Needlet-Whittle Estimates on the Unit Sphere

We study the asymptotic behaviour of needlets-based approximate maximum likelihood estimators for the spectral parameters of Gaussian and isotropic spherical random fields. We prove consistency and asymptotic Gaussianity, in the high-frequency limit, thus generalizing earlier results by Durastanti et al. (2011) based upon standard Fourier analysis on the sphere. The asymptotic results are then illustrated by an extensive Monte Carlo study.Comment: 48 pages, 2 figure

High-Frequency Tail Index Estimation by Nearly Tight Frames

This work develops the asymptotic properties (weak consistency and Gaussianity), in the high-frequency limit, of approximate maximum likelihood estimators for the spectral parameters of Gaussian and isotropic spherical random fields. The procedure we used exploits the so-called mexican needlet construction by Geller and Mayeli in [Geller, Mayeli (2009)]. Furthermore, we propose a plug-in procedure to optimize the precision of the estimators in terms of asymptotic variance.Comment: 38 page

Asymptotics for spherical needlets

We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasi-exponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high-frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets.Comment: Published in at http://dx.doi.org/10.1214/08-AOS601 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere (Mexican needlets) which were recently introduced in the literature by Geller and Mayeli (2007). We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data.Comment: Revised version for Stochastic Processes and their Application

Mixed Needlets

The construction of needlet-type wavelets on sections of the spin line bundles over the sphere has been recently addressed in Geller and Marinucci (2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal for needlets on this spin line bundle, in which needlet coefficients arise from the usual, rather than the spin, spherical harmonics, as in the previous constructions. We label this system mixed needlets and investigate in full their properties, including localization, the exact tight frame characterization, reconstruction formula, decomposition of functional spaces, and asymptotic uncorrelation in the stochastic case. We outline astrophysical applications.Comment: 26 page

Subsampling needlet coefficients on the sphere

In a recent paper, we analyzed the properties of a new kind of spherical wavelets (called needlets) for statistical inference procedures on spherical random fields; the investigation was mainly motivated by applications to cosmological data. In the present work, we exploit the asymptotic uncorrelation of random needlet coefficients at fixed angular distances to construct subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate how such statistics can be used for isotropy tests and for bootstrap estimation of nuisance parameters, even when a single realization of the spherical random field is observed. The asymptotic theory is developed in detail in the high resolution sense.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Multi-resolution anisotropy studies of ultrahigh-energy cosmic rays detected at the Pierre Auger Observatory

We report a multi-resolution search for anisotropies in the arrival directions of cosmic rays detected at the Pierre Auger Observatory with local zenith angles up to $80^\circ$ and energies in excess of 4 EeV ($4 \times 10^{18}$ eV). This search is conducted by measuring the angular power spectrum and performing a needlet wavelet analysis in two independent energy ranges. Both analyses are complementary since the angular power spectrum achieves a better performance in identifying large-scale patterns while the needlet wavelet analysis, considering the parameters used in this work, presents a higher efficiency in detecting smaller-scale anisotropies, potentially providing directional information on any observed anisotropies. No deviation from isotropy is observed on any angular scale in the energy range between 4 and 8 EeV. Above 8 EeV, an indication for a dipole moment is captured; while no other deviation from isotropy is observed for moments beyond the dipole one. The corresponding $p$-values obtained after accounting for searches blindly performed at several angular scales, are $1.3 \times 10^{-5}$ in the case of the angular power spectrum, and $2.5 \times 10^{-3}$ in the case of the needlet analysis. While these results are consistent with previous reports making use of the same data set, they provide extensions of the previous works through the thorough scans of the angular scales.Comment: Published version. Added journal reference and DOI. Added Report Numbe