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Function estimation via wavelet shrinkage for long-memory data. (English) Zbl 0859.62042
Summary: 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.

MSC:
62G07 Density estimation
62C20 Minimax procedures in statistical decision theory
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
62M99 Inference from stochastic processes
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