A New Extended Rayleigh Model and Its Modelling of Two Engineering Scenarios

Abstract

This study introduces a new three-parameter log-gamma–Rayleigh (LGR) distribution as a flexible extension of the classical Rayleigh model for lifetime and reliability data. By embedding the Rayleigh distribution within a log-gamma generating framework, the proposed model is capable of capturing a wide spectrum of distributional shapes, including left- and right-skewed patterns, as well as diverse hazard rate forms such as increasing, unimodal, and bathtub-shaped behaviors. A central theoretical contribution is expressing the LGR density as a mixture of Rayleigh densities, which enables tractable derivations of key properties. Explicit expressions are obtained for the quantile function, moments, generating function, and information measures including Rényi and Shannon entropies. Order statistics and their associated densities are also derived through a convergent series approach. Maximum likelihood estimation is employed for parameter inference, supported by closed-form expressions for the observed Fisher information matrix, which yield approximate confidence intervals. The superiority of the LGR distribution is demonstrated through two engineering datasets concerning carbon-fiber breaking stresses and ball-bearing fatigue life. In both applications, the LGR model outperforms classical competitors across multiple goodness-of-fit criteria. The results establish the LGR model as a powerful and versatile tool for modelling reliability and survival data.