Wavelet Thresholding for Non Necessarily Gaussian Noise: Idealism

For various types of noise (exponential, normal mixture, compactly supported, ...) wavelet tresholding methods are studied. Problems linked to the existence of optimal thresholds are tackled, and minimaxity properties of the methods also analyzed. A coefficient dependent method for choosing thresholds is also briefly presented. 1. Introduction. A common underlying assumption in non-parametric curve/surface/signal estimation is that the function to estimate has some redundancy ; and this is often reflected by the hypothesis that it belongs to a particular functional class. A similar prior assumption is that limited information is present in this curve/surface/signal. For example, it could be discontinuous but only at a limited number of places, or the function to estimate is assumed to only have one mode or to be monotone. Then, the heuristic for the use of wavelets in non-parametric estimation is that the expansion of such a function in a wavelet basis is sparse, i.e., only a few of the wavelet coefficients are big and the rest are small and thus negligible. Hence, in order to estimate the function, one has to estimate the large wavelet coefficients and discard the rest. This approach has proved useful and successful as shown, in recent years, by various authors Since we do not review the theory of wavelets here, we refer the reader to the books of Daubechies [8] and Meyer [31], [32] for an introduction to the subject. Nevertheless, let us just say that using a multiresolution approach there is a large family of wavelets with compact support generating orthonormal bases. Moreover, properties of compactly supported wavelets are at the root of a very efficient analog of the fast Fourier transform, the so called fast wavelet transform. With this in mind and from now on, we use an orthonormal wavelet basis from a multiresolution analysis adapted to an interval. Next, non-parametric estimation via wavelet methods is usually divided into two steps. The first step transforms the data into something which can be input into the fast wavelet transform, i.e., noisy versions (denoted by c j0,k) of the scaling coefficients c j0,k , with j 0 large. The fast wavelet transform is then applied to this data giving noisy versions of the wavelet coefficients d j,k (denoted by d j,k and called the empirical wavelet coefficients). In the second step, estimates d j,k of the