Analysis of Conditional Randomisation and Permutation schemes with application to conditional independence testing

We study properties of two resampling scenarios: Conditional Randomisation and Conditional Permutation scheme which are relevant for conditional independence testing of discrete random variables $X$ and $Y$ given random variable $Z$. Namely, we investigate asymptotic behaviour of estimates of a vector of probabilities in such settings, establish their asymptotic normality and ordering between asymptotic covariance matrices. The results are used to derive asymptotic distributions of empirical Conditional Mutual Information in these set-ups. Somewhat unexpectedly, the distributions coincide for the two scenarios, despite differences in asymptotic distribution of estimates of probabilities. We also prove validity of permutation p-values for Conditional Permutation scheme. The above results justify consideration of conditional independence tests based on re-sampled p-values and on asymptotic chi square distribution with adjusted number of degrees of freedom. We show in numerical experiments that when the ratio of the sample size to the number of possible values of the triple exceeds 0.5, the test based on the asymptotic distribution with the adjustment made on limited number of permutations is a viable alternative to the exact test for both Conditional Permutation and Conditional Randomisation scenarios. Moreover, there is no significant difference between performance of exact tests for Conditional Permutation and Randomisation scheme, the latter requiring knowledge of conditional distribution of $X$ given $Z$, and the same conclusion is true for both adaptive tests.