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Title: Nonparametric Pitman efficient distribution-free testing using optimal transport Authors:  Nabarun Deb - Columbia University (United States) [presenting]
Abstract: In recent years, the problem of optimal transport has received significant attention in statistics and machine learning due to its powerful geometric properties. We introduce the optimal transport problem and present concrete applications of this theory in statistics. In particular, we will propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of optimal transport. We demonstrate the applicability of this approach by constructing exactly distribution-free tests for two classical nonparametric problems: (i) testing for the equality of two multivariate distributions, and (ii) testing for mutual independence between two random vectors. We investigate the consistency and asymptotic distributions of these tests, both under the null and local contiguous alternatives. We further study their local power and asymptotic (Pitman) efficiency, and show that a subclass of these tests achieve attractive efficiency lower bounds that mimic previous remarkable efficiency results. Finally, we also study the rates of convergence of the estimated optimal transport maps and show that the natural plugin estimators for these maps achieve minimax optimal rates of convergence without any tuning parameters.