A conditional distribution function-based measure for independence and K-sample tests in multivariate data

We introduce a new index to measure the degree of dependence and test for independence between two random vectors. The index is obtained by generalizing the Cramér–von Mises distances between the conditional and marginal distribution functions via the projection-averaging technique. If one of the random vectors is categorical with $K$ categories, we propose slicing estimators to estimate our index. We conduct an asymptotic analysis for the slicing estimators, considering both situations where $K$ is fixed and where $K$ is allowed to increase with the sample size. When both random vectors are continuous, we introduce a kernel regression estimator for the proposed index, demonstrating that its asymptotic null distribution follows a normal distribution and conducting a local power analysis for the kernel estimator-based independence test. The proposed tests are studied via simulations, with a real data application presented to illustrate our methods.