Testing conditional independence using maximal nonlinear conditional correlation

TL;DR

This paper introduces a new measure called maximal nonlinear conditional correlation to test for conditional independence between two variables given a third, providing a statistically sound and consistent testing procedure.

Contribution

It defines a novel measure of conditional association and develops an asymptotically normal test for conditional independence based on this measure.

Findings

The test statistic is asymptotically normal under the null hypothesis.

The proposed test is consistent in detecting conditional dependence.

The measure captures nonlinear relationships effectively.

Abstract

In this paper, the maximal nonlinear conditional correlation of two random vectors $X$ and $Y$ given another random vector $Z$, denoted by $\rho_1(X,Y|Z)$, is defined as a measure of conditional association, which satisfies certain desirable properties. When $Z$ is continuous, a test for testing the conditional independence of $X$ and $Y$ given $Z$ is constructed based on the estimator of a weighted average of the form $\sum_{k=1}^{n_Z}f_Z(z_k)\rho^2_1(X,Y|Z=z_k)$, where $f_Z$ is the probability density function of $Z$ and the $z_k$'s are some points in the range of $Z$. Under some conditions, it is shown that the test statistic is asymptotically normal under conditional independence, and the test is consistent.