A q ‐Logarithmic Deformation of the Unit Burr‐XII Distribution for Bounded Data: Properties, Inference, and Quantile Regression With Biomedical Applications

Bounded data defined on the unit interval commonly arise in biostatistics, epidemiology, environmental sciences, economics, and risk modeling. Classical unit distributions are widely useful for modeling such data. Such models may provide limited flexibility when the data exhibit pronounced skewness or complex hazard rate behavior. This paper introduces a new three‐parameter distribution, called the q ‐unit Burr‐XII distribution, by incorporating the q ‐logarithmic function from nonextensive statistical mechanics into the unit Burr‐XII distribution. The deformation parameter q adds more control on the skewness and the shape of the distribution, consequently improving the ability to model diverse bounded data patterns. Several properties of the proposed model are investigated. Different methods are considered for estimating the model parameters, and their performance is assessed through a simulation study. To model bounded responses in the presence of covariates, a parametric quantile regression model is presented based on the proposed model. The methodology is illustrated using two real datasets, including a biomedical application involving android fat percentage as the response variable, with age, body mass index, gender, and physical activity level as explanatory variables. Comparative analyses against established alternatives, such as the beta, Kumaraswamy, unit Weibull, and unit Burr‐XII models, indicate that the proposed model provides a more flexible fit in several settings. These findings highlight the usefulness of the q‐unit Burr‐XII distribution as an alternative framework for modeling bounded data and quantile‐based regression analysis.