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On minimax wavelet estimators. (English) Zbl 0865.62023

Summary: Minimax rates of convergence for wavelet estimators are studied. The estimators are based on the shrinkage of empirical coefficients \(\widehat\beta_{jk}\) of wavelet decompositions of unknown functions with thresholds \(\lambda_j\). These thresholds depend on the regularity of the function to be estimated. In the problem of density estimation and nonparametric regression we establish upper rates of convergence over a large range of functional classes and global error measures. The constructed estimate is minimax (up to constant) for all \(L_\pi\) error measures, \(0<\pi\leq\infty\), simultaneously.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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