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On the consistency of the spacings test for multivariate uniformity, including on manifolds

Part of: Nonparametric inference Multivariate analysis

Published online by Cambridge University Press:  26 July 2018

Norbert Henze
Norbert Henze*
Affiliation:
Karlsruhe Institute of Technology
*
* Postal address: Institute of Stochastics, Karlsruhe Institute of Technology, Englerstr. 2, D-76133 Karlsruhe, Germany. Email address: [email protected]
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Abstract

We give a simple conceptual proof of the consistency of a test for multivariate uniformity in a bounded set K ⊂ ℝd that is based on the maximal spacing generated by independent and identically distributed points X1, . . ., Xn 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.

Keywords

Multivariate spacingtest for multivariate uniformityconsistencyextreme-value distributionRiemannian manifold

MSC classification

Primary: 62H15: Hypothesis testing
Secondary: 62G20: Asymptotic properties
Type
Short Communications
Information
Journal of Applied Probability , Volume 55 , Issue 2 , June 2018 , pp. 659 - 665
Copyright
Copyright © Applied Probability Trust 2018 

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