Statistica Sinica 19 (2009), 785-800

OPTIMAL BOUNDS FOR INVERSE PROBLEMS

WITH JACOBI-TYPE EIGENFUNCTIONS

Thomas Willer

Université de Provence

Abstract: We consider inverse problems where one wishes to recover an unknown function from the observation of a transformation of it by a linear operator, corrupted by an additive Gaussian white noise perturbation. We assume that the operator admits a singular value decomposition where the eigenvalues decay in a polynomial way, and where Jacobi polynomials appear as eigenfunctions. This includes, as an application, the well known Wicksell's problem. We establish asymptotic lower bounds for the minimax risk in a wide framework (i.e., with losses and Besov-like regularity spaces), which shows that the estimator of Kerkyacharian, Picard, Petrushev, and Willer (2007) is quasi-optimal, and thus yields the minimax rates. We also establish some new results on the needlets introduced by Petrushev and Xu (2005) which appear as essential tools in this setting. Lastly we discuss the interest of the results concerning the treatment of inverse problems by wavelet procedures.

Key words and phrases: Minimax estimation, second-generation wavelets, statistical inverse problems.