The new generalized odd Median Based Unit Rayleigh - Attia, Iman M. (Frontiers in Applied Mathematics and Statistics, 2025)
IntroductionIn this paper, the author presents the generalized form of the Median-Based Unit Rayleigh (MBUR) distribution, a novel statistical distribution that is specifically defined within the interval (0, 1). This generalization adds a new parameter to the MBUR distribution that significantly addresses the unique characteristics of data represented as ratios and proportions. The author considers a distinct technique for appending a new parameter to the unit distribution consuming the general formula for the order statistics.MethodsThe paper offers a thorough and meticulous derivation of the (PDF) for this distribution, illuminating each phase of the process with clarity and precision. It delves deep into the intricacies of the generalized odd MBUR (GOMBUR) distribution's properties, presenting a rigorous examination of the accompanying functions that are vital for robust statistical evaluation. These functions-comprising the (CDF), survival function, hazard rate, reversed hazard rate function, and raw moments.Results and discussionThe paper discusses real data analysis and how the generalization improves such analysis. The author conducts a comparative analysis of the Generalized Odd Median Base Unit Rayleigh (GOMBUR) and the Median Based Unit Rayleigh (MBUR). The primary objective is to evaluate the additional benefits provided by the new shape parameter in the estimation process, focusing on various validity indices, goodness-of-fit statistics, estimated variances of the parameters, and their corresponding standard errors. Parameter estimation is performed using the Maximum Likelihood Estimator (MLE), with the Nelder-Mead optimization method employed for this purpose. The results obtained from this study can be summarized in the following points. (i) Incorporating new parameters into the MBUR model significantly enhances its flexibility, enabling it to accommodate a variety of data shapes with differing characteristics, such as skewness and kurtosis. (ii) This added parameter enhances the estimation process, resulting in improved validity indices, including (AIC), (CAIC), (BIC), and (HQIC). Additionally, it enhances the goodness of fit by reducing test statistics such as the (AD), (CVM), and (KS) tests, while increasing the Log-Likelihood. (iii) The two forms of the model yield different values for the parameter (n) but provide the same value for the parameter (alpha). The variances of the estimated (alpha) are identical, and the covariance between the parameters is minimal—significantly lower than that observed when fitting other distributions like the Beta and Kumaraswamy. Furthermore, the determinant of the estimated variance-covariance matrix from fitting the GOMBUR-1 model is among the lowest compared to those from the Beta and Kumaraswamy distributions.