Nonparametric Bayesian estimation.

Summary In the context of a wavelet analysis, we are interested in the statistical study of a particular class of Lorentz spaces: the weak Besov spaces which appear naturally in the context of the maxiset theory. With "white Gaussian noise" assumptions, we show, thanks to Bayesian techniques, that the minimax velocities of the strong and weak Besov spaces are the same. The worst-case distributions that we show for each weak Besov space are constructed from Pareto laws and differ from those of the strong Besov spaces. Using simulations of these distributions, we construct visual representations of "typical enemies". Finally, we exploit these distributions to build a minimax estimation procedure, of the "thresholding" type, called ParetoThresh, which we study from a practical point of view. In a second step, we place ourselves under the heteroskedastic white Gaussian noise model and under the maxiset approach, we establish the suboptimality of linear estimators compared to adaptive thresholding procedures. Then, we investigate the best way to model the sparse character of a sequence through a Bayesian approach. To this end, we study the maxima of classical Bayesian estimators - median, mean - associated with a model built on heavy-tailed densities. The maximal spaces for these estimators are Lorentz spaces, and coincide with those associated with thresholding estimators. We extend this result in a natural way by obtaining a necessary and sufficient condition on the parameters of the model so that the a priori law is almost certainly concentrated on a specific Lorentz space.