Utilisation des ondelettes de Haar en estimation Bayesienne

A few years ago, statisticians have discovered the extraordinary potential of wavelets as a tool for statistical inference, especially in the fields of density estimation, nonparametric regression and smoothing. However, wavelets certainly offer many other possibilities, among which is the construction of simple approximations to known densities, just as the more classical orthonormal bases of functions (Hermite, Fourier, Laplace, etc.) do. This kind of approximation can be used to perform numerical integration and therefore, Bayesian calculation.; The main goal of this thesis is to study how the Haar basis of wavelets can be used efficiently to construct approximations to the posterior moments of a location parameter, denoted &thetas;, and to the marginal density of the observations. The results are obtained using the Bayesian paradigm. The proposed technique is also used to get a solution to the empirical Bayes problem of nonparametrically estimating the prior of &thetas; when it is unspecified. The resulting methodologies suggest a wide range of new interesting applications of wavelet bases to typical Bayesian problems.