Distributional Regression for Arbitrary Response Variables Using ReLU Neural Networks

This paper develops a unified framework for distributional regression with arbitrary response variables. Rather than relying on scalar thresholds or Euclidean structure, we model conditional event probabilities indexed by measurable sets, which extends distributional regression to general response spaces while retaining a direct connection to classical event-based formulations. Within this framework, we study two complementary estimators: a preliminary estimator based on separate regression of event indicators, and a constrained estimator that enforces inclusion-consistent predictions through a directed acyclic graph (DAG) over a finite set family. We establish general error bounds for both estimators and further specialize the results to dense ReLU network, yielding explicit convergence rates overcoming the curse of dimensionality. Simulation studies and real-data experiments further demonstrate the practical effectiveness of the proposed framework.