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Distribution-Free Nonparametric Inference Based on Optimal Transport: Efficiency Lower Bounds and Rank-Kernel Tests

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If you have a question about this talk, please contact Dr Sergio Bacallado.

The Wilcoxon rank-sum/Mann-Whitney test is one of the most popular distribution-free procedures for testing the equality of two univariate probability distributions. One of the main reasons for its popularity can be attributed to the remarkable result of Hodges and Lehmann (1956), which shows that the asymptotic relative efficiency of Wilcoxon’s test with respect to Student’s t-test, under location alternatives, never falls below 0.864, despite the former being exactly distribution-free in finite samples. Even more striking is the result of Chernoff and Savage (1958), which shows that the efficiency of a Gaussian score transformed Wilcoxon’s test, against the t-test, is lower bounded by 1. In this talk we will discuss multivariate versions of these celebrated results, by considering distribution-free analogues of the Hotelling T^2-test based on optimal transport. The proposed tests are consistent against a general class of alternatives and satisfy Hodges-Lehmann and Chernoff-Savage-type efficiency lower bounds over various natural families of multivariate distributions, despite being entirely agnostic to the underlying data generating mechanism. Analogous results for independence testing will also be presented. Finally, we will discuss how optimal transport based multivariate ranks can be used to obtain distribution-free kernel two-sample tests, which are universally consistent, computationally efficient, and have non-trivial asymptotic efficiency.

(Based on joint work with Nabarun Deb and Bodhisattva Sen.)

This talk is part of the Statistics series.

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