Test of partial effects for Frechet regression on Bures-Wasserstein manifolds

TL;DR

This paper introduces a new statistical test for partial effects in Frechet regression on Bures-Wasserstein manifolds, combining sample splitting, asymptotic theory, and practical validation.

Contribution

It presents a novel test procedure for Frechet regression on Bures-Wasserstein manifolds, with proven asymptotic properties and demonstrated effectiveness through simulations and real data.

Findings

Test statistic converges to a weighted mixture of chi squared distributions.

The procedure maintains nominal size asymptotically.

Finite-sample performance is validated via simulations and real data.

Abstract

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.