An application of the Ważewski principle to the characterization of continuous probability distributions

We introduce an unexpected application of the Ważewski retract principle to a probabilistic challenge: identifying probability distributions via the regression function of ordered statistical data such as order statistics and record values. In order to prevent repetitive arguments across various models, we examine a wide-ranging category of models known as generalized order statistics. We establish a precise formula for the underlying distribution using a specified regression and a solution to an additional Volterra-Stieltjes integral equation. Moreover, we identify the class of feasible regression functions by determining the necessary and sufficient conditions for a function to qualify as a regression. Our findings are supported by examples that highlight the importance of the stipulated conditions.