Maxiset in sup-norm for kernel estimators
Résumé
In the Gaussian white noise model, we study the estimation of an unknown multidimensional function $f$ in the uniform norm by using kernel methods. We determine the sets of functions that are well estimated at the rates $\left(\log n/n\right)^{\be/(2\be+d)}$ and $n^{-\be/(2\be+d)}$ by kernel estimators. These sets are called maxisets. Then, we characterize the maxisets associated to kernel estimators and to the Lepski procedure for the rate of convergence $\left(\log n/n\right)^{\be/(2\be+d)}$ in terms of Besov and Hölder spaces of regularity $\beta$. Using maxiset results, optimal choices for the bandwidth parameter of kernel rules are derived.Performances of these rules are studied from the numerical point of view.
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