Develop conditional independence inference for the conditional binomial-cut coefficient

Develop a valid inference procedure for the conditional binomial-cut coefficient \(\xi_{\mathrm{cut}}^{\mid Z}(X,Y)\) under the conditional-independence null hypothesis \(H_0:X\perp Y\mid Z\), using a conditional-randomization scheme rather than ordinary permutation, and establish the full conditional-testing framework.

Background

The paper defines a conditional variant, $$\xi_{\mathrm{cut}}^{\mid Z}(X,Y)=2\int E_{X,Z}[\phi(F(t\mid X,Z),F(t\mid Z))]dF(t\mid Z)$$, to measure residual dependence between $X$ and $Y$ after adjustment for covariates $Z$. Unlike the unconditional null, the conditional null $X\perp Y\mid Z$ cannot generally be calibrated by freely permuting $Y$, because ordinary permutation destroys the association between $Y$ and $Z$.

The paper therefore identifies the development of conditional-randomization or conditional-permutation inference for this coefficient as unresolved. Such a procedure would provide valid calibration for testing conditional independence while accounting for the kernel smoothing over $Z$.