Block Thresholding on the Sphere

Abstract

The aim of this paper is to study nonparametric regression estimators on the sphere based on needlet block thresholding. The block thresholding procedure proposed here follows the method introduced by Hall, Kerkyacharian and Picard in [27], [28], which we modifyto exploit the properties of spherical needlets. We establish convergence rates, and we show that they attain adaptivity over Besov balls in the regular region. This work is strongly motivated by issues arising in Cosmology and Astrophysics, concerning in particular the analysis of Cosmic rays.

Adaptive Density Estimation on the Circle by Nearly Tight Frames

“…Obviously, in the framework of finite sample situation, the asymptotic rate given in Theorem 2 has to be considered just as a prompt. In what follows, we have built an estimator (11) using the set to estimate F (θ ) = (2π) −1 exp (θ − π) 2 /2 by using CRAN R. Some graphical evidence can be found in Figure 3. We will focus on two main points:…”

“…As far as data on the unit q-dimensional sphere S q are concerned, many of those researches have been developed by using the constructions of second-generation wavelets on S q named spherical needlets. The spherical needlets, introduced in the literature by [26,27], feature properties fundamental to attain the minimax optimal rates of convergence of the estimates, such as their concentration in both Fourier and space domains: density estimation of directional data on S q was presented in [3], the analysis of nonparametric regression on sections of spin fiber bundles on S 2 by the means of spin needlets was proposed in [8] and, finally, nonparametric regression estimators on the sphere based respectively on needlet block and global thresholding were studied in [11] and [13].…”

Adaptive global thresholding on the sphere

“…Proof of Theorem 1.1. Following, for instance, [1,7,9,10,22,30] and as mentioned in Section 3.3, the L p -risk E( f n − f p L p (S d ) ) can be decomposed as the sum of a stochastic and a bias term. More specifically,…”

“…Baldi et al [1] established minimax rates of convergence for the L p -risk of nonlinear needlet density estimators within the hard local thresholding paradigm, while analogous results concerning regression function estimation were established by Monnier [38]. The block thresholding framework was investigated in Durastanti [9]. Furthermore, the adaptivity of nonparametric regression estimators of spin function was studied in Durastanti et al [10].…”

Nonparametric needlet estimation for partial derivatives of a probability density function on the $d$-torus

“…Needlets have also been established over general compact manifolds in [GM09, KNP12,Pes13], and over spin fiber bundles (see [GM10]). The pioneering results in the framework of nonparametric statistics, described in [BKMP09a], have established minimax rates of convergence L p -risk of needlet density estimators built by means of hard local thresholding techniques, while analogous results in the block and global thresholding framework were then presented in [Dur13,Dur16]. Nonparametric regression estimators of spin function have been discussed in [DGM11].…”