Exact minimax risk for density estimators in non-integer Sobolev classes

Clementine Dalelane (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

The $L_2$-minimax risk in Sobolev classes of densities with non-integer smoothness index is shown to have an analog form to that in integer Sobolev classes. To this end, the notion of Sobolev classes is generalized to fractional derivatives of order $\beta\in\mathbb R^+$. A minimax kernel density estimator for such a classes is found. Although there exists no corresponding proof in the literature so far, the result of this article was used implicitly in numerous papers. A certain necessity that this gap had to be filled, can thus not be denied.

Mots clés: exact asymptotics; fractional derivative; Fourier transform; minimax risk; Sobolev classes

Domaines: Statistiques [math.ST]

Clementine Dalelane. Exact minimax risk for density estimators in non-integer Sobolev classes. 2005. ⟨hal-00004754⟩