A novel arc tangent hyperbolic Weibull distribution for continuous proportional data and its quantile regression model

Proportional data, such as probabilities and rates bounded within the unit interval, frequently arise in economics, epidemiology, and environmental studies, yet existing distributions often struggle to capture their complex behaviour due to limitations in flexibility and robustness. While traditional approaches rely on logarithmic or inverse-exponential transformations, these methods may lack smoothness or fail to accommodate varying tail behaviours. To address these challenges, this paper introduces the ATHW (arc tangent hyperbolic Weibull) distribution, a novel two-parameter continuous model based on the hyperbolic tangent transformation, which provides enhanced smoothness and adaptability for proportional data analysis. We derive its key statistical properties, implement maximum likelihood estimation and hypothesis testing via the US algorithm, and propose a flexible unit quantile regression model through quantile function reparameterization. Extensive simulations demonstrate that the ATHW distribution outperforms conventional models in robustness and goodness-of-fit, particularly for skewed or light-tailed data. Empirical applications using OECD country data further validate its superior performance in real-world scenarios. This study thus provides a improvement in the statistical modelling of proportional data, with broad applicability across scientific disciplines.