Why minimax is not that pessimistic

In nonparametric statistics an optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this classical point of view is subject to controversy as it requires to look for the worst behaviour reached by an estimation procedure in a given space. The purpose of this paper is to show that this is not justified as the minimax risk often coincides with a generic one. We are here interested in the rate of convergence attained by some classical estimators on almost every, in the sense of prevalence, function in a Besov space.