Remember the Curse of Dimensionality: The Case of Goodness-of-Fit Testing in Arbitrary Dimension

Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions, there is no mention there of any curse of dimensionality. In fact, in some publications, a parametric rate is derived. As we discuss below, this is because a directional alternative is considered. Indeed, even in dimension one, Ingster (1987) has shown that the minimax rate is not parametric. In this paper, we extend his results to arbitrary dimension and confirm that the minimax rate is not only nonparametric, but exhibits a prototypical curse of dimensionality. We further extend Ingster's work to show that simple tests based on bin-counting achieve the minimax rate. Moreover, these tests adapt to the intrinsic dimensionality of the data --- when the underlying distributions are supported on a lower-dimensional surface.