  This paper is concerned with test of the conditional independence. We first
establish an equivalence between the conditional independence and the mutual
independence. Based on the equivalence, we propose an index to measure the
conditional dependence by quantifying the mutual dependence among the
transformed variables. The proposed index has several appealing properties. (a)
It is distribution free since the limiting null distribution of the proposed
index does not depend on the population distributions of the data. Hence the
critical values can be tabulated by simulations. (b) The proposed index ranges
from zero to one, and equals zero if and only if the conditional independence
holds. Thus, it has nontrivial power under the alternative hypothesis. (c) It
is robust to outliers and heavy-tailed data since it is invariant to
conditional strictly monotone transformations. (d) It has low computational
cost since it incorporates a simple closed-form expression and can be
implemented in quadratic time. (e) It is insensitive to tuning parameters
involved in the calculation of the proposed index. (f) The new index is
applicable for multivariate random vectors as well as for discrete data. All
these properties enable us to use the new index as statistical inference tools
for various data. The effectiveness of the method is illustrated through
extensive simulations and a real application on causal discovery.
