# Conditional Mean and Quantile Dependence Testing in High Dimension

## TL;DR

This paper introduces a new nonparametric test for assessing conditional mean and quantile dependence in high-dimensional data, effective for nonlinear relationships and supported by theoretical asymptotic properties.

## Contribution

It develops a novel test based on martingale difference divergence that detects departures from conditional mean independence without specific model assumptions, applicable in high dimensions.

## Key findings

- Performs well under nonlinear models, outperforming linear-based competitors.
- Establishes asymptotic normality of the test statistic under certain eigenvalue conditions.
- Provides a method for testing conditional quantile independence with theoretical justification.

## Abstract

Motivated by applications in biological science, we propose a novel test to assess the conditional mean dependence of a response variable on a large number of covariates. Our procedure is built on the martingale difference divergence recently proposed in Shao and Zhang (2014), and it is able to detect a certain type of departure from the null hypothesis of conditional mean independence without making any specific model assumptions. Theoretically, we establish the asymptotic normality of the proposed test statistic under suitable assumption on the eigenvalues of a Hermitian operator, which is constructed based on the characteristic function of the covariates. These conditions can be simplified under banded dependence structure on the covariates or Gaussian design. To account for heterogeneity within the data, we further develop a testing procedure for conditional quantile independence at a given quantile level and provide an asymptotic justification. Empirically, our test of conditional mean independence delivers comparable results to the competitor, which was constructed under the linear model framework, when the underlying model is linear. It significantly outperforms the competitor when the conditional mean admits a nonlinear form.