Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

In this talk, I will explain how to adopt the Euler characteristic curve (ECC) of a sample to perform one- and two-sample goodness of fit tests. Classical tools for this problem, like the Kolmogorov-Smirnov test, work well in one but not in higher dimensions. Our procedure, which we call TopoTests, works for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. We demonstrate that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. In this context, a necessary and sufficient condition for distributions to be distinguishable by means of the ECC is identified. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.