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

### Abstract

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

Topics: fractional derivative, Fourier transform, minimax risk, Sobolev classes, exact asymptotics, MSC 62C20, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]
Publisher: HAL CCSD
Year: 2005
OAI identifier: oai:HAL:hal-00004754v1
Provided by: Hal-Diderot