Structural properties and information-theoretic characterization of the unit power Weibull family

We introduce the unit power Weibull distribution, a novel probability family defined on the unit interval, and establish its comprehensive structural and statistical properties. Explicit forms are derived for the density, distribution, quantile, and hazard rate functions, demonstrating the distribution’s ability to capture diverse shapes, including bimodality and flexible hazard structures. Rigorous results are obtained for raw and central moments, cumulants, order statistics, and related measures. Information-theoretic characterizations, such as Shannon, Rényi, and Tsallis entropies, together with Kullback-Leibler divergence, are provided in closed form using integral transforms and special functions. The limiting behavior and connections with classical bounded families (e.g., beta and Kumaraswamy) are also explored. Furthermore, actuarial risk measures, including value-at-risk, expected shortfall, tail value-at-risk, and tail variance, are developed, supported by simulation-based estimation studies. Applications to real-world data from nephrology, genetics, and insurance claims illustrate the practical relevance and superior performance of the proposed model. The unit power Weibull distribution family thus offers a mathematically tractable and statistically flexible framework for bounded data modeling across diverse domains.