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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A decomposition for invariant tests of uniformity on the sphere


Author: Jean-Renaud Pycke
Journal: Proc. Amer. Math. Soc. 135 (2007), 2983-2993
MSC (2000): Primary 62G10, 62H11; Secondary 47G10, 20C15
Published electronically: May 8, 2007
MathSciNet review: 2317977
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Abstract: We introduce a $ U$-statistic on which can be based a test for uniformity on the sphere. It is a simple function of the geometric mean of distances between points of the sample and consistent against all alternatives. We show that this type of $ U$-statistic, whose kernel is invariant by isometries, can be separated into a set of statistics whose limiting random variables are independent. This decomposition is obtained via the so-called canonical decomposition of a group representation. The distribution of the limiting random variables of the components under the null hypothesis is given. We propose an interpretation of Watson type identities between quadratic functionals of Gaussian processes in the light of this decomposition.


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Additional Information

Jean-Renaud Pycke
Affiliation: Départment de Mathématiques, Université d’Évry Val d’Essone, Boulevard F. Mitterrand, F-91025 Evry Cedex, France
Email: jrpycke@univ-evry.fr, pycke@ccr.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08804-1
PII: S 0002-9939(07)08804-1
Keywords: Goodness of fit test, $U$-statistics, group representations
Received by editor(s): January 1, 2006
Received by editor(s) in revised form: May 30, 2006
Published electronically: May 8, 2007
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.