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De-noising by soft-thresholding. (English) Zbl 0820.62002
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:
 62B10 Statistical information theory 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 42C15 General harmonic expansions, frames
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