A Nonparametric Ordering-Based Probabilistic Regression with Rank-Adaptive Coefficient Functions

This paper introduces a novel nonparametric framework for probabilistic regression based on rank-adaptive coefficient functions. Standard parametric models, such as logistic regression, rely on a single global coefficient vector and implicitly assume homogeneous effects across the population. In contrast, the proposed method estimates a sequence of local linear equations along the ranked distribution of predicted risk, allowing coefficients to vary smoothly across population segments. By iteratively reorganizing observations to increase the concentration of negative outcomes at lower ranks and positive outcomes at higher ranks, the approach generates empirical distributions of coefficients that are summarized into smooth adaptive coefficient functions. The framework integrates ideas from local regression, moving-window estimation, and learning-to-rank methodologies while preserving the interpretability of linear models. Allowing coefficients to vary with rank enables the model to capture heterogeneous marginal effects without imposing a predefined link function or strong distributional assumptions. Extensive simulations show high predictive accuracy and strong recovery of the underlying risk ordering: across 100 simulated datasets, correlations between true and predicted probabilities exceed 0.99 on average, and rank correlations approach unity. A fixed-design Monte Carlo experiment, in which covariates remain constant, and coefficients vary across replications, yields similar results. Unlike many black-box machine learning approaches, the proposed framework remains transparent and interpretable, providing a flexible tool for applications in economics, health, education, and finance where treatment effects and risk relationships vary across the population.