Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Minimax or Maxisets ?


Code(s) de Classification MSC:

Résumé: In this paper, we provide a new way of evaluating the performances of a statistical estimation procedure. This point of view consists in investigating the maximal set where a procedure has a given rate of convergence. Although the setting is not extremely different from the minimax context, it is less pessimistic and provides a functional set which is authentically connected to the procedure and the model. We also investigates more traditional concerns about procedures: oracle inequalities. This notion becomes more difficult even to be practically defined when the loss function is not the $\bL_2$-norm. We explain the difficulties arising there, and suggest a new definition, in the cases of $\bL_p$-norms and point-wise estimation. The connections between maxisets and local oracle inequalities are investigated: we prove that verifying a local oracle inequality implies that the maxiset automatically contains a prescribed set linked with the oracle inequality. We have investigated the consequences of the previous statement on well known efficient adaptive methods: Wavelet thresholding and local bandwidth selection. We can prove local oracle inequalities for these methods and draw the conclusions about there associated maxisets.

Mots Clés: Non parametric estimation ; denoising ; minimax rate of convergence ; oracle inequalities ; saturation spaces ; wavelet thresholding ; local bandwidth selection

Date: 2000-01-05

Prépublication numéro: PMA-556