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Computational aspects of Cui-Freeden statistics for equidistribution on the sphere

Authors: Christine Choirat and Raffaello Seri
Journal: Math. Comp. 82 (2013), 2137-2156
MSC (2010): Primary 33C55, 60F05, 62E20; Secondary 86-08, 86A32, 11K45
Published electronically: April 29, 2013
MathSciNet review: 3073194
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Abstract: In this paper, we derive the asymptotic statistical properties of a class of generalized discrepancies introduced by Cui and Freeden (SIAM J. Sci. Comput., 1997) to test equidistribution on the sphere. We show that they have highly desirable properties and encompass several statistics already proposed in the literature. In particular, it turns out that the limiting distribution is an (infinite) weighted sum of chi-squared random variables. Issues concerning the approximation of this distribution are considered in detail and explicit bounds for the approximation error are given. The statistics are then applied to assess the equidistribution of Hammersley low discrepancy sequences on the sphere and the uniformity of a dataset concerning magnetic orientations.

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Additional Information

Christine Choirat
Affiliation: Department of Economics, School of Economics and Business Administration, Universidad de Navarra, Edificio Amigos, 31080 Pamplona, Spain

Raffaello Seri
Affiliation: Dipartimento di Economia, Università degli Studi dell’Insubria, Via Monte Generoso 71, 21100 Varese, Italy
MR Author ID: 710036

Keywords: Sphere, generalized discrepancy, equidistribution, approximation of distributions, quadratic forms in Gaussian random variables, low discrepancy (quasi-Monte Carlo) method
Received by editor(s): October 30, 2010
Received by editor(s) in revised form: February 9, 2012
Published electronically: April 29, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.