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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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