Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances

TL;DR

This paper introduces a unified method for goodness-of-fit testing applicable in Euclidean spaces and on embedded manifolds, using nearest neighbor distances, with proven asymptotic properties and competitive performance.

Contribution

It develops a novel, unified goodness-of-fit testing framework based on nearest neighbor distances for flat and curved spaces, with theoretical guarantees and practical validation.

Findings

Asymptotic normality of test statistics under null and alternative hypotheses.

Convergence to $oldsymbol{ extalpha}$-entropy under fixed alternatives.

Competitiveness with established goodness-of-fit tests in simulations.

Abstract

We present a unified approach to goodness-of-fit testing in $\mathbb{R}^d$ and on lower-dimensional manifolds embedded in $\mathbb{R}^d$ based on sums of powers of weighted volumes of $k$-th nearest neighbor spheres. We prove asymptotic normality of a class of test statistics under the null hypothesis and under fixed alternatives. Under such alternatives, scaled versions of the test statistics converge to the $\alpha$-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests.