The Statistics Seminar Speaker for Wednesday, January 27, is Bhaswar Bhattacharya of Stanford University. He is currently a fifth-year Ph.D. student in the Department of Statistics at Stanford University advised by Persi Diaconis. Prior to this, he received his Bachelor and Master degrees in Statistics from the Indian Statistical Institute, Kolkata.

Title: **Power of Graph-Based Two-Sample Tests**

Abstract: Testing equality of two multivariate distributions is a classical problem for which many non-parametric tests have been proposed over the years. Most of the popular tests are based either on geometric graphs constructed using inter-point distances between the observations (multivariate generalizations of the Wald-Wolfowitz's runs test) or on multivariate data-depth (generalizations of the Mann-Whitney rank test). These tests are known to be asymptotically normal under the null and consistent against all fixed alternatives. In this talk, a general framework of graph-based tests will be introduced that includes all these tests.

The asymptotic efficiency of a general graph-based test can be derived using Le Cam's theory of local asymptotic normality, which provides a theoretical basis for comparing the performances of these tests. As a consequence, it will be shown that popular tests based on geometric graphs such as the Friedman-Rafsky test (1979), the test based on the -nearest neighbor graph (1984), the minimum matching test of Rosenbaum (2005), among others have zero asymptotic (Pitman) efficiency against alternatives. On the other hand, the tests based on multivariate depth functions (the Liu-Singh rank sum statistic (1993)), which include the Tukey depth (1975) and the projection depth (2003), have non-zero asymptotic efficiency; though they might be computationally expensive when the dimension is large.

Finally, the limiting normal distribution of tests based on stabilizing random geometric graphs will be derived in the Poissonized setting. This can be used to compute the power of such tests against local alternatives, which validates the various applications of these tests and provides a way to compare between tests with zero Pitman efficiency.

*Refreshments will be served after the seminar in 1181 Comstock Hall. *