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

TL;DR

This paper introduces TopoTests, a novel topology-based method using Euler characteristic curves for goodness-of-fit testing across arbitrary dimensions, with proven error control and exponential power growth.

Contribution

It presents a new topology-driven testing procedure applicable to any dimension, demonstrating comparable power and error control compared to existing methods.

Findings

Type I error can be controlled

Type II error decreases exponentially with sample size

Power demonstrated through extensive simulations

Abstract

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests. We call our procedure TopoTests. The presented tests work for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.