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- David L. Donoho
- Stanford University USA
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- Iain M. Johnstone
- Stanford University USA
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- Gérard Kerkyacharian
- Université de Picardie , Amiens , France
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- Dominique Picard
- Université de Paris VII France
説明
<jats:title>SUMMARY</jats:title> <jats:p>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.</jats:p>
収録刊行物
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- Journal of the Royal Statistical Society Series B: Statistical Methodology
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Journal of the Royal Statistical Society Series B: Statistical Methodology 57 (2), 301-337, 1995-07-01
Oxford University Press (OUP)
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詳細情報 詳細情報について
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- CRID
- 1361699994811237376
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- ISSN
- 14679868
- 13697412
- http://id.crossref.org/issn/00359246
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- データソース種別
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- Crossref