Testing conditional independence using maximal nonlinear conditional correlation

Abstract

In this paper, the maximal nonlinear conditional correlation of two random vectors X and Y given another random vector Z, denoted by ρ₁(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 ∑_k=1^n_Zf_Z(z_k)ρ₁²(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.