This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In order to identify the singularities of the unknown signal, we introduce a new tool, “the structural intensity”, that computes the “density” of the location of the modulus maxima of a wavelet representation along various scales. This approach is shown to be an effective technique for detecting the significant singularities of a signal corrupted by noise and for removing spurious estimates. The asymptotic properties of the resulting estimators are studied and illustrated by simulations. An application to a real data set is also proposed.

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).