Machine-Learning-Assisted Comparison of Regression Functions

TL;DR

This paper introduces a new kernel-based framework for comparing regression functions that leverages machine learning for flexible estimation, overcoming limitations of traditional smoothing methods especially in high-dimensional settings.

Contribution

It proposes a generalized dependence measure and two novel tests for regression equality, with proven asymptotic properties under broad conditions.

Findings

Effective in high-dimensional regimes

No restrictive distributional assumptions needed

Demonstrated superior performance in numerical studies

Abstract

We revisit the classical problem of comparing regression functions, a fundamental question in statistical inference with broad relevance to modern applications such as data integration, transfer learning, and causal inference. Existing approaches typically rely on smoothing techniques and are thus hindered by the curse of dimensionality. We propose a generalized notion of kernel-based conditional mean dependence that provides a new characterization of the null hypothesis of equal regression functions. Building on this reformulation, we develop two novel tests that leverage modern machine learning methods for flexible estimation. We establish the asymptotic properties of the test statistics, which hold under both fixed- and high-dimensional regimes. Unlike existing methods that often require restrictive distributional assumptions, our framework only imposes mild moment conditions. The efficacy of the proposed tests is demonstrated through extensive numerical studies.