Abstract

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 (log n/n)ß/(2ß+d) and n-ß/(2ß+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 (log n/n)ß/(2ß+d) in terms of Besov and Hölder spaces of regularity ß. 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. © Sociedad de Estadística e Investigación Operativa 2008.