Test of partial effects for Frechet regression on Bures-Wasserstein manifolds
This provides a rigorous testing framework for high-dimensional covariance matrix analysis, though it is incremental as it adapts existing sample-splitting methods to a specific manifold setting.
The authors developed a statistical test to evaluate partial effects in Fréchet regression on Bures-Wasserstein manifolds, proving it achieves correct asymptotic size and uniform power convergence to one, with validation through simulations and real data.
We propose a novel test for assessing partial effects in Frechet regression on Bures Wasserstein manifolds. Our approach employs a sample splitting strategy: the first subsample is used to fit the Frechet regression model, yielding estimates of the covariance matrices and their associated optimal transport maps, while the second subsample is used to construct the test statistic. We prove that this statistic converges in distribution to a weighted mixture of chi squared components, where the weights correspond to the eigenvalues of an integral operator defined by an appropriate RKHS kernel. We establish that our procedure achieves the nominal asymptotic size and demonstrate that its worst-case power converges uniformly to one. Through extensive simulations and a real data application, we illustrate the test's finite-sample accuracy and practical utility.