Function estimation via wavelet shrinkage for long-memory data
The Annals of Statistics · 1996 · Volume 24 · Issue 2
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.
Keywords
42C15; 62C20; 62G07; cross-validation; fractional Brownian motion; fractional Gaussian noise; fractional Gaussian noise model; long-range dependence; minimax risk; Nonparametric regression; threshold; vaguelette; wavelet; wavelet-vaguelette ecomposition;
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