Conference Agenda

In recent years, statistical methodology based on optimal transport (OT) witnessed a considerable increase in practical and theoretical interest. A central reason for this trend is the ability of optimal transport to efficiently compare data in a geometrically meaningful way. This development was further amplified by computational advances spurred by the introduction of entropy regularized optimal transport (EOT). In applications, the OT or EOT cost are often estimated through an empirical plug-in approach, raising statistical questions about the performance and uncertainty of these estimators. This talk will survey recent theoretical and methodological insights to these topics and discusses future opportunities.

This talk is based on joint work with Thomas Staudt, Marcel Klatt, Michel Groppe, Alberto-Gonzáles-Sanz, Gilles Mordant, Christoph Weitkamp, and Axel Munk.

Tests for equality of distributions based on Optimal Transport functionals are often referred to as being not distribution free: asymptotic laws for tests statistics depend on the underlying true distributions, and this dependence seems unavoidable. Here we show that these tests are ``almost" distribution free, in a sense that there exist cut-offs independent of the true distributions that result in tests with given level of significance. These cut-offs are easy to compute and may serve as a rule-of-thumb-type heuristics, making Optimal Transport based tests more accessible for practical applications.

In this talk we are interested in detecting change-points of univariate nonstationary time series in a nonparametric setting. We introduce statistics based on the Wasserstein distance between local empirical distribution functions of the time series which are suitable to detect change-points. The one-dimensional Wasserstein distance is characterized by the sequential quantile process, and we show that this weakly converges to a Gaussian limit. Due to the nonlinearity of the quantile process, difficulties arise from the localization. A new Bahadur representation result is needed to address this, which allows us to consider the asymptotic behavior of the empirical process instead of the quantile process. The proof of this requires further study of the modulus of continuity of the empirical process. As the limit distributions of the test statistics depend on the unknown underlying distributions, a Gaussian multiplier bootstrap scheme is introduced. Lastly, a simulation study shows how well the significance level is retained under the null hypothesis of no change, and an outlook towards the power of the tests will be given.