## STA 290 Seminar: Bhaswar Bhattacharya

### STA 290 Seminar Series

**Wednesday, January 6th, 4:10pm, MSB 1147 (Colloquium Room)**

Refreshments at 3:30pm in MSB 4110 (Statistics Lounge)

Speaker: **Bhaswar Bhattacharya** (Stanford University)

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 performance 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 derive 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.