Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere

This paper addresses the nonparametric estimation of a spatially varying, heteroskedastic variance function on the unit sphere within a regression framework. While adaptive regression estimation is well-established on manifolds, characterizing localized noise structures presents unique theoretical obstacles due to bias propagation from the unknown mean function. To circumvent this, we propose a fully data-driven, multiresolution estimator based on localized spherical frames, namely, needlets, combined with a hard-thresholding protocol and a sample-splitting scheme. The approach exploits the excellent spatial and frequency localization properties of needlets to adaptively capture the local features of the variance function. We prove that the proposed estimator achieves the minimax-optimal rate of convergence over spherical Besov spaces under standard loss functions, exhibiting spatial adaptivity without requiring prior knowledge of the regularity of the variance function. This adaptivity highlights the efficacy of the method in analyzing spherical data characterized by complex heteroskedastic errors, with potential applications in fields such as cosmology, environmental modeling, and geophysics