Deviation bounds for wavelet shrinkage

We analyze the wavelet shrinkage algorithm of Donoho and Johnstone (1994) in order to assess the quality of the reconstruction of a signal obtained from noisy samples. We give a deviation estimate for the maximum squared error (and, consequently, for the average squared error), under the assumption that the signal comes from a Holder class, and the noise samples are independent, of zero mean, and bounded. Our main technique is Talagrandʹs (1995) isoperimetric theorem. Our result shows a better behavior of the wavelet shrinkage.