On the consistency of the spacings test for multivariate uniformity

TL;DR

This paper provides a simple proof of the consistency of a multivariate uniformity test based on maximal spacings, which measures the largest convex region in a bounded set that contains no sample points.

Contribution

It offers a new, straightforward proof of the test's consistency for multivariate uniformity using a coupling approach, extending previous results.

Findings

The test is consistent for multivariate uniformity.

The proof simplifies understanding of the test's properties.

Coupling techniques are effective in multivariate uniformity testing.

Abstract

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set $K \subset \mathbb{R}^d$ that is based on the maximal spacing generated by i.i.d. points $X_1, \ldots,X_n$ in $K$, i.e., the volume of the largest convex set of a given shape that is contained in $K$ and avoids each of these points. Since asymptotic results for the case $d > 1$ are only availabe under uniformity, a key element of the proof is a suitable coupling.