Minimax estimation via wavelet shrinkage

We attempt to recover an unknown function from noisy, sampled data. Using o rthonormal bases of compactly supported wavelets, we develop a nonlinear me thod which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficient. The shrinkage can be tuned to be nearly min imax over any member of a wide range of Triebel- and Besov-type smoothness constraints and asymptotically minimax over Besov bodies with p less than o r equal to q. Linear estimates cannot achieve even the minimax rates over T riebel and Besov classes with p < 2, so the method can significantly outper form every linear method (e.g., kernel, smoothing spline, sieve) in a minim ax sense. Variants of our method based on simple threshold nonlinear estima tors are nearly minimax. Our method possesses the interpretation of spatial adaptivity; it reconstructs using a kernel which may vary in shape and ban dwidth from point to point, depending on the data. Least favorable distribu tions for certain of the Triebel and Besov scales generate objects with spa rse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical probl ems. Sequels to this paper, which was first drafted in November 1990, discu ss practical implementation, spatial adaptation properties, universal near minimaxity and applications to inverse problems.