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Universal Near Minimaxity of Wavelet Shrinkage

In: Festschrift for Lucien Le Cam

Author

Listed:
  • D. L. Donoho

    (Stanford University)

  • I. M. Johnstone

    (Stanford University)

  • G. Kerkyacharian

    (Université de Picardie)

  • D. Picard

    (Université de Paris VII)

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).

Suggested Citation

  • D. L. Donoho & I. M. Johnstone & G. Kerkyacharian & D. Picard, 1997. "Universal Near Minimaxity of Wavelet Shrinkage," Springer Books, in: David Pollard & Erik Torgersen & Grace L. Yang (ed.), Festschrift for Lucien Le Cam, chapter 12, pages 183-218, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-1880-7_12
    DOI: 10.1007/978-1-4612-1880-7_12
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