A decomposition for invariant tests of uniformity on the sphere

Pycke, Jean-Renaud

doi:10.1090/S0002-9939-07-08804-1

A decomposition for invariant tests of uniformity on the sphere
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by Jean-Renaud Pycke PDF

Proc. Amer. Math. Soc. 135 (2007), 2983-2993 Request permission

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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