Bernoulli

  • Bernoulli
  • Volume 10, Number 6 (2004), 1053-1105.

Regression in random design and warped wavelets

Gérard Kerkyacharian and Dominique Picard

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Abstract

We consider the problem of estimating an unknown function f in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis { ψ jk (G),j,k} warped with the design. This allows us to employ a very stable and computable thresholding algorithm. We investigate the properties of this new basis. In particular, we prove that if the design has a property of Muckenhoupt type, this new basis behaves quite similarly to a regular wavelet basis. This enables us to prove that the associated thresholding procedure achieves rates of convergence which have been proved to be minimax in the uniform design case.

Article information

Source
Bernoulli, Volume 10, Number 6 (2004), 1053-1105.

Dates
First available in Project Euclid: 21 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.bj/1106314850

Digital Object Identifier
doi:10.3150/bj/1106314850

Mathematical Reviews number (MathSciNet)
MR2108043

Zentralblatt MATH identifier
1067.62039

Keywords
maxisets Muckenhoupt weights nonparametric regression random design warped wavelets wavelet thresholding

Citation

Kerkyacharian, Gérard; Picard, Dominique. Regression in random design and warped wavelets. Bernoulli 10 (2004), no. 6, 1053--1105. doi:10.3150/bj/1106314850. https://projecteuclid.org/euclid.bj/1106314850


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