Gaussian semiparametric estimates on the unit sphere

Abstract

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

Block Thresholding on the Sphere

“…Hence, we obtain The procedure follows these guidelines: we have to split (19) into four terms: in one of them, Aa, both the A js;p and A js;p are supposed to be bigger than the respective threshold; in another one, U u, they are both smaller and in the last two of them, Au and U a, the distance between A js;p and A js;p is shown to be bigger than a suitable threshold. In the first two cases, in order to achieve the minimax rate of convergence, we will split these terms into two parts and we will show the convergence of each part by using mainly (4), (15) and (16). The convergence of the last two terms will be instead proved by applying (17).…”

“…This approach has been extended to more general manifolds by [23], [24], [25], while their generalization to spin fiber bundles on the sphere were described in [21], [22]. Most of these researches can be motivated by applications to Cosmology and Astrophysics: for instance, a huge amount of spherical data, concerning the Cosmic Microwave Background radiation, are being provided by satellite missions WMAP and Planck, see [42], [38], [43], [19], [44], [45], [12], [46], [16] and [17] for more details. The applications mentioned here, however, do not concern thresholding estimation, but rather they can be related to the study of random fields on the sphere, such as angular power spectrum estimation, higher-order spectra, testing for Gaussianity and isotropy, and several others (see also [10]).…”

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

“…The concentration property in the spatial domain of the needlets (see (9)) allows one to focus on any subset of a given manifold without having to take into account the entire structure. Some of the most remarkable statistical applications are discussed and studied in [BKMP09b,BKMP09a,CM15,DLM14].…”

High-Frequency Tail Index Estimation by Nearly Tight Frames

“…We employ a procedure based on the so-called mexican needlet construction by Geller and Mayeli in [21]. Furthermore, we develop a plug-in procedure aimed to merge and to optimize these results with the achievements pursued in [12], [13], see also [14], where the asymptotic behaviour of Whittle-like estimates were studied respectively in the harmonic and standard needlet analysis frameworks.…”