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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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References

[1]Berrendero, J. R., Cuevas, A. and Pateiro-López, B. (2012). A multivariate uniformity test for the case of unknown support. Statist. Comput. 22, 259–271. Google Scholar
[2]Berrendero, J. R., Cuevas, A. and Vázquez-Grande, F. (2006). Testing multivariate uniformity: the distance-to-boundary method. Canad. J. Statist. 34, 693–707. Google Scholar
[3]Bickel, P. J. and Breiman, L. (1983). Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Prob. 11, 185–214. Google Scholar
[4]Deheuvels, P. (1983). Strong bounds for multidimensional spacings. Z. Wahrscheinlichkeitsth. 64, 411–424. Google Scholar
[5]Dette, H. and Henze, N. (1989). The limit distribution of the largest nearest-neighbour link in the unit d-cube. J. Appl. Prob. 26, 67–80. Google Scholar
[6]Dette, H. and Henze, N. (1990). Some peculiar boundary phenomena for extremes of rth nearest neighbor links. Statist. Prob. Lett. 10, 381–390. Google Scholar
[7]Ebner, B., Henze, N. and Yukich, J. E. (2018). Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances. J. Multivariate Anal. 165, 231–242. Google Scholar
[8]Giné, E. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3, 1243–1266. Google Scholar
[9]Henze, N. (1982). The limit distribution for maxima of 'weighted' rth-nearest-neighbour distances. J. Appl. Prob. 19, 344–354. Google Scholar
[10]Henze, N. (1983). Ein asymptotischer Satz über den maximalen Minimalabstand von unabhängigen Zufallsvektoren mit Anwendung auf einen Anpassungstest im ℝP und auf der Kugel. Metrika 30, 245–259. Google Scholar
[11]Janson, S. (1986). Random coverings in several dimensions. Acta Math. 156, 83–118. Google Scholar
[12]Janson, S. (1987). Maximal spacings in several dimensions. Ann. Prob. 15, 274–280. Google Scholar
[13]Jupp, P.E. (2008). Data-driven Sobolev tests of uniformity on compact Riemannian manifolds. Ann. Statist. 36, 1246–1260. Google Scholar
[14]Justel, A., Peña, D. and Zamar, R. (1997). A multivariate Kolmogorov-Smirnov test of goodness of fit. Statist. Prob. Lett. 35, 251–259. Google Scholar
[15]Liang, J.-J., Fang, K.-T., Hickernell, F. T. and Li, R. (2001). Testing multivariate uniformity and its applications. Math. Comput. 70, 337–355. Google Scholar
[16]Petrie, A. and Willemain, T. R. (2013). An empirical study of tests for uniformity in multidimensional data. Comput. Statist. Data Anal. 64, 253–268. Google Scholar
[17]Steele, J. M. and Tierney, L. (1986). Boundary domination and the distribution of the largest nearest-neighbor link in higher dimensions. J. Appl. Prob. 23, 524–528. Google Scholar
[18]Weiss, L. (1960). A test of fit based on the largest sample spacing. J. Soc. Indust. Appl. Math. 8, 295–299. Google Scholar
[19]Yang, M. and Modarres, R. (2017). Multivariate tests of uniformity. Statist. Papers 58, 627–639. Google Scholar
[20]Zhou, S. and Jammalamadaka, S.R. (1993). Goodness of fit in multidimensions based on nearest neighbour distances. J. Nonparametric Statist. 2, 271–284. Google Scholar