Quantile Regression for Bounded Biomedical Outcomes via the Unit Power Maxwell–Boltzmann Distribution: A COVID‐19 Mortality Case Study

Bounded lifetimes on (0,1), such as mortality proportions, are prevalent in biomedical and public health research; however, mean-based models can overlook tail behavior and heterogeneity. We introduce the unit power Maxwell distribution (UPMD) defined on (0,1) and develop a quantile regression framework that captures covariate effects across the outcomes. The UPMD is generated based on the log power transformation of the Maxwell-Boltzmann distribution, producing a tractable quantile and flexible shapes with skewness and heavy tails. This work examines several key distributional properties, including the quantile function, order statistics, residual life, and Rényi entropy. It also investigates inference via maximum likelihood, maximum product spacing, least squares, and Bayesian estimation. Extensive simulation analysis is performed to evaluate the efficiency of various estimation methods using bias, mean squared error, and interval probability coverage (CPs). For empirical evaluation, we analyze COVID-19 mortality proportions, where UPMD-based quantile regression delivers improved fit in upper and lower tails and better calibration relative to alternatives, as assessed by information criteria, formal goodness of fit tests, and residual diagnostics such as randomized quantile residuals and Cox-Snell checks. The results indicate that the UPMD provides a practical, interpretable tool for bounded biomedical outcomes.