Universal Near Minimaxity of Wavelet Shrinkage

D. L. Donoho, I. M. Johnstone, G. Kerkyacharian and D. Picard
Additional contact information
D. L. Donoho: Stanford University
I. M. Johnstone: Stanford University
G. Kerkyacharian: Université de Picardie
D. Picard: Université de Paris VII

Chapter 12 in Festschrift for Lucien Le Cam, 1997, pp 183-218 from Springer

Abstract: Abstract We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coefficients towards the origin by an amount $$ \sqrt {{2\log \left( n \right)}} \cdot \sigma /\sqrt {n}$$ The method is nearly minimax for a wide variety of loss functions-e.g. 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 Hölder classes, Sobolev classes, and Bounded Variation. This is a broader near-optimality than anything previously proposed in the minimax literature. The theory underlying the method exploits a correspondence between statistical questions and questions of optimal recovery and information-based complexity. This paper contains a detailed proof of the result announced in Donoho, Johnstone, Kerkyacharian & Picard (1995).

Keywords: Sequence Space; Wavelet Coefficient; Besov Space; Critical Case; Unconditional Basis (search for similar items in EconPapers)
Date: 1997
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1880-7_12

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DOI: 10.1007/978-1-4612-1880-7_12

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