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On the consistency of the spacings test for multivariate uniformity, including on manifolds. (English) Zbl 1401.62089
Summary: 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 independent and identically distributed points \(X_{1},\dots, X_n\) 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 \(d > 1\) case are only availabe under uniformity, a key element of the proof is a suitable coupling. The proof is general enough to cover the case of testing for uniformity on compact Riemannian manifolds with spacings defined by geodesic balls.

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
60D05 Geometric probability and stochastic geometry
Full Text: DOI
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