Risk Bounds For Distributional Regression

TL;DR

This paper derives risk bounds for nonparametric distributional regression estimators, providing theoretical guarantees and convergence rates, with applications to isotonic, trend filtering, and neural network-based methods, validated through experiments.

Contribution

It introduces new risk bounds for distributional regression under convex and non-convex constraints, including neural network estimators, with practical validation.

Findings

Established upper bounds for CRPS and MSE in convex-constrained regression.

Derived convergence rates for isotonic and trend filtering distributional regression.

Validated theoretical results through experiments on simulated and real data.

Abstract

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.