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Testing conditional independence using maximal nonlinear conditional correlation

In this paper, the maximal nonlinear conditional correlation of two random vectors $X$ and $Y$ given another random vector $Z$, denoted by $ρ_1(X,Y|Z)$, 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 $\sum_{k=1}^{n_Z}f_Z(z_k)ρ^2_1(X,Y|Z=z_k)$, 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.