High-Frequency tail Index estimation by nearly tight frames

The authors study isotropic Gaussian random fields on the unit sphere. They investigate the asymptotic behaviour of Whittle-like approximate maximum likelihood estimates of the spectral parameters of the random fields. Weak consistency and a central limit theorem are obtained under the hypothesis of Gaussianity and some smoothness conditions on the behaviour of the angular power spectrum. The asymptotic framework used in the article assumes that observations are collecting at higher and higher frequencies on a fixed domain. The main tool is a Mexican needlet Whittle-like approximation to the log-likelihood function of isotropic and Gaussian random fields. The authors also suggest a plug-in procedure to optimize the precision of the estimators in terms of asymptotic variance. The results obtained can find various applications in cosmology and astrophysics.