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 L(p) loss and linear estimators is the Besov space B(p)sinfinity. More precisely, it is shown that if a function space included in L(p) 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 B(p)sinfinity. 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