A Novel Generalization of Weibull and Generalized Rayleigh Probability Distributions With Flexible Hazard Rates Applicable to Life and Reliability Data

In this paper, we propose a novel and highly flexible lifetime distribution called the Additive Weibull Generalized Rayleigh distribution, derived by combining the cumulative hazard functions of the Additive Weibull and Generalized Rayleigh distributions. The submodels of this distribution are illustrated, encompassing both those newly derived from it and existing lifetime distributions achieved through particular parameter constraints. Numerous shapes of the novel distribution are investigated, including the probability density function, cumulative distribution function, survival function, and hazard functions. Distinct and varied shapes of the hazard rate, such as increasing, decreasing, strictly increasing, constant, and bathtub curves, are revealed. Furthermore, both unimodal and bimodal density patterns are illustrated. Key mathematical properties of the new distribution are derived, and parameter estimation is conducted using the maximum likelihood method. A simulation study is carried out utilizing the acceptance–rejection algorithm to generate random observations and evaluate the performance of the new distribution. The novel distribution exhibited a superior fit when compared to seven other competing distributions across three real-world datasets, as confirmed by standard model selection criteria. The statistical inferences, which include fitting a model to data, parameter estimation, and simulation, are thoroughly investigated, offering new perspectives on applied probability and statistical modeling, particularly in the areas of lifetime and reliability analysis. This finding establishes a basis for upcoming studies in statistics and related fields.