alternative to existing Rayleigh based extensions and enriches the toolkit for modelling skewed, heavy-tailed data.