The Annals of Statistics

Wavelet shrinkage for nonequispaced samples

Lawrence D. Brown and T. Tony Cai

Full-text: Open access

Abstract

Standard wavelet shrinkage procedures for nonparametric regression are restricted to equispaced samples. There, data are transformed into empirical wavelet coefficients and threshold rules are applied to the coefficients. The estimators are obtained via the inverse transform of the denoised wavelet coefficients. In many applications, however, the samples are nonequispaced. It can be shown that these procedures would produce suboptimal estimators if they were applied directly to nonequispaced samples.

We propose a wavelet shrinkage procedure for nonequispaced samples. We show that the estimate is adaptive and near optimal. For global estimation, the estimate is within a logarithmic factor of the minimax risk over a wide range of piecewise Hölder classes, indeed with a number of discontinuities that grows polynomially fast with the sample size. For estimating a target function at a point, the estimate is optimally adaptive to unknown degree of smoothness within a constant. In addition, the estimate enjoys a smoothness property: if the target function is the zero function, then with probability tending to 1 the estimate is also the zero function.

Article information

Source
Ann. Statist., Volume 26, Number 5 (1998), 1783-1799.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1024691357

Digital Object Identifier
doi:10.1214/aos/1024691357

Mathematical Reviews number (MathSciNet)
MR1673278

Zentralblatt MATH identifier
0929.62047

Subjects
Primary: 62G07
Secondary: 62G20

Keywords
Wavelets multiresolution approximation nonparametric regression minimax adaptivity piecewise Hölder class

Citation

Cai, T. Tony; Brown, Lawrence D. Wavelet shrinkage for nonequispaced samples. Ann. Statist. 26 (1998), no. 5, 1783--1799. doi:10.1214/aos/1024691357. https://projecteuclid.org/euclid.aos/1024691357


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