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A decomposition for invariant tests of uniformity on the sphere
Author(s):
Jean-Renaud
Pycke
Journal:
Proc. Amer. Math. Soc.
135
(2007),
2983-2993.
MSC (2000):
Primary 62G10, 62H11;
Secondary 47G10, 20C15
Posted:
May 8, 2007
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Abstract:
We introduce a  -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  -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:
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
Posted:
May 8, 2007
Communicated by:
Edward C. Waymire
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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