Asymptotic normality of an estimator of kernel-based conditional mean dependence measure

TL;DR

This paper introduces a new kernel-based estimator for conditional mean dependence, proves its asymptotic normality under various hypotheses, and develops a novel test for conditional mean independence in Hilbert space-valued variables.

Contribution

The paper presents a modified estimator with proven asymptotic normality and a new test for conditional mean independence in Hilbert spaces, advancing statistical dependence measures.

Findings

Estimator achieves asymptotic normality under null and alternative hypotheses.

New test effectively detects conditional mean independence in Hilbert space variables.

Method improves upon naive estimators with better theoretical properties.

Abstract

We propose an estimator of the kernel-based conditional mean dependence measure obtained from an appropriate modification of a naive estimator based on usual empirical estimators. We then get asymptotic normality of this estimator both under conditional mean independence hypothesis and under the alternative hypothesis. A new test for conditional mean independence of random variables valued into Hilbert spaces is then introduced.