A new extension of the Rayleigh distribution: Properties, different methods of estimation, and an application to medical data

Statistical distributions play a crucial role in modeling and analyzing real data with complex behavior. Modifying or extending traditional distributions to better capture the complex pattern of a real-world phenomenon have attracted researchers' attention. In this paper, we propose a distribution adaptable to different types of medical data: the exponentiated generalized Weibull–Rayleigh (EGWR) distribution. Its hazard function exhibits different shapes, demonstrating high flexibility in modeling different patterns. For the proposed distribution, some statistical properties, such as moments, Rényi entropy, and order statistics, are discussed. Different methods of estimation—maximum likelihood, least squares, maximum product of spacing, and Cramér–von Mises—were employed to estimate the distribution parameters. The efficiency of these methods in estimating the distribution parameters was compared in three simulation studies and three medical datasets. Furthermore, the goodness of the proposed distribution in fitting real data was examined, and the results demonstrated the efficiency and flexibility of the EGWR distribution in modeling medical data compared to other distributions.