Title:          Distribution-Free Nonparametric Inference Based on Optimal Transport and Kernel Methods 
Abstract: The Wilcoxon rank-sum (or Mann–Whitney) test is one of the most widely used tools for comparing two groups without making assumptions about the underlying data distribution. One of the reasons for its enduring popularity is a 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 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 other words, the Gaussian score transformed Wilcoxon test uniformly dominates the t-test in terms of efficiency, while also remaining distribution-free.

In this talk we will discuss multivariate versions of these celebrated results, by considering distribution-free analogues of the Hotelling T²-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. We will also discuss how optimal-transport-based multivariate ranks can be used to construct distribution-free analogues of the celebrated kernel two-sample test that enjoy a trifecta of desirable properties: universal consistency, efficient computation, and nontrivial asymptotic efficiency. 

Short Bio: Bhaswar B. Bhattacharya is an Associate Professor in the Department of Statistics and Data Science at the Wharton School, University of Pennsylvania. He received his Ph.D. from the Department of Statistics at Stanford University in 2016. Prior to that, he obtained his Bachelor and Master degrees in Statistics from the Indian Statistical Institute, Kolkata in 2009 and 2011, respectively. His research interests include non-parametric statistics, combinatorial probability, and discrete and computational geometry.