A New Transmuted Rayleigh Model with Application to Reliability Data

Abstract

The classical Rayleigh distribution, while useful in many fields, is often limited by its fixed shape and hazard function when modeling complex real-world data. To overcome these limitations, this paper introduces the Transmuted Rayleigh Distribution (TRD), a more flexible generalization developed using the Quadratic Rank Transmutation Map (QRTM) framework. The TRD incorporates a distortion parameter, , which facilitates a smooth transition between the standard Rayleigh distribution and a Weibull distribution, thereby enhancing its capability to model diverse data behaviors, including varying levels of skewness and tail weight. We derive and discuss key statistical properties of the TRD, including its moments, quantile function, entropy, and hazard rate. The parameters of the distribution are estimated using the maximum likelihood method, and a comprehensive simulation study is conducted to assess the performance of these estimators. The practical utility and superiority of the TRD are demonstrated through an application to real-world analgesic relief time data, where it is shown to provide a significantly better fit compared to several competing distributions, including the standard Rayleigh, Weibull, Gamma, and Log-Normal models, as confirmed by goodness-of-fit tests and model selection criteria.