On the properties of the beta-kumaraswamy and kumaraswamy-beta distributions: a comprehensive study in order statistics, parameter estimation, and L-moments

This paper provides a comprehensive theoretical and empirical investigation of two flexibleprobability distributions derived from the composition of Beta and Kumaraswamy generators: theBeta-Kumaraswamy (BKu) and Kumaraswamy-Beta (KB) distributions. We systematicallyderive their fundamental statistical properties including probability density functions, cumulativedistribution functions, moment-generating functions, order statistics, and L-moments. The methodof Maximum Likelihood Estimation is developed for parameter inference for both distributions.A thorough comparative analysis reveals that while both distributions offer substantial flexibilityin modeling various distribution shapes including symmetric, skewed, U-shaped, and J-shapedconfigurations, the BKu distribution demonstrates superior computational tractability due to itssimpler functional form. Empirical validation using two real datasets—petroleum rock samplesand milk production data—confirms the practical utility of both distributions. The BKudistribution consistently achieved better model fit as measured by log-likelihood and AIC values,with approximately 40% faster computation time and more stable parameter estimation acrossboth applications. These findings suggest that the BKu distribution is generally preferable forpractical applications where computational efficiency and estimation stability are paramount.Keywords: Beta Distribution; Generated Families; Kumaraswamy Distribution; L-Moments;Maximum Likelihood Estimation; Order Statistics;