Minimax estimation via wavelets for indirect long-memory data

In this paper, we model linear inverse problems with long-range dependence by a fractional Gaussian noise model and study function estimation based on observations from the model. By using two wavelet-vaguelette decompositions, one for the inverse problem which simultaneously quasi-diagonalizes both the operator and the prior information, and one for long-range dependence which decorrelates fractional Gaussian noise, we establish asymptotics for minimax risks, and show that the wavelet shrinkage estimate can be tuned to achieve the minimax convergence rate and significantly outperform linear estimates.