Block Thresholding on the Sphere

The aim of this paper is to study the nonparametric regression estimators on the sphere built by the needlet block thresholding. The block thresholding procedure proposed here follows the method introduced by Hall, Kerkyacharian and Picard in [Hall, Kerkyacharian, Picard, (1998), (1999)], modified to exploit the properties of the spherical standard needlets. Therefore, we will investigate on their convergence rates, attaining their adaptive properties over the Besov balls. This work is strongly motivated by issues arising in Cosmology and Astrophysics, concerning in particular the analysis of cosmic rays.Comment: 25 page

Quantitative central limit theorems for Mexican needlet coefficients on circular Poisson fields

The aim of this paper is to establish rates of convergence to Gaussianity for wavelet coefficients on circular Poisson random fields. This result is established by using the Stein-Malliavin techniques introduced by Peccati and Zheng (2011) and the concentration properties of so-called Mexican needlets on the circleComment: 26 pages, 4 figure

On high-frequency limits of UU-statistics in Besov spaces over compact manifolds

In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.Comment: 19 page

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

Gaussian semiparametric estimates on the unit sphere

We study the weak convergence (in the high-frequency limit) of the parameter estimators of power spectrum coefficients associated with Gaussian, spherical and isotropic random fields. In particular, we introduce a Whittle-type approximate maximum likelihood estimator and we investigate its asympotic weak consistency and Gaussianity, in both parametric and semiparametric cases.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ475 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Adaptive Nonparametric Regression on Spin Fiber Bundles

The construction of adaptive nonparametric procedures by means of wavelet thresholding techniques is now a classical topic in modern mathematical statistics. In this paper, we extend this framework to the analysis of nonparametric regression on sections of spin fiber bundles defined on the sphere. This can be viewed as a regression problem where the function to be estimated takes as its values algebraic curves (for instance, ellipses) rather than scalars, as usual. The problem is motivated by many important astrophysical applications, concerning for instance the analysis of the weak gravitational lensing effect, i.e. the distortion effect of gravity on the images of distant galaxies. We propose a thresholding procedure based upon the (mixed) spin needlets construction recently advocated by Geller and Marinucci (2008,2010) and Geller et al. (2008,2009), and we investigate their rates of convergence and their adaptive properties over spin Besov balls.Comment: 40 page

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

Localisation of directional scale-discretised wavelets on the sphere

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients exactly, in theory and practice (to machine precision). Scale-discretised wavelets are closely related to spherical needlets (both were developed independently at about the same time) but relax the axisymmetric property of needlets so that directional signal content can be probed. Needlets have been shown to satisfy important quasi-exponential localisation and asymptotic uncorrelation properties. We show that these properties also hold for directional scale-discretised wavelets on the sphere and derive similar localisation and uncorrelation bounds in both the scalar and spin settings. Scale-discretised wavelets can thus be considered as directional needlets.Comment: 28 pages, 8 figures, minor changes to match version accepted for publication by ACH