  Despite a substantial literature on nonparametric two-sample goodness-of-fit
testing in arbitrary dimensions spanning decades, there is no mention there of
any curse of dimensionality. Only more recently Ramdas et al. (2015) have
discussed this issue in the context of kernel methods by showing that their
performance degrades with the dimension even when the underlying distributions
are isotropic Gaussians. We take a minimax perspective and follow in the
footsteps of Ingster (1987) to derive the minimax rate in arbitrary dimension
when the discrepancy is measured in the L2 metric. That rate is revealed to be
nonparametric and exhibit a prototypical curse of dimensionality. We further
extend Ingster's work to show that the chi-squared test achieves the minimax
rate. Moreover, we show that the test can be made to work when the
distributions have support of low intrinsic dimension. Finally, inspired by
Ingster (2000), we consider a multiscale version of the chi-square test which
can adapt to unknown smoothness and/or unknown intrinsic dimensionality without
much loss in power.
