Nonlinear estimation over weak Besov spaces and minimax Bayes method

Weak Besov spaces play important roles in statistics as maxisets of classical procedures or for measuring the sparsity of signals. The goal of this paper is to study weak Besov balls ${\cal WB}_{s,p,q}(C)$ from the statistical point of view by using the minimax Bayes method. In particular, we compare weak and strong Besov balls statistically. By building an optimal Bayes wavelet thresholding rule, we first establish that, under suitable conditions, the rate of convergence of the minimax risk for ${\cal WB}_{s,p,q}(C)$ is the same as for the strong Besov ball ${\cal B}_{s,p,q}(C)$ that is contained by ${\cal WB}_{s,p,q}(C)$. However, we show that the asymptotically least favorable priors of ${\cal WB}_{s,p,q}(C)$ that are based on Pareto distributions cannot be asymptotically least favorable priors for ${\cal B}_{s,p,q}(C)$. Finally, we present sample paths of such priors that provide representations of the worst functions to be estimated for classical procedures and we give an interpretation of the roles of the parameters $s$, $p$ and $q$ of ${\cal WB}_{s,p,q}(C)$.