Two new unit-interval distributions with applications to COVID-19 and reservoir capacity data

Accurate modeling of data restricted to the unit interval is essential across many applied disciplines. This paper proposes two new single-parameter probability distributions tailored for variables supported on (0,1). The models are mathematically tractable and exhibit notable flexibility, with analytical expressions derived for their survival and hazard functions, moments, and parameter-driven behavior. Their single-parameter structure enables closed-form maximum likelihood and Bayesian estimators, avoiding the heavy numerical optimization typically required by existing unit-interval models. The maximum likelihood estimators are shown to be global solutions, and the Bayesian framework benefits from the conjugacy of the gamma prior, allowing for efficient posterior inference. The performance of the proposed distributions is assessed through extensive simulations and applications to four real datasets. These datasets were selected to encompass both positively and negatively skewed structures, allowing for an evaluation of model performance across diverse distributional shapes. The empirical results demonstrate that the proposed models provide an excellent fit for all skewness patterns and consistently outperform ten established benchmark models based on standard goodness-of-fit criteria.