On Testing Conditional Mean Independence for Manifold-Valued Data

This paper introduces a nonparametric test for conditional mean independence between a manifold‑valued $Y$ and Euclidean predictors $X$. The test is built on a new measure called the Manifold Martingale Difference Divergence (MMDD), which characterizes conditional mean dependence by projecting observations onto the tangent space via the logarithmic map. We provide an empirical estimator for the MMDD, establish its asymptotic null distribution, and implement a wild bootstrap procedure for finite‑sample inference. Simulations on three representative manifolds demonstrate that the proposed test maintains correct size under the null even when the distribution of $Y$ depends on $X$, in contrast to the severe size distortion exhibited by the distance covariance (dCov) test. At the same time, it achieves competitive power across a range of alternatives. An application to real data illustrates its practical utility.