Risk Bounds For Distributional Regression

OSCAR HERNAN MADRID PADILLA (University of California, Los Angeles) · Carlos Misael Madrid Padilla (Washington University, Saint Louis) · Sabyasachi Chatterjee (University of Illinois at Urbana-Champaign)
comprehensive experimentscontinuous ranked probability scoreconvergence ratesconvex-constrained regressionisotonic regressionmean estimationneural network-based estimatorsnon-convex constraintsnonparametric distributional regressionpractical effectivenessrisk boundssimulated datatheoretical resultstrend filteringworst-case mean squared error

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.