In this paper, we consider a probabilistic setting where the probability measures are considered to be random objects. We propose a procedure of construction non-asymptotic confidence sets for empirical barycenters in 2-Wasserstein space and develop the idea further to construction of a non-parametric two-sample test that is then applied to the detection of structural breaks in data with complex geometry. Both procedures mainly rely on the idea of multiplier bootstrap (Spokoiny and Zhilova (2015), Chernozhukov et al. (2014)). The main focus lies on probability measures that have commuting covariance matrices and belong to the same scatter-location family: we proof the validity of a bootstrap procedure that allows to compute confidence sets and critical values for a Wasserstein-based two-sample test.
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