Optimal Spherical Deconvolution

This paper addresses the issue of optimal deconvolution density estimation on the 2-sphere. Indeed, by using the transitive group action of the rotation matrices on the 2-dimensional unit sphere, rotational errors can be introduced analogous to the Euclidean case. The resulting density turns out to be convolution in the Lie group sense and so the statistical problem is to recover the true underlying density. This recovery can be done by deconvolution; however, as in the Euclidean case, the difficulty of the deconvolution turns out to depend on the spectral properties of the rotational error distribution. This therefore leads us to define smooth and super-smooth classes and optimal rates of convergence are obtained for these smoothness classes.