A Flexible Approach to Quantile Regression Modeling With Unit Burr-XII-Poisson and Its Applications to Cancer, Chemotherapy, and Energy Data

This article introduces the unit-Burr XII-Poisson (UBXIIP) distribution, a flexible model for bounded data in the unit interval. Unlike many existing alternatives, the UBXIIP offers enhanced versatility in modeling unit-domain phenomena. We further develop a quantile-based regression framework by reparameterizing the UBXIIP, enabling the direct interpretation of its parameters as quantiles. The regression coefficients are linked to the median of the response variable, providing intuitive and meaningful inference. Our approach provides a robust and interpretable method for analyzing relationships between predictors and bounded responses. The key statistical properties of the model are examined, including explicit closed-form expressions for the $r\text{th}$ moments, the quantile function, and the Shannon entropy. Parameter estimation for the UBXIIP distribution is performed using the maximum likelihood estimation (MLE) method. The efficiency of this estimation approach is assessed through Monte Carlo simulation studies by observing the behavior of the mean square error of the estimates. Furthermore, MLEs of the UBXIIP-quantile regression model is observed by comprehensive simulation studies through residual analysis. Three real-world applications are illustrated: modeling cell recovery rates post-chemotherapy, fitting remission times of bladder cancer patients, and assessing wind energy data. These case studies highlight the versatility and robustness of the UBXIIP distribution and its quantile regression counterpart, emphasizing their potential for diverse applications in medical research and renewable energy analysis. Likewise, they demonstrate their superior performance over the standard unit Burr XII and other competing distributions in terms of fit and predictive accuracy.