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Zbl 0820.62002
Donoho, David L.
De-noising by soft-thresholding. (English)
[J] IEEE Trans. Inf. Theory 41, No.3, 613-627 (1995). ISSN 0018-9448

Summary: The author and {\it I. M. Johnstone} [Biometrika 81, No. 3, 425--455 (1994; Zbl 0815.62019)] proposed a method for reconstructing an unknown function $f$ on $[0,1]$ from noisy data $d\sb i = f(t\sb i) + \sigma z\sb i$, $i = 0, \dots, n-1$, $t\sb i = i/n$, where the $z\sb i$ are independent and identically distributed standard Gaussian random variables. The reconstruction $\widehat f\sb n\sp*$ is defined in the wavelet domain by translating all the empirical wavelet coefficients of $d$ toward $0$ by an amount $\sigma \cdot \sqrt {2 \log (n)/n}$. We prove two results about this type of estimator. [Smooth]: With high probability $\widehat f\sp*\sb n$ is at least as smooth as $f$, in any of a wide variety of smoothness measures. [Adapt]: The estimator comes nearly as close in mean square to $f$ as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.

MSC 2000:: *62B10 Statistical information theory
94A12 Signal theory
42C15 Series and expansions in general function systems

Keywords: thresholding; mean squared error; interpolation; denoising; density estimation; empirical wavelet transform; minimax estimation; adaptive estimation; reconstruction; smoothness measures; optimal recovery model

Citations: Zbl 0815.62019

Cited in: Zbl 1099.94516 Zbl 1043.94002

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