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.Comment: Published in at http://dx.doi.org/10.1214/09-AOS770 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org