Asymptotic performance of projection estimators in standard and hyperbolic wavelet bases

We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and non-linear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.