Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

Construction of Non-asymptotic Confidence Sets in 2 -Wasserstein Space

Johannes Ebert
Vladimir Spokoiny
Alexandra Suvorikova


Abstract
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 [29],
Chernozhukov, Chetverikov and Kato [13]). 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.


Keywords:
Wasserstein barycenters, hypothesis testing, multiplier bootstrap,
change point detection, confidence sets.

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