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June 1998 Minimax estimation via wavelet shrinkage
David L. Donoho, Iain M. Johnstone
Ann. Statist. 26(3): 879-921 (June 1998). DOI: 10.1214/aos/1024691081


We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets, we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints and asymptotically mini-max over Besov bodies with $p \leq q$. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with $p<2$, so the method can significantly outperform every linear method (e.g., kernel, smoothing spline, sieve in a minimax sense). Variants of our method based on simple threshold nonlinear estimators are nearly minimax. Our method possesses the interpretation of spatial adaptivity; it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper, which was first drafted in November 1990, discuss practical implementation, spatial adaptation properties, universal near minimaxity and applications to inverse problems.


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David L. Donoho. Iain M. Johnstone. "Minimax estimation via wavelet shrinkage." Ann. Statist. 26 (3) 879 - 921, June 1998.


Published: June 1998
First available in Project Euclid: 21 June 2002

zbMATH: 0935.62041
MathSciNet: MR1635414
Digital Object Identifier: 10.1214/aos/1024691081

Primary: 62C20, 62G07
Secondary: 41A30, 62G20

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.26 • No. 3 • June 1998
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