Annals of Statistics

Function estimation via wavelet shrinkage for long-memory data

Yazhen Wang

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Abstract

In this article we study function estimation via wavelet shrinkage for data with long-range dependence. We propose a fractional Gaussian noise model to approximate nonparametric regression with long-range dependence and establish asymptotics for minimax risks. Because of long-range dependence, the minimax risk and the minimax linear risk converge to 0 at rates that differ from those for data with independence or short-range dependence. Wavelet estimates with best selection of resolution level-dependent threshold achieve minimax rates over a wide range of spaces. Cross-validation for dependent data is proposed to select the optimal threshold. The wavelet estimates significantly outperform linear estimates. The key to proving the asymptotic results is a wavelet-vaguelette decomposition which decorrelates fractional Gaussian noise. Such wavelet-vaguelette decomposition is also very useful in fractal signal processing.

Article information

Source
Ann. Statist., Volume 24, Number 2 (1996), 466-484.

Dates
First available in Project Euclid: 24 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032894449

Digital Object Identifier
doi:10.1214/aos/1032894449

Mathematical Reviews number (MathSciNet)
MR1394972

Zentralblatt MATH identifier
0859.62042

Subjects
Primary: 62G07: Density estimation 62C20: Minimax procedures 42C15: General harmonic expansions, frames

Keywords
Long-range dependence fractional Brownian motion fractional Gaussian noise fractional Gaussian noise model nonparametric regression minimax risk vaguelette wavelet wavelet-vaguelette ecomposition threshold cross-validation

Citation

Wang, Yazhen. Function estimation via wavelet shrinkage for long-memory data. Ann. Statist. 24 (1996), no. 2, 466--484. doi:10.1214/aos/1032894449. https://projecteuclid.org/euclid.aos/1032894449


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