Generalized Kernel Two-Sample Tests

Kernel two-sample tests have been widely used for multivariate data in testing equal distribution. However, existing tests based on mapping distributions into a reproducing kernel Hilbert space are mainly targeted at specific alternatives and do not work well for some scenarios when the dimension of the data is moderate to high due to the curse of dimensionality. We propose a new test statistic that makes use of a common pattern under moderate and high dimensions and achieves substantial power improvements over existing kernel two-sample tests for a wide range of alternatives. We also propose alternative testing procedures that maintain high power with low computational cost, offering easy off-the-shelf tools for large datasets. The new approaches are compared to other state-of-the-art tests under various settings and show good performance. The new approaches are illustrated on two applications: The comparison of musks and non-musks using the shape of molecules, and the comparison of taxi trips started from John F.Kennedy airport in consecutive months. All proposed methods are implemented in an R package kerTests.