# Testing multivariate uniformity based on random geometric graphs

## TL;DR

This paper introduces new goodness-of-fit tests for uniformity in multivariate data using random geometric graphs, demonstrating their effectiveness through theoretical proofs, simulations, and real data applications.

## Contribution

The paper proposes novel uniformity tests based on edge lengths of random geometric graphs, with proven asymptotic normality and superior performance in small samples.

## Key findings

- Tests are asymptotically normal under null and alternative hypotheses.
- The procedures are consistent and can detect non-uniformity in small samples.
- Simulations show they outperform or match established tests.

## Abstract

We present new families of goodness-of-fit tests of uniformity on a full-dimensional set $W\subset\R^d$ based on statistics related to edge lengths of random geometric graphs. Asymptotic normality of these statistics is proven under the null hypothesis as well as under fixed alternatives. The derived tests are consistent and their behaviour for some contiguous alternatives can be controlled. A simulation study suggests that the procedures can compete with or are better than established goodness-of-fit tests. We show with a real data example that the new tests can detect non-uniformity of a small sample data set, where most of the competitors fail.