Adaptive estimators with structured parsimony.

Summary This thesis presents new statistical procedures in a non-parametric framework and studies both their theoretical and empirical properties. Our work focuses on two different topics which have in common the indirect observation of the unknown functional parameter. The first part is devoted to a block thresholding estimation procedure for the white Gaussian noise model. We focus on the adaptive estimation of a signal f and its derivatives from n fuzzy and noisy versions of this signal and prove that our estimator reaches a (quasi-)optimal convergence speed on a large class of Besov balls. Then, we propose an adaptive estimation procedure allowing to select the parameters of the estimator in an optimal way by minimizing an unbiased risk estimator. In the second part, we place ourselves in the framework of the density model in which the unknown function undergoes a given transformation before being observed. We construct and study an adaptive estimator based on a plug-in approach and a hard block wavelet thresholding. Finally, we study the problem of estimating the m-order convolution of a density from i.i.d. observations drawn from the underlying law. We propose an adaptive estimator based on kernels, Fourier analysis and Lepski's method. We study its quadratic risk and new and fast speeds are obtained, for a large class of unknown functions. Each estimation method studied is numerically analyzed by simulations, both on simulated data and on real data.