Wavelet Shrinkage: Asymptopia?
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ID: 291681
1995
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Abstract
SUMMARY Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects—curves, densities, spectral densities, images—from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons—among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount √(2 log n)σ/√n. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions—pointwise error, global error measured in Lp-norms, pointwise and global error in estimation of derivatives—and for a wide range of smoothness classes, including standard Holder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity.
| Reference Key |
openalex_W191129667
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|---|---|
| Authors | David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, Dominique Picard |
| Journal | Journal of the Royal Statistical Society Series B (Statistical Methodology) |
| Year | 1995 |
| DOI |
10.1111/j.2517-6161.1995.tb02032.x
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| URL | |
| Keywords | Keywords not found |
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