Density estimation by kernel and wavelets methods: Optimality of Besov spaces

This paper is showing that the saturation space of the minimax rate associated to a Lp loss and linear estimators is the Besov space Bs[infinity]p. More precisely, it is shown that if a function space included in Lp is such that its minimax rate is the usual one s/(1 + 2s) and if this rate is attained by a sequence of linear estimators, then this space is included in a ball of the space Bs[infinity]p. This implies, for example, that the minimax rates that have been estimated for the Sobolev balls are in fact only a consequence of their inclusions in such Besov balls