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 flexible
probability distributions derived from the composition of Beta and Kumaraswamy generators: the
Beta-Kumaraswamy (BKu) and Kumaraswamy-Beta (KB) distributions. We systematically
derive their fundamental statistical properties including probability density functions, cumulative
distribution functions, moment-generating functions, order statistics, and L-moments. The method
of Maximum Likelihood Estimation is developed for parameter inference for both distributions.
A thorough comparative analysis reveals that while both distributions offer substantial flexibility
in modeling various distribution shapes including symmetric, skewed, U-shaped, and J-shaped
configurations, the BKu distribution demonstrates superior computational tractability due to its
simpler functional form. Empirical validation using two real datasets—petroleum rock samples
and milk production data—confirms the practical utility of both distributions. The BKu
distribution consistently achieved better model fit as measured by log-likelihood and AIC values,
with approximately 40% faster computation time and more stable parameter estimation across
both applications. These findings suggest that the BKu distribution is generally preferable for
practical applications where computational efficiency and estimation stability are paramount.
Keywords: Beta Distribution; Generated Families; Kumaraswamy Distribution; L-Moments;
Maximum Likelihood Estimation; Order Statistics;