A New Quantile Differential Family with an Explicit Quadratic Distribution: Theory, Statistical Properties, and Applications

Quantile functions provide a natural representation of probability distributions and offer direct advantages for random variate generation, simulation, and quantile-based statistical analysis. In this paper, we develop a quantile-differential construction in which a probability model is characterized through a differential equation satisfied directly by its quantile function. Within this framework, we introduce a one-parameter continuous model, termed the quadratic quantile distribution (QQD), whose explicit quantile representation leads to a simple and analytically tractable probability model. Closed-form expressions are obtained for the cumulative distribution function, probability density function, survival function, hazard rate, and cumulative hazard function. Several structural properties are also investigated, including shape characteristics, stochastic ordering, entropy measures, fractional moments, and quantile-based descriptive measures. Particular attention is devoted to the upper-tail behavior. The survival function is regularly varying with index −1, so that the QQD has a Pareto-type tail with tail index one. Consequently, the ordinary mean is infinite, whereas fractional moments exist only for orders less than one. This feature makes quantile-, survival-, and tail-based summaries more appropriate than conventional moment-based descriptions. Maximum likelihood estimation is developed for the model parameter, and its finite-sample behavior is examined through an extensive Monte Carlo study. The simulation results show a progressive reduction in bias, root mean squared error, and mean absolute error as the sample size increases, providing numerical evidence consistent with the expected large-sample behavior of the estimator without treating simulation as a proof of asymptotic consistency. The empirical performance of the QQD is examined using three right-skewed datasets from reliability and biomedical applications. Comparisons with established one- and two-parameter distributions show that the QQD can provide a competitive likelihood-based fit while retaining the simplicity of a single unknown parameter. Graphical comparisons further complement the numerical criteria, particularly in assessing the behavior of the fitted models in the upper tail. Overall, the results suggest that the QQD offers a parsimonious and analytically convenient alternative for selected strongly right-skewed positive data, while its suitability should be assessed on a case-by-case basis in view of its heavy-tailed structure.