Open Access
June 2013 Testing the isotropy of high energy cosmic rays using spherical needlets
Gilles Faÿ, Jacques Delabrouille, Gérard Kerkyacharian, Dominique Picard
Ann. Appl. Stat. 7(2): 1040-1073 (June 2013). DOI: 10.1214/12-AOAS619


For many decades, ultrahigh energy charged particles of unknown origin that can be observed from the ground have been a puzzle for particle physicists and astrophysicists. As an attempt to discriminate among several possible production scenarios, astrophysicists try to test the statistical isotropy of the directions of arrival of these cosmic rays. At the highest energies, they are supposed to point toward their sources with good accuracy. However, the observations are so rare that testing the distribution of such samples of directional data on the sphere is nontrivial. In this paper, we choose a nonparametric framework that makes weak hypotheses on the alternative distributions and allows in turn to detect various and possibly unexpected forms of anisotropy. We explore two particular procedures. Both are derived from fitting the empirical distribution with wavelet expansions of densities. We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574–594 (electronic)], the so-called needlets. The expansions are truncated at scale indices no larger than some $J^{\star}$, and the $L^{p}$ distances between those estimates and the null density are computed. One family of tests (called MULTIPLE) is based on the idea of testing the distance from the null for each choice of $J=1,\ldots,J^{\star}$, whereas the so-called PLUGIN approach is based on the single full $J^{\star}$ expansion, but with thresholded wavelet coefficients. We describe the practical implementation of these two procedures and compare them to other methods in the literature. As alternatives to isotropy, we consider both very simple toy models and more realistic nonisotropic models based on Physics-inspired simulations. The Monte Carlo study shows good performance of the MULTIPLE test, even at moderate sample size, for a wide sample of alternative hypotheses and for different choices of the parameter $J^{\star}$. On the 69 most energetic events published by the Pierre Auger Collaboration, the needlet-based procedures suggest statistical evidence for anisotropy. Using several values for the parameters of the methods, our procedures yield $p$-values below 1%, but with uncontrolled multiplicity issues. The flexibility of this method and the possibility to modify it to take into account a large variety of extensions of the problem make it an interesting option for future investigation of the origin of ultrahigh energy cosmic rays.


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Gilles Faÿ. Jacques Delabrouille. Gérard Kerkyacharian. Dominique Picard. "Testing the isotropy of high energy cosmic rays using spherical needlets." Ann. Appl. Stat. 7 (2) 1040 - 1073, June 2013.


Published: June 2013
First available in Project Euclid: 27 June 2013

zbMATH: 06279864
MathSciNet: MR3113500
Digital Object Identifier: 10.1214/12-AOAS619

Keywords: isotropy test , multiple tests , Nonparametric test , ultrahigh energy cosmic rays , wavelet procedure

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.7 • No. 2 • June 2013
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