Generalized bootstrap in the Bures-Wasserstein space

This study proposes a bootstrap-based method for uncertainty quantification in two important statistical scenarios.

First, we approximate the sampling distribution of empirical barycenters under the Bures--Wasserstein metric using a reweighted estimator.

Our theoretical results guarantee the accuracy of this approximation and enable the construction of data-driven confidence sets.

The methodology is validated through experiments on graph-structured data, including stochastic block models and brain connectomes.

Additionally, we compare bootstrap-based confidence sets with the asymptotic confidence sets obtained in arXiv:1901.00226v2, evaluating both their statistical performance and computational complexity.

Second, we investigate the generalized bootstrap framework for $M$-estimators without requiring a specific resampling scheme, thus covering both weighted and resampling methods under mild conditions.

Both contributions rely on a novel Gaussian approximation result for $M$-estimators.