Center-Outward Ranks and Signs for Testing Conditional Quantile Independence

This article treats the problem of constructing nonparametric asymptotically distribution-free tests for conditional quantile independence in multidimensions.Our approach combines the quantile martingale difference divergence with a notion of the recently introduced multivariate center-outward ranks and signs.We derive the asymptotic null representation of the proposed test statistics by exploiting the degenerate V -type and U -type structures of the quantile martingale difference divergence and the Glivenko-Cantelli strong consistency and distribution-freeness of the center-outward ranks and signs.This representation permits direct calculation of limiting null distributions without requiring bootstrap calibration.We further show that our center-outward versions of the quantile martingale difference divergence tests are consistent against all fixed alternatives.A local power analysis provides strong support for the center-outward approach by establishing the nontrivial power of our center-outward rank-based tests over root-n neighborhoods.Moreover, the proposed tests are computationally feasible and well-defined without any moment assumptions.We illustrate the advantages of the proposed methods via extensive simulation studies and a Statistica Sinica: Newly accepted Paper gene expression dataset analysis.