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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.

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